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?