- #1

etotheipi

I've learned that ##W = -\int{P_{ext} dV}##, and only during a reversible/quasi-static process where ##P_{int} = P_{ext}## can we write the work done on the gas in terms of the internal pressure (and consequently use ##PV=nRT## etc. which apply to the internal gas).

However, a lot of sources, when showing derivations of various basic thermodynamic equations, don't refer to ##P_{ext} dV## and use a general ##P dV##, which I assume refers to the internal pressure.

As an example, the change in enthalpy (at constant pressure) ##dH## is defined as$$dH = dU + d(PV) = dU + P_{int} dV$$Knowing that ##dQ = dU + P_{ext} dV##, this apparently proves that ##dH = dQ## at constant pressure. However, the two pressure are clearly different so there must be more to it?

Another example is specific heat at constant pressure. If ##dQ = C_{P}dT## then ##dU = C_{P}dT - P_{ext}dV##. They then quote that from the ideal gas law, at constant pressure, ##PdV = nRdT##, which when substituted in gives Meyer's relation. Except this is only the case if ##P = P_{ext}##, which doesn't appear to be true in this example considering the ideal gas law deals with the internal pressure.

Both of these have used the ##P## term quite ambiguously and I can't follow why. I was wondering if someone could clear things up - do these equations only "work" (no pun intended...) if the internal and external pressures are identical? Thank you!

However, a lot of sources, when showing derivations of various basic thermodynamic equations, don't refer to ##P_{ext} dV## and use a general ##P dV##, which I assume refers to the internal pressure.

As an example, the change in enthalpy (at constant pressure) ##dH## is defined as$$dH = dU + d(PV) = dU + P_{int} dV$$Knowing that ##dQ = dU + P_{ext} dV##, this apparently proves that ##dH = dQ## at constant pressure. However, the two pressure are clearly different so there must be more to it?

Another example is specific heat at constant pressure. If ##dQ = C_{P}dT## then ##dU = C_{P}dT - P_{ext}dV##. They then quote that from the ideal gas law, at constant pressure, ##PdV = nRdT##, which when substituted in gives Meyer's relation. Except this is only the case if ##P = P_{ext}##, which doesn't appear to be true in this example considering the ideal gas law deals with the internal pressure.

Both of these have used the ##P## term quite ambiguously and I can't follow why. I was wondering if someone could clear things up - do these equations only "work" (no pun intended...) if the internal and external pressures are identical? Thank you!

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