A physical point of view on the arithmetic and geometric mean inequality

We propose a simple experiment designed to justify the arithmetic and geometric mean inequality by means of the laws of thermodynamics. The experiment consists in measuring the entropy variation ΔS in the thermodynamic irreversible process of cooling a metal in water. By considering the metal and water as a single isolated system, the arithmetic and geometric theorem is seen to hold by noticing that ΔS is positive for this irreversible transformation. These interdisciplinary activities may be used to reinforce basic concepts in thermodynamics in high school or first-year college students.


Introduction
The use of physics arguments to justify mathematical statements can be a useful method both in mathematics and in physics teaching [1].In fact, by starting from a physics experiment related to a mathematical theorem, students can empirically approach topics that are typically regarded as purely conceptual.As an example, one can consider the arithmetic and geometric mean inequality and notice that direct application of the laws of thermodynamics can provide an empirical justification of this theorem [2,3].In the present work we propose a simple experiment in which the zeroth, the first and second law of thermodynamics are involved.The formal correspondence between the laws of thermodynamics and the arithmetic and geometric mean inequality is then utilized to justify the latter by an empirical approach.This hands-on approach has a double didactical valence.The first is given by the interdisciplinary character of these activities bridging the fields of thermodynamics and mathematics.The second is given by the experimental measurement of entropy variation, which directly exposes students to the concept of entropy increase in isolated systems, not always well interpreted by high school students [4].Once a solid understanding of thermodynamics concepts is established, a more comprehensive interpretation of entropy change can be explored by referring to [5][6][7][8].
In particular, the experiment is based on the estimation of entropy in an isolated system.We start by considering two bodies at different temperatures: T A for body A and T B for body B. After the two objects are set in contact, they attain, after a transient, the equilibrium temperature T E expressed as follows [9]: The above expression is the weighted arithmetic mean of T A and T , B where the weights k A and k B are the heat capacity of the two bodies.For the composite system A + B, the variation of entropy S D can be calculated as: In this way, it is possible to evaluate S D by knowledge of T , A T , B and T .E Furthermore, by considering the system A + B as isolated, by the second law of thermodynamics we may set: Taking into account equations (1) and (2), this inequality gives, by introducing dimensionless where k 0 is some arbitrary reference heat capacity value, in place of the weights k A and k B in equation (2), the following important mathematical relation, known as the arithmetic and geometric mean inequality: The above introductory concepts thus establish a clear link between thermodynamic laws and the arithmetic and geometric mean inequality.Taking advantage of the formal connection between these two topics, in the present work we propose an experimental procedure to justify the arithmetic and geometric mean inequality.We notice that the proposed experimental procedure can be extended to a number N 2 > of bodies with temperatures T T T , , .
N 1 2 ¼ This extension corresponds to the more general form of the arithmetic and geometric mean inequality.We prefer to treat the simplest case for clarity reasons.We remark that the link between physics and mathematics here is only formal and that it is not possible to give an interpretation of entropy concepts by means of the theorem at hand.
The work, which can be addressed either to advanced high school students or to first-year college students, is organized as follows.In the following section, the experimental procedure to measure the variation of entropy S D in a rather simple context is explained in detail.The collected data are used, in the third section, to justify the validity of the inequalities in equations (3) and (4).Conclusions are drawn in the last section.

The experimental procedure
The experiment is realized by means of common material.Referring to what is illustrated in the Introduction, we use a quantity of water as body A and an aluminum cylinder as body B. After having obtained the masses m A and m B of the two objects by means of a digital scale, their heat capacities k A and k B are calculated as follows: where C 4186 is the specific heat of water, and C 880 is the specific heat of the aluminum cylinder.Using m 400.0gA = and m 47.5g, We then warm up the aluminum cylinder through an electrical oven.The temperature T B cannot be measured by means of the same thermometer for various reasons, the main being that the high temperature of the aluminum cylinder can easily exceed the full-scale of the thermometer.
After having warmed up the aluminum cylinder, the latter is immersed in water using clips to avoid direct contact with this hot object.The main steps of the experimental procedure are shown in figure 1.The contact between water and hot aluminum makes a portion of superficial water to evaporate.The consequent reduction of the water mass needs to be measured at the end of the experiment to control that the mass variation is below the instrumental error.
Waiting for a sufficiently long time, the temperature of the combined system, water plus aluminum, will reach the equilibrium temperature T .
E One can monitor the increase in the water temperature until thermal equilibrium is reached at T T .

E =
The direct measurements of the temperatures T A and T E are gathered in the first two columns of table 1.By using equation (1), the temperature T B can be calculated, so that: The calculated values of T B are reported in the third column of table 1.We remark that equation ( 6) is a consequence of the first law of thermodynamics.The entropy variation is then found, by considering equations (2) and (6), in terms of the ratio x 1 For each experiment the ratio x is reported in the fourth column of table 1, the corresponding value of S D in the fifth column and the absolute error e on the quantity S D in the last column.The inequality in equation (3) can be justified by noticing that only nonnegative values of S D are obtained within the experimental limits.The absolute error e is calculated in the appendix.

The second law of thermodynamics and the arithmetic and geometric mean inequality
The experimental steps illustrated in the previous section have been followed by different groups of high school students during a summer school held at the University of Salerno in September 2019.The collected data are presented in table 1, where the calculated values of T B and S, D corresponding to the direct measurements of the temperatures T A and T , E are reported.Being S 0 D > for all measurements, the second law of thermodynamics can be justified, as specified before.
A graphical representation of the experimental points (in blue) gathered during the summer activities in Salerno is given in figure 2. In this figure, the quantity S D is plotted as a function of T A and T .B Along with the blue experimental points, the plot also shows the analytic expression of S, D given by equation (2), where T E is given by equation (1).We notice that the points are well distributed over the meshed layer representing the analytic expression of S. D We also notice that all points lie above the S 0 D = plane, thus justifying the second law of thermodynamics.
As already pointed out in the Introduction, starting from equation (2), the arithmetic and geometric mean inequality can be obtained.However, some attention to the dimension of physical quantities needs to be paid.In fact, in order to obtain equation (4), it is necessary to make the coefficients of the log terms, k A and k , B dimensionless.In this work, we divide both members of equation (2) by the constant k 1.0 .
In this way, we may rewrite equation (2) as follows: Graphical representation of the experimental points in table 1 (points appear blue if they are above the meshed layer and dark blue if they are beneath).The meshed layer represents the analytic expression of the entropy variation S D of the system as a function of the temperatures T A and T .B Table 1.Resulting data from the experimental procedure illustrated in the previous section.The error on the measured quantities T A and T E are taken as the accuracy of the thermometer (±0.5 K).The error.e on S D has been calculated by considering the procedure illustrated in the appendix.where k The dimensionless heat capacities thus allow us to write equation (8) in the following form: Because of the second law of thermodynamics in equation (2), we have S 0,  ¢ D so that, by equations (1) and (9), we obtain the arithmetic geometric mean inequality as expressed by equation (4).All sets of experimental points gathered in table 1 and represented in figure 2 confirm this inequality.

Conclusions
In this work, we present a simple experiment in thermodynamics by which the entropy variation S D of the irreversible process of cooling a piece of metal into water can be measured.The particular analytic expression of S D for this process allowed us to emphasize the close link between the laws of thermodynamics and the arithmetic and geometric mean theorem.In this respect, the proposed laboratory activity not only conveys important physical insight into irreversible processes, but also provides a rather unconventional approach to justify a mathematical theorem.
Common material has been used to carry out the experiment, which was proposed to high school students during a summer school held at the University of Salerno in 2019.Given the rather advanced character of the subject, the same laboratory activities can be addressed to first-year college students.After having reinforced classical thermodynamics concepts, one might introduce more advanced topics in statistical mechanics, as in [5], or further discussions on the meaning of entropy as in [6][7][8].
Some critical aspects of this experimental activity should be mentioned.The first is the hypothesis that the combined system of the two substances is an isolated one.In this respect, in order to avoid heat transfer to the environment, one could carefully choose the water container, using, for example, insulating cups instead of beakers on a piece of wood, as done in our experiment.The second, as already mentioned in section 2, is the evaporation of part of the water during the cooling process.By using larger amounts of water, however, the percentage of the evaporated fluid with respect to its initial quantity in the container decreases.Nevertheless, besides these aspects, the simplicity of the experiment and its link to a mathematical theorem could be useful to teachers who would like to propose interdisciplinary topics during physics laboratory lectures.equation (7), the absolute error e on the quantity S, D neglecting the errors on k A and k , B can be written as follows:

Figure 1 .
Figure 1.Four simple experimental steps to obtain the temperature T B of body B by measuring the temperature T A of body A and the equilibrium temperature T .
Since we were able to measure the temperatures T A and T E with an accuracy T 0Notice that the expression for e given above depends solely on the measured temperatures T A and T E and on the known accuracy T 0.5K.=  ∆ ORCID iDs R De Luca https://orcid.org/0000-0001-8137-6904