Lithologic Controls on Silicate Weathering Regimes of Temperate Planets

Continental weathering occurs in continental soils and saprolite where runoff (water discharge per unit surface area) facilitates fluid–rock weathering reactions (Figure 1). Seafloor weathering occurs in pores and veins of the oceanic crust (hereafter, pore space) where seawater reacts with basalts. Analogous to runoff, the fluid flow in off-axis low-temperature hydrothermal systems facilitates seafloor-weathering reactions (Stein & Stein 1994; Coogan & Gillis 2018). The fluid flow rate q (either runoff or hydrothermal fluid flow rate) is a free parameter in CHILI (Table 1). On Earth, the present-day continental runoff varies between 0.01−3 m yr⁻¹ with a mean of approximately 0.3 m yr⁻¹ (Gaillardet et al. 1999; Fekete et al. 2002). On the seafloor, the non-porosity-corrected hydrothermal fluid flow rates are between 0.001−0.7 m yr⁻¹ with a mean of about 0.05 m yr⁻¹ (Stein & Stein 1994; Johnson & Pruis 2003; Hasterok 2013).

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Table 1. Control Parameters and Their Reference Values for Modern Earth

Symbol	Description	Reference
${P}_{{\mathrm{CO}}_{2}}$	CO2 partial pressure	280 μbar
T	Surface temperature	288 K
P	Surface pressure	1 bar
q	Runoff or fluid flow rate	0.3 m yr−1
ts	Soil or pore-space age	105 yr
ψ	Dimensionless pore-space parameter	222 750

Note. T is not a control parameter for some calculations ($T=f({P}_{{\mathrm{CO}}_{2}})$, Section 2.5).

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In the fluid-transport-controlled model of continental weathering, the weathering flux w (mol m⁻² yr⁻¹) is the product of the concentration of a solute of interest C and runoff q (e.g., Maher & Chamberlain 2014),

$$\begin{eqnarray}&&w=C\ q.\end{eqnarray} \tag{ 3 }$$

We also apply the same approach to seafloor weathering. The total silicate-weathering rate ⁶ W (mol yr⁻¹) is the sum of the continental (W_cont) and seafloor-weathering rates (W_seaf). These rates are given by the product of the continental (w_cont) or seafloor-weathering flux (w_seaf) and the continental (f A_s) or seafloor surface area ((1 − f)A_s), where A_s is the planet surface area and f is the continental area fraction:

$$\begin{eqnarray}&&W={w}_{\mathrm{cont}}\,f\,{A}_{{\rm{s}}}+{w}_{\mathrm{seaf}}\,(1-f)\,{A}_{{\rm{s}}}.\end{eqnarray} \tag{ 4 }$$

Table 2. Computed Quantities

Symbol	Description	Units
w	Weathering flux	mol m−2 yr−1
W	Weathering rate	mol yr−1
${T}^{{\prime} }$	Seafloor pore-space temperature	K
A	Carbonate alkalinity	mol dm−3
pH	Negative logarithm of H+ activity	⋯
C	Concentration	mol dm−3
[C]	Concentration of a species C	mol dm−3
aC	Activity of a species C	⋯
Dw	Damköhler coefficient	m yr−1
K	Equilibrium constant	⋯
keff	Kinetic rate coefficient	mol m−2 yr−1
E	Activation energy	kJ mol−1
Eth	Thermodynamic activation energy	kJ mol−1
β	Power-law exponent	⋯
βth	Thermodynamic power-law exponent	⋯

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To model silicate weathering, the aqueous bicarbonate ion concentration [${\mathrm{HCO}}_{3}^{-}]$ is normally used as a proxy for weathering because it is a common byproduct of silicate weathering (e.g., anorthite weathering, Winnick & Maher 2018),

$$\begin{eqnarray}&&\begin{array}{l}{\mathrm{CaAl}}_{2}{\mathrm{Si}}_{2}{{\rm{O}}}_{8}+2{\mathrm{CO}}_{2}+3{{\rm{H}}}_{2}{\rm{O}}\\ \quad \to 2{\mathrm{HCO}}_{3}^{-}+{\mathrm{Ca}}^{2+}+{\mathrm{Al}}_{2}{\mathrm{Si}}_{2}{{\rm{O}}}_{5}{\left(\mathrm{OH}\right)}_{4}.\end{array}\end{eqnarray} \tag{ 5 }$$

This is a good assumption under modern Earth conditions as ${\mathrm{HCO}}_{3}^{-}$ is the primary CO₂-rich product of continental silicate weathering that is carried to oceans by rivers and reacts with Ca²⁺ to precipitate Ca-rich carbonates on the seafloor. We are interested in modeling weathering under conditions more diverse than those on modern Earth. In highly alkaline conditions, the carbonate ion ${\mathrm{CO}}_{3}^{2-}$ is produced in amounts similar to or exceeding [${\mathrm{HCO}}_{3}^{-}$]. Both ${\mathrm{CO}}_{3}^{2-}$ and ${\mathrm{HCO}}_{3}^{-}$ have the potential to contribute to Ca–Mg–Fe carbonate precipitation on the seafloor (e.g., calcite precipitation, Plummer et al. 1978),

$$\begin{eqnarray}\begin{array}{rcl}2{\mathrm{HCO}}_{3}^{-}+{\mathrm{Ca}}^{2+} & \to & {\mathrm{CaCO}}_{3}+{\mathrm{CO}}_{2}+{{\rm{H}}}_{2}{\rm{O}},\\ {\mathrm{CO}}_{3}^{2-}+{\mathrm{Ca}}^{2+} & \to & {\mathrm{CaCO}}_{3}.\end{array}\end{eqnarray} \tag{ 6 }$$

Equation (6) exemplifies that, in addition to bicarbonate and carbonate ions, divalent cations are required to drive the flux of CO₂ out of the atmosphere–ocean system. Nonetheless, because twice as many ${\mathrm{HCO}}_{3}^{-}$ ions as ${\mathrm{CO}}_{3}^{2-}$ ions are needed to precipitate 1 mole of carbonate (Krissansen-Totton & Catling 2017), in addition to ${\mathrm{HCO}}_{3}^{-}$ ($C=[{\mathrm{HCO}}_{3}^{-}]$ in Equation (3)), we consider carbonate alkalinity A resulting from reactions between silicate rocks and fluids as a proxy for weathering (C = A in Equation (3)), where

$$\begin{eqnarray}&&A=[{\mathrm{HCO}}_{3}^{-}]+2[{\mathrm{CO}}_{3}^{2-}].\end{eqnarray} \tag{ 7 }$$

Previous studies applying the fluid-transport-controlled model to continental weathering limit the lithology to monomineralic (e.g., oligoclase feldspar, Graham & Pierrehumbert 2020) or one rock type (e.g., granite, Maher & Chamberlain 2014; Winnick & Maher 2018). However, the chemistry of common silicate rocks ranges from ultramafic to felsic, with increasing SiO₂ content. To model the surface mineralogy of terrestrial exoplanets, we approximate an ultramafic, a mafic, and a felsic lithology with major mineralogical compositions of peridotite, basalt, and granite, respectively. Peridotites are upper-mantle rocks rich in MgO and FeO and poor in SiO₂ relative to basalts and granites. Basalts are igneous rocks that are common examples of rocks present in the oceanic crust of Earth, lunar mare on the Moon, and the crust of Mars. Granites are present on the modern continental crust of Earth. These are highly differentiated rocks that are rich in SiO₂, Na₂O, and K₂O due to partial melting and crystallization processes.

The three rock types (peridotite, basalt, and granite) considered in this study are assumed to be composed of two to three major mineral groups and can thus be projected onto binary or ternary diagrams spanned by the major mineral groups considered (Figure 2(a)). Each mineral group is similarly defined by one to three endmember minerals (Figure 2(b)). For instance, olivine is a solid solution of two endmember minerals, forsterite, and fayalite. In the reduced set of mineral groups with only endmember minerals, we represent peridotite using pyroxenes (wollastonite and enstatite) and olivines (forsterite and fayalite), basalt using plagioclase feldspars (anorthite and albite), and pyroxenes (wollastonite, enstatite, and ferrosilite), and granite using alkali feldspars (K-feldspar and albite), quartz and biotite micas (phlogopite and annite). Halloysite, a phyllosilicate mineral, is a byproduct (secondary mineral) of the weathering of basalt and granite but not of peridotite (see Appendix B). Kaolinite, another phyllosilicate secondary mineral, is also considered but no secondary minerals containing divalent and monovalent cations are modeled. The choice of endmember primary minerals to define these rocks makes them idealized compared to natural rocks. Moreover, for a rock, not all endmember minerals of a mineral group can be modeled due to the unity activity assumption for endmember minerals (Section 2.3). Only major minerals in typical rock types are considered, and minor minerals such as magnetite, hematite, and pyrite are not discussed. Because the net contribution of carbonate weathering on the carbon cycle is small on timescales of the order of 100 kyr for Earth (e.g., Sleep & Zahnle 2001), the weathering of carbonate minerals is ignored.

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Concentrations of products of weathering reactions at chemical equilibrium allow one to calculate a thermodynamic upper limit to weathering (maximum weathering). To calculate concentrations of silicate-weathering products, a number of chemical reactions need to be considered. Reactions between water, CO₂, and silicate minerals produce the bicarbonate ion ${\mathrm{HCO}}_{3}^{-}$. The bicarbonate ion further dissociates into the carbonate ion ${\mathrm{CO}}_{3}^{2-}$ and H⁺. Water dissociates into H⁺ and OH⁻. The relations between equilibrium constants of these reactions and thermodynamic activities of reactants and products are given in Appendix B. Thermodynamic activities quantify the energetics of mixing of constituent components in solid or aqueous solutions (DeVoe 2001). Because no nonideal solid-solution behaviors are considered, the endmember compositions of the minerals are used for calculations, and thus, the activities of minerals are set at unity. Furthermore, in a dilute solution, the activity of liquid water is approximately unity.

The activity of an aqueous species (e.g., ${\mathrm{HCO}}_{3}^{-}({aq})$) is given by the product of its concentration [${\mathrm{HCO}}_{3}^{-}({aq})]$ and the activity coefficient γ normalized to the standard state of concentration (C₀ = 1 mol dm⁻³), ${a}_{{\mathrm{HCO}}_{3}^{-}({aq})}=\tfrac{\gamma \ [{\mathrm{HCO}}_{3}^{-}({aq})]}{{C}_{0}}$ (DeVoe 2001). In an ideal, dilute solution, γ → 1 and $[{\mathrm{HCO}}_{3}^{-}({aq})]={a}_{{\mathrm{HCO}}_{3}^{-}({aq})}\,{C}_{0}$, an assumption made throughout the study for all aqueous species. The activity of a gaseous species such as CO₂(g) is given by the ratio of its fugacity in the gas mixture to the fugacity of pure CO₂(g) at the total pressure P (i.e., surface pressure in this study). Thus, ${a}_{{\mathrm{CO}}_{2}(g)}=\tfrac{{f}_{{\mathrm{CO}}_{2}}}{{f}_{{\mathrm{CO}}_{2}}^{\mathrm{tot}}}=\tfrac{{\rm{\Gamma }}{P}_{{\mathrm{CO}}_{2}}}{{{\rm{\Gamma }}}^{\mathrm{tot}}P}$, where ${P}_{{\mathrm{CO}}_{2}}$ is the CO₂ partial pressure, P is the total surface pressure, and Γ and Γ^tot are fugacity coefficients for the CO₂ component and pure CO₂, respectively. The fugacity coefficients give a correction factor for the nonideal behavior due to mixing and/or pressure effects. For CO₂(g), the fugacity coefficient varies between 0.5 and 1 up to pressures of 200 bar (Spycher & Reed 1988). Our assumptions of unity fugacity coefficients and P = 1 bar lead to ${a}_{{\mathrm{CO}}_{2}(g)}={P}_{{\mathrm{CO}}_{2}}$.

Equilibrium constants depend on temperature and pressure (see Appendix A). Chemical reactions on continents are characterized by the surface temperature T and surface pressure P (Table 1). Seafloor-weathering reactions are characterized by the seafloor pore-space temperature ${T}^{{\prime} }$ and pressure ${P}^{{\prime} }$. In the temperature range 273–373 K, data suggest that ${T}^{{\prime} }$ is within 1% of T (Krissansen-Totton & Catling 2017, and references therein) and hence ${T}^{{\prime} }=T$ is assumed. Moreover, pressure in the range of 0.01–1000 bar has a negligible effect on equilibrium constants and resulting concentrations (see Figures A1 and B3). However, seafloor pressure affects carbonate stability. Nonetheless, ${P}^{{\prime} }$ is fixed to 200 bar, which is the pressure at approximately 2 km depth in Earth's present-day oceans. When extending the continental weathering model to seafloor weathering, we consider a freshwater ocean, in which ocean chemistry does not limit the production of cations and carbonate alkalinity.

To demonstrate the computation of thermodynamic solute concentrations resulting from fluid-rock reactions, weathering of peridotite is considered. Because two pyroxene endmembers (wollastonite and enstatite) and two olivine endmembers (forsterite and fayalite) are considered to constitute peridotite, dissolution reactions of these four minerals are of interest (Appendix B, Table B1, rows: (a), (b), (d), (e)). Besides, reactions in the water–bicarbonate system are needed (Appendix B, Table B1, rows: (o), (p), (q), (r)). These eight chemical reactions result in 8 equations and 10 unknowns. The eight equations are given by the relations between activities and equilibrium constants (Appendix B). The 10 unknowns are the activities of CO₂(g), CO₂(aq), SiO₂(aq), Ca²⁺, Mg²⁺, Fe²⁺, H⁺, OH⁻, ${\mathrm{HCO}}_{3}^{-}$, and ${\mathrm{CO}}_{3}^{2-}$. An additional equation is given by balancing the charges of all cations and anions present in the solution. These 9 equations are further reduced to one polynomial equation with two unknowns, activities of ${\mathrm{HCO}}_{3}^{-}$ and CO₂(g) (see Appendix B, Table B2). If ferrosilite, the third endmember of pyroxene, is also considered, an additional equation is introduced but the unknowns remain the same and the system is overdetermined. This scenario stems from the assumption of endmember minerals with unity activities instead of solid solutions. In the case of solid solutions, the activities of endmember minerals become unknowns rather than being fixed at unity. This increases the number of unknowns for a given set of equations, and requires more equations that can be derived from additional reactions (e.g., see Galvez et al. 2015, for a treatment of solid solutions). Although mineral solid solutions are commonplace in rocks, endmember considerations allow us to establish a simple framework in which the effects of various lithologies can be explored on the weathering of exoplanets. Therefore, the presence of ferrosilite in peridotite is ignored. Moreover, the presence of K-feldspar and quartz in basalt and anorthite in granite is not considered. For a given ${P}_{{\mathrm{CO}}_{2}}$, ${a}_{{\mathrm{HCO}}_{3}^{-}}$ is calculated by finding the sole physical root of such a polynomial equation. The bicarbonate ion concentration at chemical equilibrium is obtained from the standard concentration, [${\mathrm{HCO}}_{3}^{-}$]_eq = ${a}_{{\mathrm{HCO}}_{3}^{-}}$× 1 mol dm⁻³. Similarly, activities of all aqueous species are converted to concentrations using the standard concentration. Subsequently, all unknowns are calculated from the relations between activities and equilibrium constants (Table B1).

At chemical equilibrium, the carbonate alkalinity increases, and the pH of the solution decreases with an increase in ${P}_{{\mathrm{CO}}_{2}}$ at T = 288 K (Figure 3(a)). As ${P}_{{\mathrm{CO}}_{2}}$ increases, [${\mathrm{HCO}}_{3}^{-}$]_eq and [${\mathrm{CO}}_{3}^{2-}$]_eq increase monotonically, thereby increasing A_eq, whereas pH_eq decreases monotonically. For all except ${P}_{{\mathrm{CO}}_{2}}\lt 20$ μbar, [${\mathrm{HCO}}_{3}^{-}$]_eq contributes significantly to A_eq. The weathering flux (Equation (3)) resulting from thermodynamic solute concentrations gives the maximum weathering flux. Because no secondary minerals are modeled for peridotite weathering, the weathering flux for ${P}_{{\mathrm{CO}}_{2}}\gt 0.1\,\mathrm{bar}$ is likely an overestimate (e.g., aqueous alteration of olivine produces secondary minerals; Kite & Melwani Daswani 2019). Figure 3(b) shows that the thermodynamic weathering flux calculations using [${\mathrm{HCO}}_{3}^{-}$]_eq and A_eq increase monotonically with ${P}_{{\mathrm{CO}}_{2}}$. We follow the same procedure as for peridotite to compute the maximum solute concentrations and the maximum weathering flux for basalt, granite, or individual endmember silicate minerals by solving polynomial equations given in Appendix B and the corresponding charge balance equations.

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In natural environments, fluid–rock reactions may not have the time to attain chemical equilibrium due to high fluid flow rates. The maximum weathering model (Section 2.3) does not accurately represent the weathering flux when chemical equilibrium for mineral dissolution reactions is not attained. We present the generalized weathering model by extending the fluid-transport-controlled approach of Maher & Chamberlain (2014). Mineral dissolution reactions, being rate limiting, essentially regulate the concentration of reaction products including [${\mathrm{HCO}}_{3}^{-}$]. In this limit, kinetics plays a more dominant role than thermodynamics. Maher & Chamberlain (2014) provide a solute transport equation to calculate such a transport-buffered (dilute) solute concentration from its value at chemical equilibrium, fluid flow rate (runoff) q, and the Damköhler coefficient D_w , which gives the "net reaction rate" (see below). The solute transport equation with ${\mathrm{HCO}}_{3}^{-}$ is

$$\begin{eqnarray}&&\left[{\mathrm{HCO}}_{3}^{-}\right]=\displaystyle \frac{{\left[{\mathrm{HCO}}_{3}^{-}\right]}_{\mathrm{eq}}}{1+\tfrac{q}{{D}_{w}}}.\end{eqnarray} \tag{ 8 }$$

In their formulation, Maher & Chamberlain (2014) multiply D_w by an arbitrary scaling constant τ = e² ≈ 7.389 in order to scale up the solute concentration to 88% of its equilibrium value at q = D_w . Ibarra et al. (2016) when applying the solute transport equation of Maher & Chamberlain (2014), use τ = 1 without any scaling. For simplicity, we use τ = 1, implying $[{\mathrm{HCO}}_{3}^{-}]=0.5\ {\left[{\mathrm{HCO}}_{3}^{-}\right]}_{\mathrm{eq}}$ at q = D_w . The resulting [${\mathrm{HCO}}_{3}^{-}]$ of our model at q = D_w is about 43% lower than that of the Maher & Chamberlain (2014) model, which is a much smaller difference than the 5−10 orders of magnitude range of solute concentrations explored in this study.

The quantity ${D}_{w}/q$ in Equation (8) is the Damköhler number, the ratio of the fluid residence time and chemical equilibrium timescale (Steefel & Maher 2009). The ${D}_{w}/q$ ratio is essentially the ratio of the "net reaction rate" and the fluid flow rate. When the fluid residence time exceeds the chemical equilibrium timescale, or the net reaction rate exceeds the fluid flow rate (q < D_w ), $[{\mathrm{HCO}}_{3}^{-}]\to {\left[{\mathrm{HCO}}_{3}^{-}\right]}_{\mathrm{eq}}$ and the weathering flux reaches its maximum value for a given q, $w={\left[{\mathrm{HCO}}_{3}^{-}\right]}_{\mathrm{eq}}\ q$ (Figure 4). This weathering regime is called the thermodynamically limited weathering (hereafter, thermodynamic regime), also known as runoff-limited weathering. When the chemical equilibrium timescale exceeds the fluid residence time, or the fluid flow rate exceeds the net reaction rate (q > D_w ), $[{\mathrm{HCO}}_{3}^{-}]\to {\left[{\mathrm{HCO}}_{3}^{-}\right]}_{\mathrm{eq}}\tfrac{{D}_{w}}{q}$ and $w={\left[{\mathrm{HCO}}_{3}^{-}\right]}_{\mathrm{eq}}\ {D}_{w}$, making w independent of [${\mathrm{HCO}}_{3}^{-}$]_eq because D_w is modeled to be inversely proportional to [${\mathrm{HCO}}_{3}^{-}$]_eq (see below). This regime is known as kinetically limited weathering (hereafter, kinetic regime). The transition between these two regimes occurs at q = D_w . The comparison of the timescales described here is conceptually identical to the "quenching approximation" employed in atmospheric chemistry (Prinn & Barshay 1977; Tsai et al. 2017).

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Rewriting the formulation of D_w from Maher & Chamberlain (2014),

$$\begin{eqnarray}&&{D}_{w}=\displaystyle \frac{\psi }{{\left[{\mathrm{HCO}}_{3}^{-}\right]}_{\mathrm{eq}}({k}_{\mathrm{eff}}^{-1}+{{mA}}_{\mathrm{sp}}{t}_{{\rm{s}}})},\end{eqnarray} \tag{ 9 }$$

where [${\mathrm{HCO}}_{3}^{-}$]_eq is the equilibrium solute concentration, k_eff is the effective kinetic rate coefficient given by kinetics data (Appendix A), t_s is the age of soils or pore space, A_sp is the specific reactive surface area per unit mass of the rock, m is the mean molar mass of the rock, ψ = L(1 − ϕ)ρ X_r A_sp is a dimensionless pore-space parameter that combines five parameters including flow path length L, porosity ϕ, rock density ρ, and the fraction of reactive minerals in fresh rock X_r . Although Maher & Chamberlain (2014) parameterize D_w using nine quantities, in Appendix D, we show that D_w is mostly sensitive to four of these quantities given their plausible ranges: [${\mathrm{HCO}}_{3}^{-}$]_eq, k_eff, t_s, and L (L is absorbed in ψ). The remaining five parameters are fixed to reference values (Table 3). The parameter ψ scales k_eff with dimensions of moles per unit reactive surface area of rocks per unit time to w with dimensions of moles per unit exposed continental or seafloor area for a given lithology per unit time. In Equation (9), the solid mass to fluid volume ratio ρ_sf given in the D_w formulation of Maher & Chamberlain (2014) is rewritten in terms of solid density and porosity using ρ_sf = ρ(1 − ϕ)/ϕ. The kinetic rate coefficients of mineral dissolution reactions are obtained from Palandri & Kharaka (2004) as a function of T and pH (see Appendix A). The solution pH at chemical equilibrium is used to calculate k_eff. For minerals with no k_eff data, the k_eff of a corresponding endmember mineral from the same mineral group is adopted. For rocks, we adopt k_eff given by the minimum k_eff among constituent minerals because the slowest reaction is normally rate limiting; but see, e.g., Milliken et al. (2009), where the fastest dissolving mineral sets the pace of rock dissolution.

Table 3. Fixed Parameters

Symbol	Description	Value
ϕ	Porosity	0.175 a
L	Flow path length	1 m a
Xr	Fraction of reactive minerals	1 c
	in fresh rock
Asp	Specific surface area	100 m2 kg−1 a
	of mineral or rock
ρ	Density of mineral or rock	2700 kg m−3 a
m	Mean molar mass of rock	0.27 kg mol−1 a
As	Planet surface area	510.1 Mm2 b
f	Continental area fraction	0.3 b
S	Stellar flux	1360 W m−2 b
α	Planetary albedo	0.3 b
${P}^{{\prime} }$	Seafloor pore-space pressure	200 bar b

Notes.

^a Maher & Chamberlain (2014). ^b Present-day Earth. ^c All considered minerals are reactive.

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In Equation (9), when t_s = 0 (young soils), ${D}_{w}=\tfrac{{k}_{\mathrm{eff}}\psi }{{\left[{\mathrm{HCO}}_{3}^{-}\right]}_{\mathrm{eq}}}$ and w = k_eff ψ. In this "fast kinetic" regime, the weathering flux is directly proportional to the kinetic rate coefficient, as assumed in traditional kinetic weathering models (e.g., Walker et al. 1981). When ${t}_{{\rm{s}}}\gg \tfrac{1}{{k}_{\mathrm{eff}}{{mA}}_{\mathrm{sp}}}$ (old soils), ${D}_{w}=\tfrac{\psi }{{\left[{\mathrm{HCO}}_{3}^{-}\right]}_{\mathrm{eq}}{{mA}}_{\mathrm{sp}}{t}_{{\rm{s}}}}$, and $w=\tfrac{\psi }{{{mA}}_{\mathrm{sp}}{t}_{{\rm{s}}}}$. This regime may be termed as the "slow kinetic" regime; however, an accepted terminology is supply-limited weathering (hereafter, supply regime), which is limited by the supply of fresh rocks in the weathering zone (Riebe et al. 2003; West et al. 2005). An increase in t_s decreases the influence of chemical kinetics on weathering. Although t_s depends on the soil production and physical erosion rates, which in turn are sensitive to climate, topography, and fluid flow properties, there is no consensus on the formulation of soil production and physical erosion rates (Riebe et al. 2003; West et al. 2005; Gabet & Mudd 2009; West 2012; Maher & Chamberlain 2014; Foley 2015). Treating t_s as a free parameter makes it possible to model both the age of continental soils and seafloor pore space. The transition between the kinetic and supply regimes can be defined at ${t}_{{\rm{s}}}=\tfrac{1}{{k}_{\mathrm{eff}}{{mA}}_{\mathrm{sp}}}$, where the kinetic reaction rate equals the supply rate of fresh rocks (see Equation (9)).

In the fluid-transport-controlled model, the mineral dissolution reactions (Appendix B, rows (a)−(n) in Table B1) do not attain chemical equilibrium, and their reaction products are given by the solute transport equation (Equation (8)). On the other hand, reactions in the water–bicarbonate system (Appendix B, rows (o)−(r) in Table B1), are considered to reach chemical equilibrium. This assumption is justified because the equilibrium timescale of chemical reactions in the water–bicarbonate system is of the order of milliseconds to days, unlike mineral dissolution reactions that may take months to thousands of years (e.g., Palandri & Kharaka 2004; Schulz et al. 2006). Thus, the generalized concentrations of aqueous species in the water–bicarbonate system such as [${\mathrm{CO}}_{3}^{2-}$], [H⁺], and [OH⁻] are calculated from the transport-buffered [${\mathrm{HCO}}_{3}^{-}]$ and equilibrium constants (see Appendix B, Figure B2 for the flow chart of the methodology). Rewriting [${\mathrm{CO}}_{3}^{2-}]$ in terms of [${\mathrm{HCO}}_{3}^{-}$], ${P}_{{\mathrm{CO}}_{2}}$, and the equilibrium constants in Equation (7),

$$\begin{eqnarray}&&A=[{\mathrm{HCO}}_{3}^{-}]+2\displaystyle \frac{{K}_{\mathrm{Car}}{\left[{\mathrm{HCO}}_{3}^{-}\right]}^{2}}{{K}_{\mathrm{Bic}}{P}_{{\mathrm{CO}}_{2}}}.\end{eqnarray} \tag{ 10 }$$

Figure 4(a) shows the carbonate alkalinity and the solution pH for the generalized model of peridotite weathering as a function of ${P}_{{\mathrm{CO}}_{2}}$ at a constant surface temperature (T = 288 K). For ${P}_{{\mathrm{CO}}_{2}}\lt 1$ μbar, [${\mathrm{HCO}}_{3}^{-}$], [${\mathrm{CO}}_{3}^{2-}]$, and pH in Figure 4(a) exhibit the same behavior as in Figure 3(a) because q < D_w , reiterating that chemical equilibrium calculations are valid in the thermodynamic regime resulting in the maximum weathering flux. The maximum ([${\mathrm{HCO}}_{3}^{-}$]_eq and A_eq, Figure 3(b)) and generalized ([${\mathrm{HCO}}_{3}^{-}]$ and A, Figure 4(b)) weathering fluxes diverge from each other beyond the thermodynamic to kinetic regime transition that occurs at ${P}_{{\mathrm{CO}}_{2}}\sim 1$ μbar. The choice of q determines this transition. At a higher q, the regime transition would shift to a lower ${P}_{{\mathrm{CO}}_{2}}$ and vice versa. For ${P}_{{\mathrm{CO}}_{2}}\gt 1$ μbar, [${\mathrm{HCO}}_{3}^{-}]$ becomes independent of ${P}_{{\mathrm{CO}}_{2}}$ in the kinetic weathering regime. This is because the kinetic rate coefficient of the slowest reaction (fayalite dissolution) is constant at a fixed T and does not vary when the pH is basic (see Appendix A). However, the results shown in Figure 4 are expected to change when the temperature is not held constant and depends on ${P}_{{\mathrm{CO}}_{2}}$, where the relation between the two is given by a climate model (Section 2.5).

In the kinetic weathering regime, [${\mathrm{CO}}_{3}^{2-}]$ decreases with ${P}_{{\mathrm{CO}}_{2}}$ (Figure 4(a)) because [${\mathrm{CO}}_{3}^{2-}]$ is directly proportional to [${\mathrm{HCO}}_{3}^{-}]$ and inversely proportional to ${P}_{{\mathrm{CO}}_{2}}$ (Equation (10)). Because [${\mathrm{HCO}}_{3}^{-}]$ is constant in the kinetic regime, [${\mathrm{CO}}_{3}^{2-}]$ shows a strong decrease because of its sole dependence on ${P}_{{\mathrm{CO}}_{2}}$. We follow the same procedure as for peridotite to compute the generalized concentrations for the weathering of basalt, granite, or individual minerals and find that lithology strongly impacts the occurrence of weathering regimes (Appendix B, Figure B3).

The greenhouse effect of CO₂ exerts a strong control on the planetary surface temperature. A climate model enables one to express the surface temperature T as a function of the CO₂ partial pressure ${P}_{{\mathrm{CO}}_{2}}$ for a given planetary albedo α and top-of-atmosphere stellar flux S. Such a climate model is essential to assess the role of climate in weathering on temperate planets. CHILI provides the functionality to couple T and ${P}_{{\mathrm{CO}}_{2}}$ using any climate model. Previous studies provide formulations of climate models (e.g., Walker et al. 1981; Kasting et al. 1993). Recently, Kopparapu et al. (2013, 2014) performed 1D radiative-convective calculations to obtain T as a function of ${P}_{{\mathrm{CO}}_{2}}$, α, and S. Studies such as Haqq-Misra et al. (2016) and Kadoya & Tajika (2019) provide fitting functions to the models of Kopparapu et al. (2013, 2014). We use the fitting function provided by Kadoya & Tajika (2019) to couple T in the range 150–350 K with ${P}_{{\mathrm{CO}}_{2}}$ in the range 10⁻⁵ − 10 bar. For α = 0.3 (present-day albedo of Earth) and S = 1360 W m⁻² (present-day solar flux), the Kadoya & Tajika (2019) fitting function results in T between 280 and 350 K for ${P}_{{\mathrm{CO}}_{2}}$ between 10 μbar and 0.5 bar (see Appendix E for details).

The thermodynamic solute concentrations provide an upper limit to weathering (e.g., peridotite weathering; Figure 3). To evaluate the case of maximum weathering on temperate planets, in Figure 5, we provide the [${\mathrm{HCO}}_{3}^{-}$]_eq weathering flux of three common rocks (basalt, peridotite, granite) as a function of climate properties, ${P}_{{\mathrm{CO}}_{2}}$ and T, at the present-day mean runoff of 0.3 m yr⁻¹. The dependence of weathering on total surface pressure is negligible (see Appendix B). The [${\mathrm{HCO}}_{3}^{-}$]_eq weathering flux for all rocks increases monotonically with ${P}_{{\mathrm{CO}}_{2}}$ at a constant temperature because weathering intensifies as ${P}_{{\mathrm{CO}}_{2}}$ increases (Figure 5(a)). This is a direct consequence of the calculations of solute concentrations at chemical equilibrium (Section 2.3). Table 4 gives fitting parameters of the thermodynamic weathering flux of [${\mathrm{HCO}}_{3}^{-}$]_eq to the kinetic weathering expression (Equation (2)) for the silicate rocks and minerals considered in this study. Such a fit, although not the best approximation of the calculated values, provides a way to compare the sensitivity of thermodynamic weathering to climate properties with studies assuming kinetic weathering (e.g., Walker et al. 1981; Sleep & Zahnle 2001; Foley 2015).

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Fluid–rock reactions produce both monovalent (e.g., K⁺, Na⁺) and divalent cations (e.g., Ca²⁺, Mg²⁺). The sensitivity of [${\mathrm{HCO}}_{3}^{-}$]_eq flux to ${P}_{{\mathrm{CO}}_{2}}$ depends on the capacity of a rock to produce divalent cations. More divalent cations in the solution require more ${\mathrm{HCO}}_{3}^{-}$ and ${\mathrm{CO}}_{3}^{2-}$ ions to balance the charges. Consequently, the higher the fraction of divalent cations in a rock, the higher the thermodynamic ${P}_{{\mathrm{CO}}_{2}}$ sensitivity (β_th). For instance, peridotite, which produces only divalent cations, exhibits the highest β_th among the three rocks (Figure 3). Granite produces more monovalent cations than peridotite and basalt, and therefore has the lowest β_th. This effect has been discussed for feldspar minerals by Ibarra et al. (2016); Winnick & Maher (2018).

The β_th values for endmember minerals within the same mineral group (pyroxene, olivine, mica, and amphibole), with the exception of feldspars, are similar to each other (Table 4). This result is again attributed to the presence of monovalent or divalent cations. The divalent-cation-producing pyroxene, olivine, mica, and amphibole endmembers exhibit β_th ∼ 0.5, and monovalent-cation-producing albite and K-feldspar exhibit β_th ∼ 0.25. The deviation from these ideal values of 0.5 and 0.25 is a result of the simultaneous consideration of the mineral dissolution reaction and the water–bicarbonate reactions. For instance, Winnick & Maher (2018) show that reaction stoichiometry controls β_th values by considering the dissolution reactions of individual feldspar minerals and find β_th = 0.25 for albite and K-feldspar. However, we find that the presence of ions produced by the water–bicarbonate system makes β_th dependent on equilibrium constants of all reactions considered in addition to stoichiometry and hence β_th = 0.26 for albite and K-feldspar (Table 4). The β_th value of the divalent-cation-producing anorthite is 0.69, considerably higher than other divalent-cation-producing minerals. This is again attributed to reaction stoichiometry as demonstrated by Winnick & Maher (2018), although their study results in a slightly smaller β_th = 0.67 because reactions in the water–bicarbonate system are ignored.

The weathering flux produced by the rock with a higher β_th is higher (Figure 3(a)). However, the choice of secondary minerals produced during weathering influences this result. For instance, if kaolinite is considered instead of halloysite, the weathering flux of basalt is higher by more than an order of magnitude, which is even higher than that of peridotite (see Table 4). The effect of a secondary mineral on the weathering flux of granite is smaller than that of basalt. This difference is attributed to the impact of the secondary mineral on weathering reactions of feldspar endmember minerals. Anorthite, a constituent of basalt in our model, exhibits an order of magnitude increase in the weathering flux when kaolinite is used instead of halloysite (Table 4), whereas the weathering flux of albite and K-feldspar, constituents of granite in our model, increases by a factor of 3 for the same change. Moreover, the nonconsideration of secondary minerals for peridotite weathering results in high weathering flux at high ${P}_{{\mathrm{CO}}_{2}}$, which in reality may be lower (e.g., Kite & Melwani Daswani 2019). Thus, the choice of secondary mineral strongly affects the w and β_th of rocks. However, this effect is small for the weathering of individual minerals (Table 4).

Contrary to observations of the increase in kinetic weathering flux with temperature (e.g., Walker et al. 1981), the thermodynamic weathering flux for rocks decreases with the temperature at a fixed ${P}_{{\mathrm{CO}}_{2}}$ (Figure 5(b)). The fitting parameter, the activation energy of thermodynamic weathering, provides a scaling relation between weathering and temperature. Unlike kinetic weathering, E_th is negative for the weathering of all rocks and minerals except albite and K-feldspar (Table 4). This result is a consequence of the decrease in equilibrium constants as a function of temperature for all mineral dissolution reactions except those of albite and K-feldspar (see Appendix A). The choice of secondary mineral influences the magnitude of E_th but not its sign. Winnick & Maher (2018) observe this effect for plagioclase feldspars, which contain anorthite in addition to albite. This effect is discussed further in Section 4.2.

The fitting parameters provided in Table 4 should be used with caution as β_th depends on T and E_th depends on ${P}_{{\mathrm{CO}}_{2}}$. For example, β_th for peridotite at ${P}_{{\mathrm{CO}}_{2}}$ = 280 μbar varies between 0.71 at T = 273 K and 0.77 at T = 373 K, whereas E_th for peridotite at T = 288 K varies between −55 kJ mol⁻¹ at ${P}_{{\mathrm{CO}}_{2}}$ = 1 μbar and −46 kJ mol⁻¹ at ${P}_{{\mathrm{CO}}_{2}}$ = 1 bar. This provides further reason to calculate thermodynamic concentrations consistently by considering all necessary reactions simultaneously, as formulated in this study.

Figure 6 shows the sensitivity of both maximum and generalized weathering fluxes of peridotite to CO₂ partial pressure and surface temperature. The maximum weathering model gives an upper limit to the weathering flux. The generalized weathering flux is either equal to or smaller than the maximum weathering flux depending on if the generalized model encounters the thermodynamic regime or not. The generalized weathering flux in the kinetic regime is lower than that in the thermodynamic regime and higher than that in the supply regime. Because the weathering flux in the thermodynamic regime is an upper limit to weathering, the kinetic flux cannot exceed the thermodynamic flux. To isolate the effect of ${P}_{{\mathrm{CO}}_{2}}$ on weathering, we present the peridotite-weathering flux as a function of ${P}_{{\mathrm{CO}}_{2}}$ at a fixed surface temperature (Figure 6(a)). The sensitivity of weathering to temperature is demonstrated at two ${P}_{{\mathrm{CO}}_{2}}$ values (Figures 6(b) and (d)). In reality, T depends on ${P}_{{\mathrm{CO}}_{2}}$ due to the greenhouse effect of CO₂, which is normally modeled using a climate model (Section 2.5). The strength of the coupling between T and ${P}_{{\mathrm{CO}}_{2}}$ is sensitive to numerous parameters including incident solar flux and planetary albedo that are fixed to present-day values (Table 3). Because the solar flux is held constant, the coupling between T and ${P}_{{\mathrm{CO}}_{2}}$ becomes stronger than that during Earth's early history where the solar flux dropped to about 70% of its present-day value. Figure 6(c) demonstrates peridotite weathering under the limit of strong coupling between T and ${P}_{{\mathrm{CO}}_{2}}$.

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The maximum (thermodynamic) carbonate alkalinity and bicarbonate ion fluxes increase monotonically with ${P}_{{\mathrm{CO}}_{2}}$ at T = 288 K (Figure 6(a)). This monotonic behavior is a direct result of chemical equilibrium calculations with ${P}_{{\mathrm{CO}}_{2}}$ as a free parameter at a fixed temperature. The difference between the maximum A and [${\mathrm{HCO}}_{3}^{-}]$ fluxes at low ${P}_{{\mathrm{CO}}_{2}}$ is due to the excess contribution of [${\mathrm{CO}}_{3}^{2-}]$ to A when the solution pH is basic (see Figure 3). The thermodynamic weathering flux decreases with T at a fixed ${P}_{{\mathrm{CO}}_{2}}$ (Figures 6(b) and (d)). As explained in Section 3.1 and later discussed in Section 4.2, this decrease is a result of the decrease in equilibrium constants of mineral dissolution reactions as a function of temperature (see Appendix A). The thermodynamic flux at ${P}_{{\mathrm{CO}}_{2}}$ = 0.1 bar is higher than that at ${P}_{{\mathrm{CO}}_{2}}$ = 280 μbar by about a factor of 30. Unlike at ${P}_{{\mathrm{CO}}_{2}}$ = 280 μbar, at ${P}_{{\mathrm{CO}}_{2}}$ = 0.1 bar, there is a negligible difference between the thermodynamic A and [${\mathrm{HCO}}_{3}^{-}]$ fluxes. When T depends on ${P}_{{\mathrm{CO}}_{2}}$ via a climate model, the behavior of thermodynamic weathering as a function of ${P}_{{\mathrm{CO}}_{2}}$ depends on the trade-off between the individual effects of T and ${P}_{{\mathrm{CO}}_{2}}$ (Figure 6(c)). Up to ${P}_{{\mathrm{CO}}_{2}}=0.01\,\mathrm{bar}$, the thermodynamic fluxes increase with ${P}_{{\mathrm{CO}}_{2}}$ because the ${P}_{{\mathrm{CO}}_{2}}$ effect dominates, whereas beyond ${P}_{{\mathrm{CO}}_{2}}=0.05\,\mathrm{bar}$, there is a small decrease in the thermodynamic fluxes because the effect of T takes over.

When soils are young (soil age is zero, t_s = 0), the generalized [${\mathrm{HCO}}_{3}^{-}]$ flux at T = 288 K, which is in the kinetic regime, is almost constant for the given ${P}_{{\mathrm{CO}}_{2}}$ range (Figure 6(a)). The kinetic rate coefficient depends on T as well as the solution pH which in turn depends on ${P}_{{\mathrm{CO}}_{2}}$ (Section 2.4). However, the kinetic weathering flux is independent of ${P}_{{\mathrm{CO}}_{2}}$ at a fixed temperature because k_eff of peridotite is independent of ${P}_{{\mathrm{CO}}_{2}}$ when the solution pH is basic (Figure 6(a)). This k_eff is determined by the fayalite dissolution reaction as it is rate limiting among the considered mineral dissolution reactions for peridotite (see Appendix A). As a function of T at constant ${P}_{{\mathrm{CO}}_{2}}$, kinetic weathering exhibits a strong dependence on temperature as seen in Figures 6(b) and (d). For the ${P}_{{\mathrm{CO}}_{2}}$ = 280 μbar case, the generalized (t_s = 0) model switches from the kinetic to the thermodynamic regime at ∼310 K where the limit of maximum weathering is encountered. For the ${P}_{{\mathrm{CO}}_{2}}$ = 0.1 bar case, this transition temperature increases to ∼340 K. The transition from kinetic to thermodynamic regime occurs at high T and low ${P}_{{\mathrm{CO}}_{2}}$. The sequence of this transition is in contrast to the switch from the thermodynamic to the kinetic regime that occurs at ${P}_{{\mathrm{CO}}_{2}}=1$ μbar as a function of ${P}_{{\mathrm{CO}}_{2}}$ (see Figure 4). If the climate model is invoked, the generalized (t_s = 0) weathering flux increases steeply with ${P}_{{\mathrm{CO}}_{2}}$ and encounters the thermodynamic regime at about ${P}_{{\mathrm{CO}}_{2}}=0.1\,\mathrm{bar}$, a result of the kinetic temperature dependence (Figure 6(c)). When T is strongly coupled to ${P}_{{\mathrm{CO}}_{2}}$, the generalized model may switch from the thermodynamic to kinetic regime at low ${P}_{{\mathrm{CO}}_{2}}$ and from the kinetic regime back to the thermodynamic regime at high ${P}_{{\mathrm{CO}}_{2}}$.

When soils are old (present-day characteristic soil age, t_s = 100 kyr), the [${\mathrm{HCO}}_{3}^{-}]$ flux at T = 288 K as a function of ${P}_{{\mathrm{CO}}_{2}}$ is constant and in the supply regime (Figure 6(a)). This regime is limited by the supply of fresh rocks, and the influence of chemical kinetics on weathering is small compared to when soils are young (t_s is small). This regime is almost independent of T (Figures 6(b) and (d)). Consequently, unlike the kinetic weathering flux, the supply-limited weathering flux is independent of ${P}_{{\mathrm{CO}}_{2}}$ when T dependence via the climate model is invoked (Figure 6(c)). In the supply regime, the weathering flux depends on the age of soils, which is held constant. There is almost no difference between the generalized (t_s = 100 kyr) A and [${\mathrm{HCO}}_{3}^{-}]$ fluxes between Figures 6(a) and (c) because the inclusion of the T effect via the climate model is not strong enough to escape from the supply regime. In contrast, for the generalized (t_s = 0) model, invoking the climate model increases the weathering fluxes in the kinetic regime enough to enter the thermodynamic regime at about ${P}_{{\mathrm{CO}}_{2}}=0.1\,\mathrm{bar}$.

We apply the generalized weathering model to both continental (Figures 7(a) and (b)) and seafloor silicate weathering (Figures 7(c) and (d)) for diverse cases that may represent weathering scenarios on temperate rocky exoplanets. The climate model is used to couple T with ${P}_{{\mathrm{CO}}_{2}}$ by fixing the stellar flux and planetary albedo to present-day Earth values (Section 2.5). This implies a strong coupling between T and ${P}_{{\mathrm{CO}}_{2}}$ such that T varies from 280 to 350 K when ${P}_{{\mathrm{CO}}_{2}}$ varies from 10 μbar to 0.5 bar. The lithology and pore-space properties of continents and seafloor on present-day Earth are different from each other. This presents an opportunity to test the generalized weathering model in an extended parameter space beyond applications to continental weathering. We show the results for basalt and granite in Figure 7. The sensitivity of the carbonate alkalinity weathering flux to ${P}_{{\mathrm{CO}}_{2}}$ is a complex function of climate, fluid flow rate, and rock and pore-space properties. This result implies that the weathering flux cannot be simply approximated by Equation (2) assuming either kinetic weathering (e.g., Walker et al. 1981; Berner et al. 1983; Sleep & Zahnle 2001; Foley 2015) or thermodynamic weathering (Winnick & Maher 2018, and Section 3.1, this study).

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In Figure 7(a), the continental A flux is calculated for three values of runoff that are representative of arid, modern mean, and humid climates. Gaillardet et al. (1999) report regional variations in the present-day runoff from 0.01 to 3 m yr⁻¹. Because the choice of humid runoff in our model is higher than the arid runoff by three orders of magnitude, the corresponding weathering fluxes should differ by the same amount if the weathering fluxes are in the thermodynamic regime. However, the differences are smaller in the given ${P}_{{\mathrm{CO}}_{2}}$ range. This is because nonthermodynamic regimes exhibit smaller weathering fluxes than the thermodynamic regime. For example, up to ${P}_{{\mathrm{CO}}_{2}}$ = 0.2 bar, the arid model of granite is in the thermodynamic regime (upper limit for this model), whereas the humid model of granite is in the supply regime (lower limit for this model). Thus, the differences between the two models are smaller than the maximum possible differences. For more arid climates, lithology plays an even more important role as the weathering becomes thermodynamically limited for the whole ${P}_{{\mathrm{CO}}_{2}}$ range. In contrast, the supply regime is largely independent of lithology. For this reason, lithology has a negligible impact on the modern mean and humid cases.

The age of soils is a key parameter that determines if the weathering is limited by reaction kinetics or limited by the supply of fresh rocks. Figure 7(b) shows that lithology has no influence on the weathering flux of old soils (t_s = 100 kyr and t_s = 1 Myr) as opposed to young soils (t_s = 0). The old-soil models are in the supply regimes. Granite and basalt in young soils are in different weathering regimes. Granite is in the thermodynamic regime for the whole ${P}_{{\mathrm{CO}}_{2}}$ range because the net reaction rate (D_w = 1–3 m yr⁻¹) is higher than the fluid flow rate (q = 0.3 m yr⁻¹). In contrast, D_w (0.05–4.5 m yr⁻¹) of basalt is lower than q = 0.3 m yr⁻¹ up to ${P}_{{\mathrm{CO}}_{2}}=0.5\,\mathrm{bar}$, implying a transition from kinetic to thermodynamic regime above this ${P}_{{\mathrm{CO}}_{2}}$.

On Earth, the continental and seafloor pore space differ in terms of pore-space properties besides the differences in lithology. The dimensionless pore-space parameter ψ depends on the flow path length L that is normally assumed to be of the order of the regolith thickness (Section 2.4). Because the thickness of the oceanic crust where seafloor weathering occurs is of the order of 100 m (e.g., Alt et al. 1986; Coogan & Gillis 2018), we assume L = 100 m and ψ = 100ψ₀. Moreover, the average age of the oceanic crust on Earth at the present day is approximately equal to 50 Myr, about 500 times the characteristic age of continental soils.

In Figure 7(d), we compare the seafloor A flux for characteristic seafloor values of t_s = 50 Myr and ψ = 100ψ₀ with two models, one with t_s = 0 and another with ψ = ψ₀. The present-day seafloor A flux is in the supply regime, making it independent of lithology. The supply-limited fluxes of seafloor weathering are smaller than those of continental weathering by a factor of 5 because $w\propto \psi /{t}_{{\rm{s}}}$ in this regime. When ψ is lowered from 100ψ₀ to ψ₀, the supply-limited weathering flux decreases by two orders of magnitude at a much lower ${P}_{{\mathrm{CO}}_{2}}$. For the young seafloor pore-space case, the A fluxes of basalt and granite are in the thermodynamic regime, and consequently, the impact of lithology is pronounced. Compared to the t_s = 0 basalt model in Figure 7(b) that is in the kinetic regime, the t_s = 0 basalt model in Figure 7(d) is in the thermodynamic regime. This difference arises due to the choice of ψ that makes the net reaction rate (D_w ) higher than the fluid flow rate, pushing basalt into the thermodynamic regime. Being in the thermodynamic regime, chemical equilibrium controls the weathering flux of basalt. At high ${P}_{{\mathrm{CO}}_{2}}$, the thermodynamic weathering flux of basalt decreases slightly as the decreasing effect of T takes over the increasing effect of ${P}_{{\mathrm{CO}}_{2}}$. This decreasing effect of T is due to the decrease in equilibrium constants of weathering reactions as a function of temperature (Appendix A). The effect of T on the weathering flux of granite is small and hence there is no net decrease in the weathering flux at high ${P}_{{\mathrm{CO}}_{2}}$.

In Figure 7(c), we calculate the seafloor A flux for three hydrothermal fluid flow rates, where the two extreme q values differ by three orders of magnitude, similar to the strategy in Figure 7(a). Because the fluid flow rates are directly proportional to the hydrothermal heat flux (Stein & Stein 1994; Coogan & Gillis 2013), a variation in the hydrothermal heat flux implies a variation in the fluid flow rate. Depending on the age of oceanic crust, present-day non-porosity-corrected fluid flow rates are observed between 0.001 and 0.7 m yr⁻¹ (Johnson & Pruis 2003). The three cases of seafloor weathering shown in Figure 7(c) are broadly similar to their continental counterparts. The modern mean and high fluid flow rates result in lithology-independent weathering fluxes that are in the supply-limited regimes. The low hydrothermal fluid flow rate causes the granite model to be in the thermodynamic regime for the full ${P}_{{\mathrm{CO}}_{2}}$ range, although the basalt model transitions from the thermodynamic to supply regime at ${P}_{{\mathrm{CO}}_{2}}=0.6$ mbar.

In this study, silicate-weathering rates are computed by the simultaneous consideration of dissolution reactions of all minerals present in a rock as well as reactions in the water–bicarbonate system. Three common silicate rocks (peridotite, basalt, and granite) are examined. We develop the maximum weathering model (Section 2.3) presuming chemical equilibrium and the generalized weathering model (Section 2.4) that applies to both equilibrium and nonequilibrium conditions. The generalized weathering model allows us to explore weathering in three different regimes (Figure 8). To simulate the transport-based dilution of equilibrium concentrations of weathering products, the solute transport equation of Maher & Chamberlain (2014) is implemented. This equation is based on the interplay between the fluid flow rate (q) and the net reaction rate (D_w ). When q < D_w , reactions are limited by thermodynamics or transport (runoff) and the weathering is thermodynamically limited (Figure 8(a)). In contrast, when q > D_w , reactions are limited by kinetics (and independent of runoff), and the weathering is kinetically limited. Another parameter, the age of soils (t_s), is introduced by Maher & Chamberlain (2014) to model the effect of a limited supply of fresh rocks on the net reaction rate. The higher the t_s, the lower is the D_w . This gives rise to another regime, supply-limited weathering (Figure 8(a)).

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As a function of transport and climate properties, the carbonate alkalinity weathering flux shows a strong dependence on lithology in the thermodynamic and kinetic regimes and a weak dependence on lithology in the supply regime (Figure 8). This impact of lithology for the three weathering regimes as a function of q is seen in Figure 8(a). There is approximately an order of magnitude difference between the thermodynamic fluxes of peridotite and basalt as well as those of basalt and granite. The thermodynamic weathering flux is proportional to the equilibrium carbonate alkalinity, which is strongly sensitive to the mineralogy considered for a given rock. In the kinetic regime, the differences are smaller but significant. The kinetic weathering flux is proportional to the effective rate coefficient of a given rock. In contrast, in the supply-limited regime, the A fluxes almost overlap with each other because there is no impact of lithology. The supply regime is strongly sensitive to pore-space properties that are held constant.

The climate sensitivity of the weathering flux showcases how the weathering regime may depend on lithology. Figure 8(b) shows that the generalized t_s = 0 model of the weathering fluxes of peridotite and basalt increase steeply with ${P}_{{\mathrm{CO}}_{2}}$ up to ∼0.1 bar where $T=f({P}_{{\mathrm{CO}}_{2}})$ is obtained from the climate model. Beyond this ${P}_{{\mathrm{CO}}_{2}}$ value, peridotite and basalt enter the thermodynamic regime. This is because the weathering flux in the kinetic regime cannot keep increasing indefinitely. As soon as the model hits the thermodynamic upper limit, the model follows the thermodynamic sensitivities of weathering that are independent of pore-space properties and depend on chemical equilibrium and lithology. It is important to note that extrapolations of kinetic weathering expressions (Equation (2)) used in previous studies (e.g., Walker et al. 1981; Berner et al. 1983; Sleep & Zahnle 2001; Foley 2015; Krissansen-Totton & Catling 2017) may incorrectly predict the weathering flux to be higher than the upper limit provided by the thermodynamic flux.

Unlike basalt and peridotite, the generalized young-soil (t_s = 0) model of granite is in the thermodynamic regime for the given ${P}_{{\mathrm{CO}}_{2}}$ range (Figure 8(b)). This is also evident from Figure 8(a) where the vertical line (q = 0.3 m yr⁻¹) falls right before the thermodynamic to kinetic regime transition for granite. Thus, the climate sensitivity of the weathering of fresh granite at q = 0.3 m yr⁻¹ is determined largely by thermodynamics instead of kinetics. On the other hand, for the generalized models at the present-day characteristic soil age of t_s = 100 kyr, the weathering fluxes of the three rocks overlap with each other. This is because this t_s value is so high that the effect of k_eff on the "net reaction rate" D_w is negligible in pushing the model out of the supply regime. In this regime, the weathering flux is independent of the climate and transport properties.

Because most laboratory measurements of kinetic rate coefficients are available for individual minerals, previous studies discuss the weathering of individual minerals instead of rocks (e.g., Walker et al. 1981; Berner et al. 1983). In reality, all minerals in rocks undergo weathering contemporaneously, rendering consideration of individual minerals in isolated systems less informative. Because the minerals in a rock are in contact with the aqueous solution, solute concentrations are buffered by the dissolution reactions of these minerals. It is essential to consider these reactions simultaneously to solve for solute concentrations. The generalized weathering model shows that the choice of individual minerals or rocks determines the weathering regime. For example, the feldspar endmember minerals (anorthite, albite, and K-feldspar) are in the thermodynamic regime for the ${P}_{{\mathrm{CO}}_{2}}$ range considered, whereas rocks exhibit both thermodynamic and kinetic regimes (see Figure B3). In their implementation of the fluid-transport-controlled model, Graham & Pierrehumbert (2020) find that weathering is largely independent of kinetics because of their choice of oligoclase (a type of plagioclase feldspar mineral) to model weathering, which is in the thermodynamic regime of weathering for a wide range of CO₂ partial pressures similar to the feldspar endmembers shown in Figure B3.

It is widely accepted that weathering intensifies with surface temperature (e.g., Walker et al. 1981; Berner et al. 1983; Kump et al. 2000; Brantley et al. 2008); but see Gaillardet et al. (1999) and Kite et al. (2011) for alternate viewpoints. This T dependence of weathering is due to the increase in kinetic rate coefficients of mineral dissolution reactions as a function of T (see Appendix A). Laboratory and field measurements of kinetic rate coefficients are fitted to the Arrhenius law, $w\propto \exp (-E/{RT})$, where E is the activation energy and R is the universal gas constant (Palandri & Kharaka 2004). The generalized weathering model based on the fluid-transport-controlled approach (Maher & Chamberlain 2014) captures the diversity of weathering regimes in a generic formulation, which is particularly useful for applications to exoplanets with potentially diverse surface environments. Studies applying the fluid-transport-controlled model find that T has a small effect on weathering (Winnick & Maher 2018; Graham & Pierrehumbert 2020). This statement holds in the thermodynamic regime of weathering for certain plagioclase feldspars including oligoclase, which coincidentally exhibits a small T sensitivity, although Winnick & Maher (2018) find that the equilibrium [${\mathrm{HCO}}_{3}^{-}]$ resulting from plagioclase feldspars decreases with T.

To isolate the effect of temperature on weathering from that of the ${P}_{{\mathrm{CO}}_{2}}$ effect, we first present weathering as a function of T at a fixed ${P}_{{\mathrm{CO}}_{2}}$ in Figure 9(a). The generalized weathering model at t_s = 0, q = 0.3 m yr⁻¹, and ${P}_{{\mathrm{CO}}_{2}}$ = 280 μbar (no climate model) shows that the carbonate alkalinity flux of rocks is in the kinetic regime at low temperatures and in the thermodynamic regime at high temperatures. In the kinetic regime, there is a steep increase in the weathering flux of granite up to ∼285 K, basalt up to ∼310 K, and peridotite up to ∼320 K because the effective kinetic rate coefficients show an exponential increase with temperature. Beyond these transition temperatures (where the net reaction rate equals the fluid flow rate), the models enter their respective thermodynamic regimes. This transition implies a switch from the negative feedback of silicate weathering to the carbon cycle to a potential positive feedback. As the thermodynamic regime gives the maximum possible weathering flux, the kinetic weathering flux cannot exceed the thermodynamic weathering flux in the generalized model. Importantly, the kinetic weathering flux of granite is higher than that of basalt and peridotite because the slowest kinetic reaction among the constituent endmember minerals of granite has a higher kinetic rate coefficient than those of basalt and peridotite. This result is based on the choice of endmember minerals to define rocks and is expected to change with the choice of other endmember minerals or mineral solid solutions.

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The kinetic weathering flux increases with T as expected; however, the thermodynamic weathering flux decreases with T (Figure 9(a)). As described in Section 3.1, if fitted to a kinetic weathering expression (Equation (2)), the thermodynamic model gives a negative value for the activation energy (E_th) for the weathering of rocks and most minerals. Such a negative value highlights that thermodynamic weathering exhibits a negative slope as a function of T. This negative exponential decrease in weathering is a result of the negative slope exhibited by equilibrium constants of mineral dissolution reactions as a function of T except for albite and K-feldspar (see Appendix A). This T dependence is a thermodynamic property of mineral dissolution reactions in the aqueous system. For rocks, constituent minerals determine the overall dependence of thermodynamic weathering on T. For example, for a rock consisting of only albite and K-feldspar, the thermodynamic weathering flux is expected to increase with T, similar to the prevalent understanding of the T dependence of kinetic weathering. However, apart from these two minerals, all other minerals considered in this study show a negative slope. Because granite consists of both albite and K-feldspar in addition to quartz, phlogopite, and annite, granite shows the least negative slope among the three rocks considered (Figure 9(a)). Moreover, the weathering of basalt in the thermodynamic regime decreases with T more steeply than peridotite because of the presence of anorthite in basalt which exhibits the most negative slope in the equilibrium constant−temperature parameter space.

When the climate model is invoked assuming present-day solar flux and present-day planetary albedo, ${P}_{{\mathrm{CO}}_{2}}$ increases from 10 μbar to 0.5 bar as a function of T varying from 280 to 350 K, and the decreasing effect of T on weathering disappears (Figure 9(b)). This is because, in the thermodynamic regime, the weathering flux shows a positive power-law dependence on ${P}_{{\mathrm{CO}}_{2}}$, which is stronger than the exponential decrease in weathering with temperature. And in the kinetic regime, the exponential increase in weathering with temperature dominates. It is interesting to note that basalt and peridotite enter the thermodynamic regime only at ∼325 K and ∼345 K, respectively, whereas granite is in the thermodynamic regime for the given temperature range. For this climate model, assuming a constant stellar flux results in a strong coupling between T and ${P}_{{\mathrm{CO}}_{2}}$, in contrast to the behavior when ${P}_{{\mathrm{CO}}_{2}}$ is held constant. Depending on the stellar flux and planetary albedo, the strength of coupling between T and ${P}_{{\mathrm{CO}}_{2}}$ is probably in between the two panels shown in Figure 9, pushing the thermodynamic to kinetic transition temperature given in Figure 9(a) to higher values. The supply-limited weathering flux, on the other hand, is independent of T (e.g., Figures 6(c) and (d)). This is because the supply-limited weathering flux depends on pore-space parameters including t_s and ψ that are assumed to be constant for a given model. In reality, the age of soils may indirectly depend on temperature through the effect of precipitation and runoff on physical erosion rates (e.g., West et al. 2005; West 2012; Foley 2015).

Previous studies have argued that because runoff depends on precipitation, which in turn depends on T, the weathering flux should increase even more strongly with T when a T-dependent runoff is assumed (e.g., Berner & Kothavala 2001). Figure 9 shows that this statement does not hold in the kinetic regime of weathering. This is because the kinetic regime is independent of runoff in the fluid-transport-controlled model (Maher & Chamberlain 2014). Considering a linear dependence of runoff on temperature (e.g., Equations (41) and (42) in Graham & Pierrehumbert 2020) instead of a constant runoff, there is no impact on weathering because the kinetic regime is independent of runoff in the generalized model.

During the Archean (2.5–4 Ga), the incident solar radiation was about 70%–80% of its present-day value, not high enough to maintain a temperate climate with present-day atmospheric CO₂ levels (Sagan & Mullen 1972; Charnay et al. 2020). Although there are no direct measurements of historical weathering rates, the Archean geological record suggests a steady decrease in ${P}_{{\mathrm{CO}}_{2}}$ from the order of 0.1 bar to modern values while maintaining surface temperatures between 280 K and 315 K (Krissansen-Totton et al. 2018, and references therein). The lower insolation during the Archean was compensated by the greenhouse effect of CO₂. As the insolation increased, climates should have become warmer than the observations suggest. Without a negative feedback of silicate weathering that allowed a decrease in CO₂ levels as insolation increased, modern climates would not have been temperate (Walker et al. 1981; Berner et al. 1983; Kasting et al. 1993). The extent of silicate weathering during the history of Earth and the contribution of seafloor weathering are debated (e.g., Sleep & Zahnle 2001; Berner & Kothavala 2001; Foley 2015; Coogan & Gillis 2018; Krissansen-Totton et al. 2018). By applying the generalized weathering model to Earth, we lay the foundation for understanding climate regulation on temperate planets.

Because present-day continents on Earth are largely felsic and the seafloor is mafic, we approximate the lithology of continents by granite and the seafloor by basalt. Continental and seafloor-weathering rates on Earth are calculated up to 4 Ga (Figures 10(c) and (d)). This is the first application of the fluid-transport-controlled weathering model of Maher & Chamberlain (2014) to seafloor weathering on Earth. Because there are no direct measurements of historical weathering rates on Earth, we compare the generalized model developed in this study with a model from Krissansen-Totton et al. (2018, hereafter KT18, their Figure 3). Instead of using a climate model, time-dependent median models of ${P}_{{\mathrm{CO}}_{2}}$ and T from the same KT18 model are used as inputs to the generalized model. The same inputs are used to obtain the continental weathering rate from Walker et al. (1981) and the seafloor weathering from Brady & Gíslason (1997), where the present-day weathering rates of both models are normalized to those of KT18 models.

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To first order, one may expect that continental and seafloor-weathering rates depend on the continental surface area (f A_s) and the seafloor surface area ((1 − f) A_s), respectively (Equation (4)). However, not all continental and seafloor surface area undergoes weathering on Earth. Fekete et al. (2002) report the value of the continental weatherable area to be equal to 93 Mm², about 60% of the modern continental area ( = 0.6 f A_s). About 147 Mm² of the seafloor area ( = 0.41 (1 − f) A_s) is expected to contribute to seafloor weathering (given by the exposed area of low-temperature hydrothermal systems; Johnson & Pruis 2003). Therefore, in Figure 10, the continental and seafloor-weathering rates are given by W_cont = w_cont × 93 Mm² and W_seaf = w_seaf × 147 Mm², respectively. A more precise definition of continental weatherable area is the area susceptible to precipitation and runoff, and that of seafloor weatherable area is the area covered by low-temperature hydrothermal systems that are younger than approximately 60 Myr (Stein & Stein 1994; Johnson & Pruis 2003; Coogan & Gillis 2018). Nonetheless, the definition given in Equation (4) assumes all surfaces are weatherable, thereby giving upper estimates of the global weathering rates on exoplanets.

The present-day continental weathering rate for young soils (t_s = 0) in Figure 10(c) is in the thermodynamic regime (see also Figure 8(a)). Similarly, the present-day seafloor-weathering rate for young pore space is in the thermodynamic regime (Figure 10(d)). As the CO₂ levels increase (prior to 0.5 Ga), continental weathering for young soils is driven by kinetics (Figure 10(c)), whereas seafloor weathering for young pore space is in the thermodynamic regime even at high ${P}_{{\mathrm{CO}}_{2}}$. The weathering rate of the young-soil granite model decreases between 1 and 2 Ga. This is because the effective kinetic rate coefficient of granite is almost constant as a function of pH in basic solutions resulting from our calculations and depends mostly on T, which decreases between 1 and 2 Ga. Seafloor weathering in the thermodynamic regime increases monotonically because the effect of ${P}_{{\mathrm{CO}}_{2}}$ on weathering wins over the T effect. In the thermodynamic regime, the choice of secondary minerals also affects the weathering rates. For example, on one hand, the granite and basalt weathering rates for t_s = 0 models would be higher by a factor of 20 and 3 when kaolinite is used as a secondary mineral instead of halloysite. On the other hand, if secondary minerals incorporating divalent cations are modeled, the weathering rates would be lower because bicarbonate and carbonate ions equivalent to cations locked up in secondary minerals would be removed from the solution (e.g., Kite & Melwani Daswani 2019).

The continental weathering rate derived from carbonate alkalinity for young soils and the seafloor-weathering rate derived from carbonate alkalinity for young pore space are, respectively, about an order of magnitude and up to three orders of magnitude higher than that of KT18 models (Figures 10(c) and (d)). The Walker et al. (1981) continental weathering rate and the Brady & Gíslason (1997) seafloor-weathering rate exhibit a monotonic rise (β = 0.3 and β = 0.23, respectively). There are multiple reasons for the discrepancy between our t_s = 0 models and other models in Figure 10. Our models are designed to provide absolute weathering rates for generality and are therefore not intrinsically calibrated to present-day weathering rates, unlike other studies. The entire exposed planetary surface area may not contain fresh rocks for weathering, and continents do not necessarily have a uniform lithology and topography. On Earth's continents, orogeny (mountain building) exposes fresh rocks to the surface that are highly susceptible to weathering (low t_s), whereas the contribution of cratons (ancient continental crust, high t_s) to weathering is smaller than mountains (Maher & Chamberlain 2014). Similarly, ridge volcanism exposes fresh basalt at midocean ridges but the majority of the seafloor area is older than a million years.

For a planet with surface conditions where all divalent and monovalent cations react with carbonate and bicarbonate ions to produce carbonates, carbonate alkalinity is a proxy for the flux of CO₂ out of the atmosphere–ocean system. However, carbonate alkalinity overestimates the global weathering rate on modern Earth, which is limited by the flux of divalent cations instead of carbonate alkalinity (France-Lanord & Derry 1997). This is because monovalent cations do not contribute to the carbonate precipitation in the present-day ocean. Figure 10(c) shows that the weathering rate derived from divalent cations in our granite-weathering model is smaller than the carbonate alkalinity weathering rate by up to a factor of 5. Alternatively, a planet with no rivers but only coastal springs that are isolated from the atmosphere would produce divalent cations but no carbonate alkalinity and effectively no CO₂ flux out of the atmosphere–ocean system. Therefore, care should be taken when extrapolating a single weathering proxy to diverse planetary conditions.

The present-day continental weathering rate derived from carbonate alkalinity matches that of KT18 when the soil age is set to the characteristic soil age (100 kyr). The soil age encapsulates several processes including tectonics, erosion, and soil production (Heimsath et al. 1997; Riebe et al. 2003; West et al. 2005; Maher & Chamberlain 2014). For example, the soil age decreases with increasing erosion or increasing mineral supply rates (Maher & Chamberlain 2014). On continents, the dimensionless pore-space parameter varies with topography, and its mean value for present-day continents is not well constrained as previous studies adopt values that span an order of magnitude on either side of the reference value used in this study (Maher 2010, 2011; Maher & Chamberlain 2014; Winnick & Maher 2018). Given the validity range in parameters and weathering proxies, our model allows the present-day global continental weathering on Earth to either be in the thermodynamic, kinetic, or supply-limited regimes of weathering.

To match the present-day seafloor-weathering rate derived from carbonate alkalinity with that of KT18, the pore-space age is increased to 50 Myr (characteristic seafloor age) and the dimensionless pore-space parameter is decreased from 100ψ₀ to 18ψ₀. Because these two pore-space parameters have never been discussed in the context of seafloor weathering, future studies must evaluate their magnitude and extent of validity. For the same parameters, the seafloor-weathering rate derived from divalent cations is smaller by less than a factor of 2. Both of these models are in the supply-limited regime of weathering and therefore exhibit almost constant weathering rates.

The present-day regional variation in continental runoff (0.01–3 m yr⁻¹, Gaillardet et al. 1999; Fekete et al. 2002) and seafloor fluid flow rate (0.001–0.7 m yr⁻¹, Stein & Stein 1994; Johnson & Pruis 2003; Hasterok 2013) is more than two orders of magnitude. At fluid flow rates lower than the mean values (arid climates or low hydrothermal heat flux; see Figure 7), the models might escape kinetic/supply regimes and enter the thermodynamic regime. These models rely on characteristic values of t_s, ψ, and q to be suitable proxies for calculating global weathering rates, which are not necessarily appropriate for present-day Earth. For example, mountains contribute an order of magnitude more weathering flux than cratons (Maher & Chamberlain 2014). Additionally, basaltic regions on continents (omitted in our calculations) may contribute a weathering flux (weathering rate per unit area) higher than that of granitic regions by a factor of 5 (Ibarra et al. 2016). On Earth, local weathering rates can be estimated from data and integrated to compute the global weathering rate for the planetary surface. However, exoplanets lack (Earth-independent) data constraints on global properties such as fluid flow rates and pore-space parameters, let alone their potential regional variations. Hence, we propose to study exoplanets by implementing the generalized weathering model to various lithologies using characteristic values of fluid flow rates and pore-space parameters.

Holding modeling parameters constant throughout Earth's dynamic history is another critical approximation. The uplift of the Himalayan-Tibetan Plateau in the past 40 million years has decreased the characteristic soil age, which has likely increased the global weathering rates (Kump et al. 2000). The lithology of continents has also evolved over time and therefore the application of one type of lithology may result in inaccurate weathering rates for certain periods of Earth's history. The continental area fraction (and consequently, the weatherable area) was smaller in the Archean than today (Dhuime et al. 2017). Even if the strength of continental weathering flux was similar at the beginning of the Archean compared to today but the continental area was significantly smaller, then the continental weathering rate must have been significantly smaller. The true contribution of the silicate-weathering flux to the carbon cycle also depends on the carbonate compensation depth in the oceans (Pytkowicz 1970; Ridgwell & Zeebe 2005) as well as reverse weathering (Mackenzie & Garrels 1966; Isson & Planavsky 2018; Krissansen-Totton & Catling 2020) that requires knowledge of ocean salinity and ocean pH, which suggests that an ocean chemistry model should form the basis of future research.

It has been proposed that measuring the gaseous abundance of CO₂ in the atmospheres of Earth-sized exoplanets will allow one to statistically distinguish between Venus-like and Earth-like climates (Bean et al. 2017; Checlair et al. 2019; Graham & Pierrehumbert 2020). Negative feedback associated with weathering is necessary to control the climatic impact of CO₂ over geological timescales on an Earth-like planet (Walker et al. 1981). One-dimensional, radiative-convective atmospheric models used to study the boundaries of the classical habitable zone usually assume the presence of the negative weathering feedback to justify the assumption of CO₂ being present as a greenhouse gas at the outer boundary of the habitable zone and having only a minor abundance at its inner edge (Kasting et al. 1993; Kopparapu et al. 2013). The effects of CO₂ on the runaway greenhouse effect associated with water are debated (Nakajima et al. 1992; Abe 1993). Moreover, if weathering is limited by the supply of fresh rocks, the negative feedback is lost (West 2012; Foley 2015). This is evident from Figure 8, where supply-limited weathering flux is seen to be insensitive to changes in T and ${P}_{{\mathrm{CO}}_{2}}$. However, if soil age indirectly depends on T and ${P}_{{\mathrm{CO}}_{2}}$ on geological timescales, the negative feedback may reinstate. Our models of thermodynamic weathering flux show that the weathering flux decreases with T and increases with ${P}_{{\mathrm{CO}}_{2}}$ (Figure 9). In the parameter space where the effect of T dominates, it is possible to have a positive climate feedback associated with weathering for lithologies tested in this study. Such a scenario may have implications on the boundaries of the habitable zone. Future studies should combine the climate and weathering models to investigate these possibilities.