Abstract
In the present chapter we shall look at the concept of energy as it developed from empirical observations to become one of the two basic concepts (the other being entropy ) of a branch of physics known as thermodynamics. The story begins with heat .
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Notes
- 1.
The large value of c water means that water does not cool easily; this is why it is used in radiators and hot water bottles.
- 2.
Today’s value is: 1 cal = 4.186 J.
- 3.
Sometimes a macroscopic variable (a physical quantity) is called a thermodynamic variable; and similarly a macroscopic state is called a thermodynamic state.
- 4.
This is true only for a gas of monoatomic molecules (e.g., Ar or He). The energy given by Eq. (9.4) derives from the translational motion of the molecules. For a gas of diatomic molecules at ordinary temperatures there is an additional contribution to the energy, equal to NkT, associated with the rotational motion of the molecules. Therefore the total energy E = 5NkT/2. In which case c P /c V  ≈ 1.4.
- 5.
By this time practically every one agreed that a gas consisted of molecules moving randomly in the space available to them.
- 6.
The isothermal compressibility of a substance (solid, liquid, or gas) is defined by:
$$ \kappa_{T } = - \frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T} $$.
- 7.
- 8.
It is worth noting that, though we used the symbol T to denote the temperature in our definition of the Carnot engine, such is not necessary and was not the case in Carnot’s statement of his theorem.
- 9.
We simplify the description if we postulate a third law of thermodynamics (suggested by Walther Nernst in 1906) which states that: The entropy of a system at the absolute zero of temperature is a universal constant, which we can put equal to zero.
- 10.
We note, in this respect, that friction, though often represented by a sigle force is not one; friction involves a number of random collisions between the surface atoms of two bodies. Each collision by itself is a reversible process but the entire process is not.
- 11.
The limits of the integration are to be understood as follows: we consider the integral I(α) = \( \int\limits_{ - \alpha }^{ + \alpha } {P(x)dx} \). Βy choosing α sufficiently large, we obtain: │I(α) − 1│ < ε, however small ε is. In other words: I is the limit (in the Weierstrass sense explained in Sect. 1.1) of I(α) as α → \( \infty \).
- 12.
The molecules in the gas are of the same type (e.g., oxygen molecules) but we assume that we can distinguish between one oxygen molecule and another as if each one had a number attached to it. We note also that if the gas consisted of N 1 molecules of one kind and N 2 molecules of another kind the distribution we shall obtain applies to them separately. This corresponds to the law of partial pressures stated in Sect. 8.3.
- 13.
That would not be the case if ε i depended on r, as it happens when the gas exists in a potential field U(r) which varies significantly over the region occupied by the gas. Disregarding the potential energy of the molecules due to the gravitational field (U = mgz) in the derivation of Eq. (9.25) implies that mgL ≪ kT, so that exp(−mgz/kT) ≈ 1, for 0 < z < L = V 1/3.
- 14.
A useful concept in this respect is the mean free path λ of a molecule. It is the average distance a molecule travels before it collides with another molecule. The average duration τ of a free path is called the collision time: τ = λ/√(2kT/m). For example: for a H2 gas at its critical point (see Fig. 9.3a) λ ≈ 10−7 cm and τ ≈ 10−11 s.
- 15.
In practice, e.g., for a H2 gas under normal conditions, any non uniformity in density or pressure or temperature, over distances of the order of 10−7 cm will be smoothed out in about 10−11 s. Variations over macroscopic distances persist over longer times. This is how, for example, the propagation of sound in air becomes possible. The establishment of local equilibrium is a prerequisite of the macroscopic variation which constitutes the sound wave.
- 16.
Otto Stern was born in 1888 in Sorau (in Silesia, Germany). He obtained his Degree in Physical Chemistry from the University of Breslau in 1912. He worked at the University of Frankfurt am Main, and Rostock, where he became a professor of Physical Chemistry in 1923. In 1933 he moved to the United States, being appointed Research Professor of Physics at the Carnegie Institute of Technology in Pittsburgh. Apart from the work mentioned here, he did important work, in cooperation with Gerlach, on the deflection of atoms by the action of magnetic fields on the magnetic moments of the atoms. He also demonstrated the wave nature of atoms by observing interference effects in rays of hydrogen and helium atoms. And he also measured the magnetic moments of sub-atomic particles including the proton. Otto Stern was awarded the Nobel Prize for Physics for the year 1943. He died in 1969.
- 17.
From now onwards we shall, when no ambiguity arises, refer to the micro-states of the system simply as the states of the system.
- 18.
Gibbs derived the same result in a somewhat different manner by considering, to begin with, a microcanonical ensemble: all members of which have the same mass, the same volume V and the same energy (between E and E + ΔE). Assuming that all microstates i of the given system occur with equal probability f(i) = 1 if E i lies between E and E + ΔE, and f(i) = 0 otherwise, one can evaluate the average value \( \overline{A} \) of any thermodynamic quantity A, in the manner of Eq. (9.33), as follows: \( \overline{A} \)  = \( \frac{1}{C} \) \( \sum\nolimits_{i} {f(i)} \) A i , where C  = \( \sum\nolimits_{i} {f(i)} \). One obtains the same average value \( \overline{A} \) as the one obtained with the ensemble we have considered in the text, known as the canonical ensemble. This is because the fluctuations about the average value of the energy, obtained in the canonical ensemble (where T rather than E is the common property of the members of the ensemble), are very small (negligible for ordinary systems of large mass). The canonical ensemble and its properties derive from those of the microcanonical ensemble by straight forward arguments which we need not present here. Finally, it is worth noting that Boltzmann considered such ensembles (in relation to substances other than gases) in a paper he wrote in 1884 but he did not elaborate on them in the way that Gibbs did some years later.
- 19.
The diffusion equation, derived here on an intuitive basis, can be derived, at least for a dilute gas, from Boltzmann’s transport equation, which would show the dependence of D on the temperature and the properties of the particles involved in the collisions that drive the phenomenon. We note that the diffusion equation applies not only to a dilute gas, but to a variety of systems as well. In most cases the dependence of D on the properties of the diffusing particles, those of the medium in which they move and on the temperature is determined experimentally.
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Modinos, A. (2014). Thermodynamics and Statistical Mechanics. In: From Aristotle to Schrödinger. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-00750-2_9
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