Question

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 $P\Delta V$, 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 $U_f$?

• A. $U_f=U_i\left(sincetotalinternalenergyisalwaysconserved\right)$
• B. $U_f=U_i+n{C}_{v}T$
• C. $U_f=U_i+n{C}_v\Delta T$
• D. $U_f=U_i+C_v\Delta T$

Answer 1

The correct option is D

$U_f=U_i+C_v\Delta T$

: In the diagram, the two lines represent isotherms for temperatures T and $(T+\Delta T)$ - 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 -

$\Delta E_{int}=n{C}_v\Delta T$

Therefore in our case,

$U_f=U_i+\Delta E_{int}$

$\Rightarrow U_f=U_i+n{C}_v\Delta T$.