A mass m of an ideal gas initially has volume Voand temperature To.  When it is kept at constant volume Vo, heat Q is required to increase its temperature to To + DT.  If the volume is not kept constant, but the gas expands from Vo to 3Vo when the temperature increases from To to (To + 2DT), then the heat supplied must be

In the first case the rise in temperature is ΔT and in the second case it is 2ΔT. The heat required for this must be 2Q. In addition to rising the temperature, the system is expanding and doing work and additional heat will be required for this, hence more than 2QNumber of moles, n = m/M.In he first case. Q = nCv(Tf - Ti)In the second case, from 1st law of thermodynamics,