When an ideal gas is heated, the change in internal energy is limited to increasing the average translational kinetic energy of the gas molecules. Thus, there is a simple relationship between ΔU of the gas and the change in temperature that occurs. Derive this relationship with the help of ideas about the kinetic–molecular theory of gases developed in Chapter 6. After doing so, obtain numerical values (in J mol^-1 K^-1) for the following molar heat capacities.

(a) The heat capacity, C_V for one mole of gas under constant-volume conditions; 

(b) The heat capacity, C_p, for one mole of gas under constant-pressure conditions.