# Partial Derivatives with gas, pressure, volume, temperature..

• Oct 9th 2008, 08:28 PM
Qt3e_M3
Partial Derivatives with gas, pressure, volume, temperature..
The gas law for a fixed mass m for an ideal gas at absolute temperature T, pressure P, and volume V is PV = mRT, where R is the gas constant.
Show that бP/бV бV/бT бT/бP = -1

So basically P = mRT/V, V = mRT/P, T = PV/mR are the different functions, and we take the partial derivative of each...but not totally sure how
• Oct 10th 2008, 08:56 AM
Qt3e_M3
Still looking for this...:(
• Oct 10th 2008, 09:02 AM
galactus
Take the partial of each by just treating the others as a constant.

$\displaystyle \frac{{\partial}P}{{\partial}V}=\frac{-mRt}{V^{2}}$

$\displaystyle \frac{{\partial}V}{{\partial}T}=\frac{mR}{P}$

$\displaystyle \frac{{\partial}T}{{\partial}P}=\frac{V}{mR}$

Multiply them together:

$\displaystyle (\frac{-mRt}{V^{2}})(\frac{mR}{P})(\frac{V}{mR})=\frac{-mRT}{PV}$

But, $\displaystyle PV=mRT$

Sub it in:

$\displaystyle \frac{-mRT}{mRT}=-1$