Pressure

$P=\frac{F}{A}$

• $F$ is force
• $A$ is area
• Units are pascals or bars or torr or in Atm or in PSI

Statistical Mechanics

$P\equiv -\left(\frac{\partial U}{\partial V}\right)=\tau {\left(\frac{\partial \sigma }{\partial V}\right)}_U=\tau \frac{\partial }{\partial V}\ln Z$

• Statistical mechanics definition of pressure. Can be thought of as like a volumetric energy density
• $U$ is the [internal energy] of the system
• $V$ is volume of the system
• $\tau$ is the [fundamental temperature] of the system
• $\sigma$ is the [fundamental entropy] of the system
• $Z$ is the [partition function]

$P=-{\left(\frac{\partial U}{\partial V}\right)}_{\tau }+\tau \left({\frac{\partial \sigma }{\partial V}}_{\tau }\right)$

• First term is energy pressure, important for solids
• Second term is entropy pressure, important for gases

$P=-{\left(\frac{\partial F}{\partial V}\right)}_{\tau }$

• $F$ is the [Helmholtz free energy]