Jozsef Garai
The First Law of Thermodynamics
and the Thermodynamic Description of Elastic Solids
Historically, the thermodynamic behavior of gasses was described first and the derived equations were adapted to solids. It can be shown that the adoption is incomplete which results in an incorrect thermodynamic description of the solids phase. The expression of the first law of thermodynamic is also interpreted as a statement of energy conservation, which is one of the fundamental laws of nature and with no doubt, is correct.
The first law of thermodynamics expresses the mechanical equivalency of heat which was demonstrated by Joule’s paddle-wheel experiment in the year of 1845. The law is usually written as the sum of the differentials of the heat [q] and work [w]
dU = δq + δw (1)
where U is the internal energy of the system. The sum of heat and work can be expressed in a various ways. Here the temperature and the volume are used which can easily be measured. The internal energy in solid phase is then:
dU = ncVdT - (p-pth)dV (2)
where n is the number of moles, cV is the constant volume molar heat capacity, T is the temperature, p is the pressure, and V is the volume. The thermal pressure (pth) can be calculated as:
(3)
where α V and KT are the volume coefficient of thermal expansion and the isothermal bulk modulus defined as:
(4)
and
(5)
Joule’s experiment demonstrated only one way transformation of mechanical work to heat in liquid phase. His postulation for the universality of the mechanical equivalency of heat has never been proven experimentally. If heat is added to a liquid system then this heat can not be completely transferred to work because the energy of the expansion work is far less than the energy supplied as heat. The same problem exists in solid phase in both directions (Fig. 1).
Figure 1. The applicability of Joule’s postulation, regarding to the mechanical equivalency of heat, as a function of phase and the direction of the energy transformation
It can be concluded that Joule’s postulate regarding to the mechanical equivalency of heat, also known as the first law of thermodynamics is not universal and does not apply to elastic solids and to liquids in heat to work direction. The lacking equivalency of the two energies raises the question should or should not be these energies summed when the internal energy of a system is counted. The applicability of Joule’s Law is discussed in detail and criterion for the application of the law is proposed1.
The other problem with the contemporary expression of first law [Eq. 3] is that the expression only counts the isobaric work. There is no term for the isothermal work. If the temperature is changed at constant volume according to equation (3) no work is done on or by the system, which is incorrect. Plotting the P-V relationship at two different temperatures, T1 and T2 it can be seen by visual inspection that isothermal work is done on the system when the temperature is increased from T1 to T2 at constant volume (Fig. 2.). The used linear relationship between the volume and the pressure for simplification does not affect the outcome of this conclusion.
Figure 2. The areas representing the isothermal work at pressures 1 and 2 and temperatures 1 and 2 respectively are shaded.
Suggestions, how the fundamental equations should be revised, are posted1. The modification of current description requires a consensus of the scientific community regarding to the followings: shall or shall not the isothermal work and heat summed and how the rest of the state function should be defined in solid phase. Comments and suggestions are appreciated.
Reference:
1 arXiv:0705.1484v5 [physics.chem-ph]