A new approach to determine the density of liquids and solids without measuring mass and volume: introducing the solidensimeter

Halit Kiriktaş; Mehmet Şahin; Sinan Eslek; İrem Kiriktaş

doi:10.1088/1361-6552/aaa91e

Introduction

The density of a substance is its mass per unit volume and is a distinctive characteristic of a matter [1]. Difficulties are encountered when teaching this concept despite its importance in helping students to comprehend this density concept when applied to a substance and its characteristics [2]. In this regard, simplifying the mathematical operations used in the process of teaching the concept of density, clarifying the relevant experimental processes and simplifying the equipment used in these processes can make it easier to teach.

The literature reveals different methods used to determine the density of solids. Considering the theoretical background of the aforementioned methods, two different tendencies are apparent. The first is the mechanism, which uses the fundamental density relation ( $d=m/v$ ). In such mechanisms, the mass (m) and volume (v) of the solids object are determined separately using appropriate measurement tools. Subsequently, the measured values of the mass and volume are substituted into the density formula ( $d=m/v$ ) and the density of the solid is calculated [3]. Using more than one measurement tool and a mathematical equality makes the process complicated and difficult to learn [2]. In the second type of mechanism, the density measurement is carried out using the Archimedes' principle and equilibrium-moment measurements. Some of these mechanisms only use Archimedes' principle [4]. In these mechanisms, the mass (m) is determined by means of water in a scale as in previous systems, while the volume is determined by water in a scaled container. The values obtained are still substituted into the formula to determine the density.

The other mechanisms are designed on the basis of the second type of mechanism that only uses Archimedes' principle. In these systems, the equilibrium-moment equation as well as Archimedes' principle are used. In such designs, an equilibrium-moment mechanism is established and an object whose mass is known is placed on one side of the balance mechanism. On the other side of the mechanism, the object, whose density is intended to be measured, is hung and submerged in the liquid. This mechanism generates a new mathematical equation using the equilibrium-moment and Archimedes' principle and uses this equation to determine the density [5–7]. In these mechanisms, although the density can be found without measuring the volume and mass, there is a need for many materials and the theoretical calculations require a comprehensive background in physics. The given mechanisms are also not very successful in determining the density of liquids. Therefore, these mechanisms are not particularly useful for young age groups and thus, it makes it difficult for students to comprehend the concept of density to achieve higher grades in a course.

In this regard, an approach that uses a practical experimental processes and a simple mathematical ratio in order to determine the density of any solid or liquid would be more useful. The theoretical background and the mechanism of the solidensimeter, which is thought to be useful for this purpose, are presented in this paper.

Theoretical background and design of the approach

This assembly was designed to calculate the density of any solids or liquids without needing to obtain the volume and mass of the material. The system is theoretically based on the Archimedes' principle.

It is stated in Archimedes' principle that [1],

(1)
A substance submerged in a liquid displaces as much liquid as its volume
(2)
Floating or hanging a substance in a liquid displaces as much liquid as its mass.

The system, designed according to these principles, is composed of two scaled glass containers (beakers) that can intertwine and contain an adequate amount of water. In the density measurement process, the glass containers are filled with water and the inner beaker placed into the outer beaker. The inner and outer beakers are filled with water. While the level of water is scaled in both beakers the following issues must be considered,

(1)
The level of water in the outer beaker must be sufficient enough to float and stand flat for the inner beaker.
(2)
The level of water in the inner beaker must be enough to allow the substance to be submerged.

The level of water in each beaker was recorded after they were filled with water. We can refer to the initial level of water in the outer container as 'b_i' and the inner container as 'a_i'. Subsequently, the solid substance or liquid whose density will be calculated is placed in the inner beaker and the final state of the liquid levels in both beakers recorded again. After placing the substance, the level of water in the outer beaker is referred to as 'b_f' and the level of water in the inner beaker as 'a_f'. At this point, according to Archimedes' principle, the difference between the initial level of water in the outer beaker (b_i) and the final water level (b_f), (b_f − b_i) is equal to the mass of the substance or the liquid (the density of the water is used as 1 g cm⁻³). As for the inner beaker, the difference between the final water level (a_f) and the initial water level (a_i), (a_f − a_i) is equal to the volume of the substance or liquid. When we use these statements in the basic density formula ( $d=m/v$ ), the equation becomes d = (b_f − b_i)/(a_f − a_i). Finally, the experimental values are used in the equation and the density of the solid or liquid was calculated. Figure 1 shows a schematic representation of the system used for solids, which was named as the solidensimeter and its use. Figure 2 shows a schematic representation of the solidensimeter used for liquids.

**Figure 1.** A schematic representation of the *solidensimeter* used for solids. (a) Before placing the solid. (b) After placing the solid.
a_i: The level of water/liquid in the inner beaker before placing the substance inside the container.
b_i: The level of water in the outer beaker before placing the substance inside the container.
a_f: The level of water/liquid in the inner beaker after placing the substance inside the container.
b_f: The level of water in the outer beaker after placing the substance inside the container.
d_o: The density of the substance.
d_w: The density of water (1 g cm⁻³). ${{\boldsymbol{d}}_{{\bf o}}}\boldsymbol{=}\frac{\boldsymbol{m}}{\boldsymbol{v}}\boldsymbol{=}\frac{{{\boldsymbol{b}}_{{\bf f}}}-{{\boldsymbol{b}}_{{\bf i}}}}{{{\boldsymbol{a}}_{{\bf f}}}-{{\boldsymbol{a}}_{{\bf i}}}}.$
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**Figure 1.** A schematic representation of the *solidensimeter* used for solids. (a) Before placing the solid. (b) After placing the solid.
a_i: The level of water/liquid in the inner beaker before placing the substance inside the container.
b_i: The level of water in the outer beaker before placing the substance inside the container.
a_f: The level of water/liquid in the inner beaker after placing the substance inside the container.
b_f: The level of water in the outer beaker after placing the substance inside the container.
d_o: The density of the substance.
d_w: The density of water (1 g cm⁻³). ${{\boldsymbol{d}}_{{\bf o}}}\boldsymbol{=}\frac{\boldsymbol{m}}{\boldsymbol{v}}\boldsymbol{=}\frac{{{\boldsymbol{b}}_{{\bf f}}}-{{\boldsymbol{b}}_{{\bf i}}}}{{{\boldsymbol{a}}_{{\bf f}}}-{{\boldsymbol{a}}_{{\bf i}}}}.$
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**Figure 2.** A schematic representation of the *solidensimeter* used for liquids. (a) Before adding the liquid. (b) After adding the liquid.
a_i: The level of water/liquid in the inner beaker before adding the liquid into the container.
b_i: The level of water in the outer beaker before adding the liquid into the container.
a_f: The level of water/liquid in the inner beaker after adding the liquid into the container.
b_f: The level of water in the outer beaker after adding the liquid into the container.
d_l: The density of the liquid.
d_w: The density of water (1 g cm⁻³). ${{\boldsymbol{d}}_{{\bf l}}}\boldsymbol{=}\frac{\boldsymbol{m}}{\boldsymbol{v}}\boldsymbol{=}\frac{{{\boldsymbol{b}}_{{\bf f}}}-{{\boldsymbol{b}}_{{\bf i}}}}{{{\boldsymbol{a}}_{{\bf f}}}-{{\boldsymbol{a}}_{{\bf i}}}}.$
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**Figure 2.** A schematic representation of the *solidensimeter* used for liquids. (a) Before adding the liquid. (b) After adding the liquid.
a_i: The level of water/liquid in the inner beaker before adding the liquid into the container.
b_i: The level of water in the outer beaker before adding the liquid into the container.
a_f: The level of water/liquid in the inner beaker after adding the liquid into the container.
b_f: The level of water in the outer beaker after adding the liquid into the container.
d_l: The density of the liquid.
d_w: The density of water (1 g cm⁻³). ${{\boldsymbol{d}}_{{\bf l}}}\boldsymbol{=}\frac{\boldsymbol{m}}{\boldsymbol{v}}\boldsymbol{=}\frac{{{\boldsymbol{b}}_{{\bf f}}}-{{\boldsymbol{b}}_{{\bf i}}}}{{{\boldsymbol{a}}_{{\bf f}}}-{{\boldsymbol{a}}_{{\bf i}}}}.$
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The following points must be considered while using the system,

The system was designed according to the Archimedes' principles. For this reason, students must comprehend the aforementioned principles in order to use the system effectively. Students must be well informed about Archimedes' principles. Accordingly, students should be made aware that assuming the density of the water as d_w = 1 g cm⁻³, height difference in the outer container will be equal to the mass of the substance whose density will be found.
The density of the water used in the system is regarded as d_w = 1 g cm⁻³
The inner beaker in which the substance is added must contain enough water/liquid to allow the substance to be submerged and the substance must be fully submerged in the water/liquid.
The substance must be submerged in the inner beaker. In this respect, the density of the substance must be larger than the density of water. If the density of the substance is smaller than the density of water, another liquid in which the substance can be submerged can be used (its density is smaller than the density of water). This situation does not influence the mathematical equation. Based on this, the process can be conducted.
While the density is being calculated, filling the outer beaker with water is enough. Optionally, the inner beaker can be filled with a liquid whose density will be calculated or the measurement can be carried out using the same equation in the presence of any other liquid, including water.
During measurement, the inner beaker should be vertically stable in the outer beaker in order to measure the water levels correctly. This may be achieved if the volumes of the containers to be used in the measurement are close to each other. The difference between the volumes of the two containers should not be too large. That is, the containers should be very close to each other in terms of volume. We need the inner container only a little smaller than the outer container so that it can fit. That way one can obtain vertically stable inner beaker (etc inner beaker: 100 cc, outer beaker: 250 cc or inner beaker: 250 cc, outer beaker: 400 cc).
The sensitivity of the glass containers used in this study is 25 ml. The size and maybe the scale of the outer container may make a difference in the precision of the measurement. Therefore, it is better to use small scale containers or smaller containers to have more precise measurements. If the sensitivity of the measurements conducted with this system is expected to be high, glass containers with higher sensitivity (10 ml, 5 ml or 1 ml) may be used.
While reading the level of the liquids in the glass containers, the liquid value is different based on whether the liquid is colored or not, in other words the liquid surface forms a concave or convex curve. According to these rules, in colorless liquids, the lowest level of the liquid surface is used while in colored liquids, the highest level of the liquid surface is accepted as the volume.
Finally, the liquid in the outer beaker must be water. In the inner beaker, any liquid whose density is larger than the solid can be used for solid samples. While calculating the density of liquids, the measurement can be carried out without any liquid in the inner beaker.(Note: Liquids with the characteristics stated above (olive oil) were used in the measurements for solids. While for liquids the density measurement was conducted without any liquid, such as water or olive oil, in the inner beaker. Similar results were obtained for each situation.)

The sample measurement procedures for liquids and solids using the solidensimeter are shown below. Figure 3 shows the sample density measurement process for solids using the solidensimeter.

**Figure 3.** Measurement of the density of a solid object using the *solidensimeter*. (a) Student conducting the density experiment. (b) Before placing the object. (c) After placing the object.b_i = 225 cm³, b_f = 247 cm³ ${{d}_{{\rm o}}}=\frac{m}{V}=\frac{{{b}_{{\rm f}}}-{{b}_{{\rm i}}}}{{{a}_{{\rm f}}}-{{a}_{{\rm i}}}}=\frac{\left(247\,{\rm c}{{{\rm m}}^{3}}-225\,{\rm c}{{{\rm m}}^{3}} \right)~\times ~1\,{\rm g}\,{\rm c}{{{\rm m}}^{-3}}}{\left(47\,{\rm c}{{{\rm m}}^{3}}-40\,{\rm c}{{{\rm m}}^{3}} \right)}.$ a_i = 40 cm³, a_f = 47 cm³ d_o = 3.14 g cm⁻³ (the density of stopper).
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Figure 4 shows a sample density measurement process for a liquid (olive oil) whose density is unknown.

**Figure 4.** Measurement of the density of a liquid substance using *solidensimeter*. (a) Student conducting the density experiment. (b) Before adding the substance. (c) After adding the substance.b_i = 300 cm³, b_f = 350 cm³ ${{d}_{{\rm l}}}=\frac{m}{V}=\frac{{{b}_{{\rm f}}}-{{b}_{{\rm i}}}}{{{a}_{{\rm f}}}-{{a}_{{\rm i}}}}=\frac{\left(350\,{\rm c}{{{\rm m}}^{3}}-300\,{\rm c}{{{\rm m}}^{3}} \right)~\times ~1\,{\rm g}\,{\rm c}{{{\rm m}}^{-3}}}{\left(150\,{\rm c}{{{\rm m}}^{3}}-75\,{\rm c}{{{\rm m}}^{3}} \right)}.$ a_i = 75 cm³, a_f = 150 cm³ d_l = 0.67 g cm⁻³(the density of olive oil).
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Conclusions

The solidensimeter is effective for calculating the density of any solid or liquid substance using a simple mathematical ratio without the need for mass and volume measurement. The system is composed of basic materials including two scaled glass containers and water, and its assembly is quite simple, therefore it can be regarded as both economical and functional. Similarly, since the system explains density in a simple and concise way, it may be effective and useful for students that have difficulty in understanding the concept in various learning environments. As a result, the solidensimeter can be considered as an economical, easy-to-use, and functional system that can be used in the learning process.

A new approach to determine the density of liquids and solids without measuring mass and volume: introducing the solidensimeter

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Abstract

Introduction

Theoretical background and design of the approach

Conclusions

A new approach to determine the density of liquids and solids without measuring mass and volume: introducing the solidensimeter

Article metrics

Submit

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Share this article

Author e-mails

Author affiliations

Author notes

Dates

Peer review information

Abstract

Introduction

Theoretical background and design of the approach

Conclusions