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
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,
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
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.
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/(
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