Pressure

#Physics

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

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

Statistical Mechanics

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

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

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

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

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

• $\displaystyle F$ is the [Helmholtz free energy]