# Can we have constant volume, and pressure in a system?

I was given a system where heat was added to a mass of Nitrogen gas in a canister at constant pressure? And given the volume of the canister as 1L, I assumed that the gas would expand to fill the container, and as the pressure is constant, it's temperature would increase, this means however that I did not know whether to use $$c_p$$ or $$c_v$$ for nitrogen, and I assume I've made a false assumption somewhere? Is it possible to keep a gas at constant temperature and pressure while adding energy to it? and what is the correct way to find it's specific heat capacity as heat is added.

• Unless you have an open system (the amount of gas varies), you only have a variance of $2$, i.e. you only need two variables to specify the state of the system (pressure and temperature for example). Therefore, in your transformation is trivial, the initial and final state is the same. If you truly maintain that the volume is constant, you have no work, so no heat (no internal energy variation). You need to weaken your assumptions (typically in a canister, only the volume is constant).
– LPZ
May 5, 2022 at 13:15
• I think, the volume is being altered to keep the pressure and perhaps they didn't explain that (or it was implied) in the question. May 5, 2022 at 13:34
• Not in a pure mathematical sense, but there are liquids with extremely low vapor pressure whose volume changes very little with a change in temperature at constant pressure. May 5, 2022 at 14:01

The ideal gas law says $$PV = Nk_B T,$$ where $$P$$ is pressure, $$V$$ is volume, $$T$$ is temperature, $$N$$ is the number of particles, and $$k_B$$ is Boltzmann's constant. The first law of thermodynamics says $$\Delta U = Q - W,$$ where $$U$$ is the internal energy, $$Q$$ is the heat added, and $$W$$ is the work done by the system.
Let's say both pressure and volume remain constant while you add heat. Since the volume didn't change the gas did no work. The heat causes an increase in internal energy, so the temperature increases. In order to maintain a constant $$PV$$, the number of particles, $$N$$, must decrease. So gas must escape from the system. If the number of particles changes during the process, the energy you add as heat won't be evenly distributed among all particles. Some will escape before you've finished adding the heat. I'm not sure it makes sense to talk about the specific heat of a system like this. Specific heat is the heat capacity per mass or per particle or per mole of particles. How much heat did you add per particle if the number of particles changed?