In the P-Vdiagram show, moles of a gas are taken through a process from state i to f, at constant pressure. Part of the heat must have gone into doing the work $P Δ V$ , and the rest in increasing the internal energy. If the molar specific heat at constant volume for the gas is $C_{V}$ , and the initial total internal energy is $U_{i}$ , what is the final internal energy $U_{f}$ ?

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Solution

The correct option is D

: In the diagram, the two lines represent isotherms for temperatures T and $(T + Δ T)$ - any point on either line will be at the same temperature. If a process takes the state of a gas from a point on one isotherm to a point on another isotherm, the change in internal energy will be related to the temperature difference only, and nothing else, given by the relation -

$Δ E_{i n t} = n C_{v} Δ T$

Therefore in our case,

$U_{f} = U_{i} + Δ E_{i n t}$

$\Rightarrow U_{f} = U_{i} + n C_{v} Δ T$ .

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