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 $\Delta T$, and expands the volume by $\Delta V$. The expanding gas displaces the container and does work, W. Choose the correct inferences from the following options

• A. $Q=n{C}_p\Delta T$
• B. $W=nR\Delta T$
• C. $\text{Increase in internal energy,}$ $\Delta E_{int}=n(C_p-R)\Delta T$
• D. $\text{Increase in internal energy,}$ $\Delta E_{int}=n{C}_v\Delta T$.

Answer 1

Answer: (A), (B), (C), (D)

$\text{Increase in internal energy,}$ $\Delta E_{int}=n{C}_v\Delta T$.

From kinetic theory, we can express the internal energy as $\Delta E_{int}=n{C}_v\Delta T$, 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,

$W=\overrightarrow{F}.\overrightarrow{s}=\left(PA\right).s=P(A.s)=P\Delta V$.

But form the ideal gas equation,

$PV=nRT$

$\Rightarrow P\Delta V+V\Delta P=nR\Delta T$

$\Rightarrow P\Delta V=nR\Delta T$ (for constant P)

$\Rightarrow W=P\Delta V=nR\Delta T$. (1)

The heat supplied to increase the temperature of n moles of the gas by $\Delta T$ must be -

$Q=n{C}_p\Delta T$. (2)
Now, applying the first law on (1) and (2)-

$\Delta E_{int}=Q-W$

$\Rightarrow \Delta E_{int}=n{C}_p\Delta T-nR\Delta T$

$\Rightarrow \Delta E_{int}=n(C_p-R)\Delta T$

$\Rightarrow \Delta E_{int}=n{C}_v\Delta T-n(C_p-R)\Delta T$

$\Rightarrow C_p-C_v=R$. (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.