[Araconan]

So I was reading over my textbook's explanation regarding the proof of the equation between enthalpy and molar heat capacity. The portion of it that I'm confused about, is that after the proof is done, it states that the proof itself was done without assuming either constant pressure or constant volume. (So basically the equation can be used at any time, regardless of pressure or volume).

The proof is as follows (for mono-atomic gases):

We know that ΔH = ΔE + ΔPV.
Because ΔE = nC_vΔT              (1)
and ΔPV = nRΔT                    (2)

ΔH = nC_vΔT = nRΔT
Since nC_p = n(C_v+R)ΔT          (3)
ΔH = nC_pΔT

However, doesn't equation 1 assume constant volume (since it's only valid when no PV work is done), and equation 3 assume constant pressure (since PV work is done)?
Therefore, doesn't this proof require both the assumption of constant volume and constant pressure, and therefore can only be used under such conditions?
Why can it be used under any conditions, and according to the textbook, doesn't require either assumption of constant volume or constant pressure?

[UG]

The internal energy, U, can be expressed as a function of temperature and volume:
dU = (δU/δT)_V δT + (δU/δV)_T δV
We know that (δU/δT)_V is just the constant volume heat capacity. For an ideal gas, the term (δU/δV)_T becomes zero because there is no potential energy of interaction between the molecules of an ideal gas, so changing the volume does not affect the internal energy. The only contribution to the internal energy is then just the kinetic energy of the molecules as the temperature changes. Thus, for an ideal gas undergoing any process, not just one at constant volume dU = (δU/δT)_V δT or just dU = C_v δT. A similar case can be made for the enthalpy change when you express it as a function of temperature and pressure. The term (δH/δP)_T = 0 so again the enthalpy change is just a function of temperature only (just for ideal gases remember) hence dH = C_p δT

[Araconan]

The internal energy, U, can be expressed as a function of temperature and volume:
dU = (δU/δT)_V δT + (δU/δV)_T δV
We know that (δU/δT)_V is just the constant volume heat capacity. For an ideal gas, the term (δU/δV)_T becomes zero because there is no potential energy of interaction between the molecules of an ideal gas, so changing the volume does not affect the internal energy. The only contribution to the internal energy is then just the kinetic energy of the molecules as the temperature changes. Thus, for an ideal gas undergoing any process, not just one at constant volume dU = (δU/δT)_V δT or just dU = C_v δT. A similar case can be made for the enthalpy change when you express it as a function of temperature and pressure. The term (δH/δP)_T = 0 so again the enthalpy change is just a function of temperature only (just for ideal gases remember) hence dH = C_p δT

I don't really have a solid background in Calculus Could you please provide a more simplistic explanation please? >.<

[UG]

Sure, I will try and simplify it. The squiggly things are meant to be partial derivatives . Basically, we know that the internal energy changes if the temperature and/or the volume of the system changes, the term (δU/δT)_V is basically saying, how fast does U change when you make a small change in the temperature (at constant volume), the other term (δU/δV)_T is describing how much the internal energy changes when you make a small change in the volume (at constant temperature). Together, these describe the total internal energy change. For an ideal gas, the term (δU/δV)_T goes to zero, for reasons that I have explained in my previous post (i.e. the internal energy of an ideal gas does not change with volume). Hence you are left with the term (δU/δT)_V, which is simply the definition of the constant volume heat capacity. Integrating dU = Cv δT gives ΔU = C_vΔT.

[Araconan]

Ahh, I see now. Thank you!