Reversibility, Enthalpy, and Heat Capacity

Objectives

This section discusses the concepts of reversibility, enthalpy, and heat capacity

Preknowledge

Please read the previous section on Derivatives

Reversibility

Reversibility is one of the hardest concepts in thermodynamics. Unfortunately it is also one of the hardest to teach students. But as you work with thermodynamics you will understand this concept rather intuitively.

Let us first start with a technical definition of reversibility:

A process is reversible if its direction can be reversed at any point by an infinitesimal change in external conditions.

For now there are just two things you need to be aware of:

1. Reversible processes are ideal processes. No real process is reversible. Reversible processes are used as a baseline.
2. The expression we had previously used, [math]dW=-PdV[/math], is applicable only for reversible processes.

Enthalpy

Let us consider a constant pressure process. We will assume reversible work and only mechanical work (usually called PV work).

From the First Law:

[math]dU = dQ + dW[/math]

solving for heat, dQ

[math]dQ = dU - dW[/math]

considering only PV work:

[math]dQ = dU + PdV[/math]

since pressure is constant it can be moved inside the derivative,

[math]dQ = dU + d(PV) = d(U + PV)[/math]

The quantity U + PV is called the Enthalpy, H:

[math]dQ = dH[/math] for a constant pressure process.

Enthalpy is probably the most common thermodynamic quantity used, especially in chemistry and engineering. Especially common is the use of the "enthalpy of reaction". Often this is called just the "heat of reaction", but note from above that it is actually the heat change at constant pressure.

Heat Capacity

Heat Capacity, C 

The change in heat per unit temperature:

[math]C \equiv \frac{\partial Q}{\partial T}[/math]

There is however a problem with this quantity; it is path dependent. That makes it difficult, if not impossible, to measure. Instead, we will use two special forms.

Heat capacity at constant pressure

We saw above that at constant pressure, the heat change is the enthalpy change. Then the heat capacity is:

[math]C_P = \left (\frac{\partial H}{\partial T} \right )_P[/math]

Enthalpy is a state function, since the sum of state functions is also a state function. Therefore, from the above equation, constant pressure heat capacity is a also state function and is path independent.

Heat capacity at constant volume

To determine the heat capacity at constant volume, let us begin again with the first law (and again considering only PV work):

[math]dQ = dU + PdV[/math]

If we have constant volume, dV = 0, therefore:

[math]dQ = dU[/math] for a constant volume process.

Then,

[math]C_V = \left (\frac{\partial U}{\partial T} \right )_V[/math]

And again, constant volume heat capacity is path independent.

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