What is R in CP CV R.

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Difference in heat capacities Cp - Cv

The temperature dependence of the internal energy and enthalpy at constant volume or pressure is combined with the heat capacities of constant volume or pressure. Since there is a connection, the question of a corresponding connection between and arises. This thermodynamic problem can be solved as follows.

The starting point of the derivation are the differential forms of the 1st law and the internal energy.

Rearrangement sets the converted heat equal to the remaining terms in Eq. (1):

We now consider the supply or removal of differential heat as an isobaric change of state, i.e. there is constant pressure equilibrium. Furthermore, we “divide” Eq. (2) by, the resulting quotient / is equal to the heat capacity in this case. Equation (2) has thus assumed the following form.

Note on "sharing"
That in the derivation of Eq. (3) “Divide” by is a mathematically incorrect expression common in chemistry. The following procedure is correct. In Eq. (2) is replaced by and by or. is not a state variable, so the difference sign cannot be used. So must be replaced by with the meaning of a small value of warmth. Now Eq. (2) divided by and the limit value formation can be carried out for approaching zero at constant pressure. Here, / formally arises from /. The total result is Eq. (3).

What is still unknown is the change in the internal energy of the substance with volume at constant temperature. Remarkably, the following relationship can be derived from the 1st and 2nd law of thermodynamics and the Maxwell equations.

Eq. (4) inserted in Eq. (3) leads to the equation we are looking for for the difference in heat capacities -.

Difference in heat capacities

The result is remarkable: The difference between two caloric variables is determined by purely thermal variables and can thus be expressed by the thermal expansion coefficient and the isothermal compressibility (see Eq. (5)). Measurements of the heat capacity at constant volume are much more difficult to carry out than at constant pressure. Therefore, usually by means of Eq. (5) calculated from.

Ideal gas

It applies

and thus