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Question

Consider the set up shown, where we have an ideal gas trapped in a cylindrical container, kept on a thermal reservoir which provides heat to the system. The weight on top maintains a constant pressure P on the gas. Supplying heat Q changes the temperature by ΔT, and expands the volume by ΔV. The expanding gas displaces the container and does work, W. Choose the correct inferences from the following options


A

Q = nCpΔT

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B

W = nRΔT

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C

Increase in internal energy, ΔEint = n(Cp R)ΔT

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D

Increase in internal energy, ΔEint = nCvΔT.

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Solution

The correct option is D

Increase in internal energy, ΔEint = nCvΔT.


From kinetic theory, we can express the internal energy as ΔEint = nCvΔT, for all processes.

In a process where the pressure is kept constant, supplying heat will increase the temperature and volume, according to Charles' law. The expansion will displace the weight kept on top by S(say), and do the following work,

W = F.s = (PA).s = P(A.s) = PΔV.

But form the ideal gas equation,

PV = nRT

PΔV + VΔP = nRΔT

PΔV = nRΔT (for constant P)

W=PΔV = nRΔT. (1)

The heat supplied to increase the temperature of n moles of the gas by ΔT must be -

Q = nCpΔT. (2)
Now, applying the first law on (1) and (2)-

ΔEint = Q W

ΔEint = nCpΔT nRΔT

ΔEint = n(Cp R)ΔT

ΔEint = nCvΔT n(Cp R)ΔT

Cp Cv = R. (3)

From equations (1) - (3), we see that all the given options are correct, and can be derived by applying the first law of thermodynamics in conjunction with the kinetic theory of gases.


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