An ideal gas has molar heat capacity C_V at constant volume. The gas undergoes a process where in the temperature changes as T=T₀ (1+ V^2 ) , where, T and V are temperature and volume respectively, T₀ and are positive constants. The molar heat capacity C of the gas is given as C=C_V+R f(V) , where, f(V) is a function of volume. The expression for f(V) is

Step-by-step solution

Given, T=T₀ (1+ V^2 ) aligned & d T d V = T ₀ 2 V & d V= d ~T ~T ₀ 2 V aligned From first law of thermodynamics, array cc & d Q=d U+d W & n C d T=n C_V d T+p d V & n C d T+n C_V d T+p ( d T T₀ 2 V ) & C=C_V+ ( p n ) ( 1 T₀ 2 V ) array Now as gas is ideal, or aligned & p V=n R T & p n = R T V = R [T₀ (1+ V^2 ) ] V aligned substituting in Eq. (ii), we get, C=C_V+R ( 1+ V^2 V^2 )