# What at "constant state variable" mean in thermodynamics?

For example we calculate change in enthalpy at constant pressure or we measure heat capacity at constant pressure. My question is does the external pressure must be constant or that of the system?

For example suppose we heat a gas inside a container at external pressure of $$1$$ atm. The differential of th enthalpy is:

$$dH=dU+pdV+Vdp$$

and for finite changes:

$$ΔH=ΔU + pΔV + VΔp$$

From the second equation because $$VΔp=0$$ (no matter if the pressure were homogeneous inside the container at all times) and by using first law of thermodynamics $$ΔH=Q$$. So the heat is the same no matter if the process is quasi-static or not because what matters is the final and initial state. Lets say we have also measured the temperature difference between these two states. Then $$C_p=\frac{ΔH}{ΔΤ}$$.

But this can't be correct because as we know heat capacity at constant $$p$$ is defined as:

$$C_p=\left(\frac{\partial H}{\partial T}\right)_p$$

Does this imply that we should calculate such quantities only at quasi-statically processes? So for finite changes in thermodynamic potentials (like enthalpy, gibbs free energy etc) "at constant state variable (e.g. pressure)" it doesn't matter if the external pressure is constant or the process takes place quasi-statically but "at constant state variable" matters if we want to calculate quantities that involve ratios? I always thought that "at constant state variable X" means $$dX=0$$.

To sum it up my question is what at "at constant state variable X" means in thermodynamics.

• If your gas is at 3 atm, and you suddenly drop the external pressure to 1 atm, and hold it at that value until the gas re-equilibrates adiabatically (and irreversibly), do you think that the change in enthalpy is zero? Do you think that the temperature changes? Apr 15 at 13:51
• Are you saying that $Q=C_p\Delta T$ for all processes in which the initial and final pressures are the same? It is not clear exactly what you are actually saying? Apr 15 at 18:10

Does this imply that we should calculate such quantities only at quasi-statically processes?

No it does not. State variables like enthalpy do not depend on the process, only on the beginning and final state values at equilibrium. So for $$\Delta p$$ to be zero, it means that the initial and final equilibrium pressure of the gas is the same for the initial and final states. It does not require that the process be quasi-static or a constant pressure process, for that matter.

If, in fact, the process is quasi-static, then the gas is in equilibrium with the external pressure throughout the process and obviously the initial and final gas pressures will have to be the same. But it is not required that the gas be in equilibrium with the external pressure throughout the process. Only that the initial and final gas pressure is the same.

For example (see FIG 1), let's say we have a constant quasi-static (reversible) external pressure expansion process between states 1 and 2. Clearly, $$\Delta P=0$$. But we could also connect the same two states by first conducting a reversible isothermal expansion from state 1 to the final volume and lower pressure (state 1a of FIG 1) and follow this with a reversible constant volume (isochoric) heat addition to increase the pressure to the original pressure (state 2). In this example $$\Delta p=0$$ but the path was not a constant pressure path.

For another alternative path (see FIG 2), in this case not involving a quasi-static (reversible) path, let the initial external pressure suddenly drop to some lower external pressure which is then maintained constant until the gas re-equilibriates with the lower pressure. The external pressure is then held constant until the gas re-equilibriates with the external pressure (state 1a). This is then followed by a constant volume heat addition to return the gas to its original pressure (state 2). Once again, $$\Delta p=0$$ without the gas pressure being constant throughout the path.

There are, for that matter, an infinite number of possible paths connecting states 1 and 2. For every such path, $$\Delta p=0$$.

To sum it up my question is what at "at constant state variable X" means in thermodynamics.

Without seeing the statement in context, it seems to mean a property (pressure, temperature, pressure, internal energy, enthalpy) of the system that does not change during a process. But as I already stated, a property during a process between two equilibrium states does not necessarily have to be constant for that property to be the same for the two states.

Hope this helps. 