COMEDK 2023 Shift 1 Paper

To find the change in internal energy of a gas during a process, we can use the relation that connects the change in internal energy with pressure and volume changes, especially under constant pressure. The change in internal energy, ∆U, for an ideal gas can be calculated using the specific heats and the first law of thermodynamics. It's given by the equation:
∆U=nCv∆T
where ∆U is the change in internal energy, n is the amount of substance, Cv is the molar specific heat capacity at constant volume, and ∆T is the change in temperature.
Since we're dealing with a process at constant pressure, we can also relate the change in temperature to the change in volume using the ideal gas law. Under constant pressure, the change in volume ∆V and the change in temperature ∆T are related by the equation:
P∆V=nR∆T
Given that the volume changes from V to 3V, we have:
∆V=3V−V=2V
Substituting this into the equation gives us:
P⋅2V=nR∆T
Now, relating the specific heats, we know that:
γ=‌

Cp

Cv

where γ is the given ratio of specific heats, Cp is the molar specific heat at constant pressure, and Cv is the molar specific heat at constant Volume. The molar specific heat capacities are related to the gas constant R by:
Cp−Cv=R
Substituting γ=‌

Cp

Cv

into the expression and solving for Cv gives:
Cv=‌

R

γ−1

Therefore, the change in internal energy can be also expressed in terms of γ,R, and ∆T :
∆U=n(‌

R

γ−1

)∆T=‌

P⋅2V

γ−1

So, the change in internal energy when the volume changes from V to 3V at constant pressure is ‌

2PV

(γ−1)

, which is Option D.