Question

Derive the general relation between the heat capacity at constant volume (Cv ) and the heat capacity at constant pressure (Cp).

a)Cv=Cp+R
b)Cv=Cp−R
c)Cv=Cp×R
d)Cv=Cp÷R

Answer 1

The general relationship between the heat capacity at constant volume (Cv) and at constant pressure (Cp) for an ideal gas is expressed by the equation Cp = Cv + R, where R is the gas constant.

Step-by-step explanation:

The relationship between the heat capacity at constant volume (Cv) and the heat capacity at constant pressure (Cp) for an ideal gas is given by the equation Cp = Cv + R. This relationship is derived from the ideal gas law and applies to dilute gases such as monatomic gases (e.g., He), diatomic gases (e.g., O2), and polyatomic gases (e.g., CO2, NH3). The expression considers that when a gas expands at constant pressure, there is an additional amount of work done and hence more heat is required, which is accounted for by the gas constant R. Each degree of freedom (translational, rotational, vibrational) contributes to the heat capacity, with vibrational degrees contributing differently due to their energetics. When comparing processes at constant volume versus constant pressure, we have to consider the work done by the gas, which impacts the amount of heat required. The first law of thermodynamics, which combines the internal energy change, heat added, and work done, is an integral part of understanding these concepts.