The First Law of Thermodynamics contains an explicit statement about the amount by which the internal energy U of a gas changes when work W or heat Q is received or given up by the system. It must be emphasized that contrary to Q and W, U is a state variable, i.e., its value depends only on the state of the system and not on how this state was attained.
The Gay-Lussac experiment consists of two containers connected by a pipe and valve (Figure 1). Container 1 is filled with an ideal gas, container 2 is completely evacuated. The system is perfectly insulated from its surroundings.
The valve is opened and the gas confined in 1 expands into vacuum 2. Pressure and volume change while the temperature remains constant. Since no work or heat are exchanged with the surrounding, the internal energy will not change during this process. Thus, the internal energy of an ideal gas is only a function of its temperature.
where V is volume and T, temperature. With
the change in internal energy is obtained from
Since for an ideal gas U is a function only of temperature, it follows from Equation (2) that the specific heat capacity cv for an ideal gas is independent of pressure and volume. Values of cv are often expressed as polynomials in T (see, for example, Reid, Prausnitz and Sherwwod).
For nonideal fluids, the following equation for pressure dependence of the internal energy can be derived from the fundamental relationship between enthalpy and internal energy:
where κ is the isothermal compressibility and β is the volume expansivity. Equations (4) and (5) are usually only applied to liquids, with κ and β being very small if the fluid can be treated as incom pressible, i.e., if it is not near the critical point.
Reid, R., Prausnitz, J., and Sherwood, T. (1977) The Properties of Gases and Liquids. McGraw-Hill Book Company.