# Why does the negative sign arise in this thermodynamic relation?

This is not a simple application of partial derivatives, since the variables that are being held constant vary here, but an instance of the triple product rule, which says that for any three quantities $x,y,z$ depending on each other, the relation $$ \left(\frac{\partial x}{\partial y}\right)_z\left(\frac{\partial y}{\partial z} \right)_x\left(\frac{\partial z}{\partial x}\right)_y = -1$$ holds.

This all starts from the basic relationship $$dP=\left(\frac{\partial P}{\partial T}\right)_VdT+\left(\frac{\partial P}{\partial V}\right)_TdV$$Since, at constant pressure, dP=0, if we solve for dT/dV at constant pressure, we obtain: $$\left(\frac{\partial T}{\partial V}\right)_P=-\frac{\left(\frac{\partial P}{\partial V}\right)_T}{\left(\frac{\partial P}{\partial T}\right)_V}$$

Other answers have given the mathematical reasons, but as these are physical functions it’s always nice to relate back to physical intuition.

Thinking of an ordinary substance – a lump of “stuff”, doesn’t matter whether it’s a solid or a gas or whatever – if you *decrease* the volume available to it, the pressure should go *up*. In fact that is exactly how we physically put a sample under pressure. So the left-hand side should be negative.

On the other hand, if you *increase* the temperature (usually) the pressure should *increase* at constant volume, or alternatively the volume should *increase* at constant pressure.^{1} So both of the factors on the right-hand side should be positive.^{2}

So we had better include a minus sign, to at least get the signs to match on both sides of the equation.

- If you’re thinking about a material with negative thermal expansion, then both of these factors should instead be negative, but again the right-hand side of the equation ends up positive.
- To be precise, I have used another identity here: that $(\partial T/\partial V)_P = (\partial V/\partial T)_P^{-1}$.