Question

Write the mayer’s relation.

Answer 1

Hint As we know mayer’s relation is $$C_{P,m}-C_{V,m}=R$$ and we know that where $$C_{P,m}$$ and $$C_{V,m}$$ are molar specific heat capacities at constant pressure and constant volume and $$R$$is universal gas constant.

Complete step by step answer As we recall about mayer’s relation it is the relation between molar specific heats at constant pressure and constant volume for any ideal gas that is
$$C_{P,m}-C_{V,m}=R$$, where $$C_{P,m}$$ and $$C_{V,m}$$ are molar specific heat capacities at constant pressure and constant volume and $$R$$is universal gas constant. For more generalisation we write it as
$$C_P-C_V=VT{\displaystyle \frac{{\left({\alpha }_V\right)}^2}{{\beta }_T}}$$, where we know that $$C_P$$is heat capacity body at constant pressure and $$C_V$$ is heat capacity of the body at constant volume and you should note that we have heat capacity rather than specific heat capacity because heat capacity is the energy transferred to body to increase the temperature where specific heat capacity is the heat required to increase the temperature of the body,$$V$$ is volume , $$T$$ is temperature, $${\alpha }_V$$ is thermal expansion coefficient and $${\beta }_T$$ is isothermal compressibility. From this relation we can consider many changes by changing the values on right hand side that is if the absolute temperature approaches to zero then differences of left hand side also approaches to zero and for incompressible substances $$C_P$$ and $$C_V$$ are identical and for substances like solids and liquids we can say that difference between specific heats is negligible.

Note Mayer’s relation is only valid for ideal gases and we can calculate this relation for different gases by changing values on the right hand side like for gas with absolute temperature approaching zero and for gases which can’t be expanded under isothermal conditions.