Pressure

Pressure ($p$) is the force per unit area applied on a surface, in a direction perpendicular to that surface, i.e. the scalar part of the stress tensor under equilibrium/hydrosatic conditions.

Thermodynamics

In thermodynamics the pressure is given by

$p=-\left.{\frac {\partial A}{\partial V}}\right\vert _{T,N}=k_{B}T\left.{\frac {\partial \ln Q}{\partial V}}\right\vert _{T,N}$

where $A$ is the Helmholtz energy function, $V$ is the volume, $k_{B}$ is the Boltzmann constant, $T$ is the temperature and $Q(N,V,T)$ is the canonical ensemble partition function.

Units

The SI units for pressure are Pascals (Pa), 1 Pa being 1 N/m², or 1 J/m³. Other frequently encountered units are bars and millibars (mbar); 1 mbar = 100 Pa = 1 hPa, 1 hectopascal. 1 bar is 10⁵ Pa by definition. This is very close to the standard atmosphere (atm), approximately equal to typical air pressure at earth mean sea level: atm, standard atmosphere = 101325 Pa = 101.325 kPa = 1013.25 hPa = 1.01325 bar

Stress

The stress is given by

${\mathbf {F} }=\sigma _{ij}{\mathbf {A} }$

where ${\mathbf {F} }$ is the force, ${\mathbf {A} }$ is the area, and $\sigma _{ij}$ is the stress tensor, given by

$\sigma _{ij}\equiv \left[{\begin{matrix}\sigma _{x}&\tau _{xy}&\tau _{xz}\\\tau _{yx}&\sigma _{y}&\tau _{yz}\\\tau _{zx}&\tau _{zy}&\sigma _{z}\\\end{matrix}}\right]$

where where $\ \sigma _{x}$, $\ \sigma _{y}$, and $\ \sigma _{z}$ are normal stresses, and $\ \tau _{xy}$, $\ \tau _{xz}$, $\ \tau _{yx}$, $\ \tau _{yz}$, $\ \tau _{zx}$, and $\ \tau _{zy}$ are shear stresess.

See also

• Pressure equation
• Virial pressure
• Test volume method

References

Related reading

• Aidan P. Thompson, Steven J. Plimpton, and William Mattson "General formulation of pressure and stress tensor for arbitrary many-body interaction potentials under periodic boundary conditions", Journal of Chemical Physics 131 154107 (2009)
• G. C. Rossi and M. Testa "The stress tensor in thermodynamics and statistical mechanics", Journal of Chemical Physics 132 074902 (2010)
• Nikhil Chandra Admal and E. B. Tadmor "Stress and heat flux for arbitrary multibody potentials: A unified framework", Journal of Chemical Physics 134 184106 (2011)
• Takenobu Nakamura, Wataru Shinoda, and Tamio Ikeshoji "Novel numerical method for calculating the pressure tensor in spherical coordinates for molecular systems", Journal of Chemical Physics 135 094106 (2011)
• Péter T. Kiss and András Baranyai "On the pressure calculation for polarizable models in computer simulation", Journal of Chemical Physics 136 104109 (2012)