# Constant Volume vs. Constant Pressure?

Approaching the following question:

Consider two experiments in which 2 moles of a monatomic ideal gas are heated from temperature $T$ to temperature $T + \Delta T$: in the first experiment the volume $V$ is kept constant, in the second experiment the pressure $p$ is kept constant. How much more heat is needed in the second experiment than in the first experiment to raise the temperature by the given amount $\Delta T$?

The answer is $2 R \Delta T$.

The origin of the problem may be found here under Question 14.

I am confused as to home to come to this conclusion. I believe I am able to utilize $pV = N k_B T$ and $VT^\alpha = const$, $pV^\gamma = const$ but I am unsure of how the two constants apply to this.

Do both constant apply to each of the experiments? How do I manipulate these equations to achieve my desired result?

• $pV^\gamma=\text{const.}$ and $VT^\alpha=\text{const.}$ are both equations for adiabatic process, not isobaric or isochoric. – Siyuan Ren Jul 23 '12 at 0:56

In the first experiment, work done is 0 as volume is constant. Using the first law of thermodynamics $q=U-w$, $q=\Delta U$. In the second case extra heat is needed due to the work done which is $\Delta(pV)=p\Delta V$, as pressure is constant. Using the ideal gas equation $p \Delta V = nR \Delta T=2R\Delta T$. Note the change in internal energy depends only on the change in temperature and is same in both the cases.