The ratio of the molar heat capacities of an ideal gas is CP/CV = 7/6. Calculate the change in internal energy of 1.0 mole of the gas when its temperature is raised by 50 K

(a) keeping the pressure constant,


(b) keeping the volume constant and


(c) adiabatically.


Given:


n = number of moles = 1,


Cv = specific heat capacity at constant volume,


Cp = specific heat capacity at constant pressure


dT = change in temperature = 50K.


γ= Ratio of molar heat capacities = CP/CV = 7/6 => Cv = 6Cp/7.


(a) Formula used:


Pressure constant: Isobaric process. For an isobaric process,


change in internal energy dU = nCvdT,


Where


n = number of moles,


Cv = specific heat at constant volume,


dT = rise in temperature


Also, Cp-Cv = R.


Cp = specific heat at constant pressure


Cv = specific heat at constant volume


R = universal gas constant = 8.314 J/mol/K


Substituting: Cp - 6Cp/7 = Cp/7 = R => Cp = 7R.


Therefore Cv = Cp - R = 6R = (6 X 8.314)J/mol/K


Therefore,


dU = 1 mol X (6 X 8.314)J/mol/K X 50K = 2494.2 J(Ans)


(b) Volume constant: Isochoric process, dV = 0(change in volume)


First law of thermodynamics gives us: dU = dQ - dW


Where dU = change in internal energy, dQ = change in heat,


dW = work done = Pressure x change in volume = PdV


Since dV = 0, dU = dQ.


Hence, dU = nCvdT since dQ = nCvdT.


Where


n = Number of moles,


Cv = Specific heat at constant volume,


dT =Change in temperature


Putting the values in the above formula, we get


Therefore, dU = 1 mol x (6 x 8.314) J/mol/K x 50K


= 2494.2 J(Ans)


(c) Adiabatic process: dQ(heat change) = 0. Therefore,


dU(change in internal energy) = dW(work done)


Since dQ = dU + dW.


For an adiabatic process, dW = dT/(

1