# Which heat capacity is used in the heat equation?

The heat equation is often written as $$\frac{\partial T}{\partial t} = \frac{\kappa}{c} \nabla^2T$$ where $$\kappa$$ is the thermal conductivity and $$c$$ is a heat capacity per volume.

I often see $$c$$ written as $$c_P$$ implying that it is the heat capacity (per unit volume) for a system held at constant pressure, but I was wondering if this was necessary of not?

I understand that in most 'everyday' experimental examples, pressure will be the variable that is held constant, and that for liquids and solids there isn't much difference between $$c_P$$ and $$c_V$$ anyway. However, in theory, can the heat capacity in this equation be with whatever variable you want to hold constant (so could be either $$c_P$$ or $$c_V$$ depending on your situation?

I also found this similar question but I couldn't find a definitnive answer to my question in their answers: In deriving the heat transfer equation, why do we use heat capacity at constant pressure?.

• The HE (Fourier) should be $\frac{\partial T}{\partial t} = \frac{\kappa}{\rho C_p} \nabla^2T$. Note the partials: it is a PDE. $\rho$ is the material's density.
– Gert
Oct 5, 2020 at 19:13
• Apologies for the typo Oct 5, 2020 at 19:14
• Funny, I had the exact same question since over the last 3 years and I have been told that the answer is "whatever is being measured", which is essentially in agreement with alpehzero's answer. Oct 5, 2020 at 19:18
• @AlexGower: you're welcome.
– Gert
Oct 5, 2020 at 19:28

Both options ($$C_p$$ or $$C_v$$) are approximations.
The simplest assumption is that the object is not mechanically constrained and that the non-uniform internal strain energy is small. Therefore the internal pressure in the solid remains approximately constant and $$C_p$$ is the relevant thermal capacity.
If the object is constrained to have constant volume, $$C_v$$ would be a better approximation, since it takes account of the internal strain energy (assuming the internal stress field is uniform, of course).