Internal pressure is a measure of how the internal energy of a system changes when it expands or contracts at constant temperature. It has the same dimensions as pressure, the SI unit of which is the pascal. Internal pressure is usually given the symbol π T {\displaystyle \pi _{T}} . It is defined as a partial derivative of internal energy with respect to volume at constant temperature:
π T = ( ∂ U ∂ V ) T {\displaystyle \pi _{T}=\left({\frac {\partial U}{\partial V}}\right)_{T}}
Thermodynamic equation of state Internal pressure can be expressed in terms of temperature, pressure and their mutual dependence:
π T = T ( ∂ p ∂ T ) V − p {\displaystyle \pi _{T}=T\left({\frac {\partial p}{\partial T}}\right)_{V}-p}
This equation is one of the simplest thermodynamic equations. More precisely, it is a thermodynamic property relation, since it holds true for any system and connects the equation of state to one or more thermodynamic energy properties. Here we refer to it as a "thermodynamic equation of state."
Derivation of the thermodynamic equation of state The fundamental thermodynamic equation states for the exact differential of the internal energy:
d U = T d S − p d V {\displaystyle \operatorname {d} U=T\operatorname {d} S-p\operatorname {d} V}
Dividing this equation by d V {\displaystyle \operatorname {d} V} at constant temperature gives:
( ∂ U ∂ V ) T = T ( ∂ S ∂ V ) T − p {\displaystyle \left({\frac {\partial U}{\partial V}}\right)_{T}=T\left({\frac {\partial S}{\partial V}}\right)_{T}-p}
And using one of the Maxwell relations:
( ∂ S ∂ V ) T = ( ∂ p ∂ T ) V {\displaystyle \left({\frac {\partial S}{\partial V}}\right)_{T}=\left({\frac {\partial p}{\partial T}}\right)_{V}\ } , this gives
π T = T ( ∂ p ∂ T ) V − p {\displaystyle \pi _{T}=T\left({\frac {\partial p}{\partial T}}\right)_{V}-p}
Perfect gas In a perfect gas, there are no potential energy interactions between the particles, so any change in the internal energy of the gas is directly proportional to the change in the kinetic energy of its constituent species and therefore also to the change in temperature:
d U ∝ d T {\displaystyle \operatorname {d} U\propto \operatorname {d} T} . The internal pressure is taken to be at constant temperature, therefore
d T = 0 {\displaystyle dT=0} , which implies d U = 0 {\displaystyle dU=0} and finally π T = 0 {\displaystyle \pi _{T}=0} , i.e. the internal energy of a perfect gas is independent of the volume it occupies. The above relation can be used as a definition of a perfect gas. The relation π T = 0 {\displaystyle \pi _{T}=0} can be proved without the need to invoke any molecular arguments. It follows directly from the thermodynamic equation of state if we use the ideal gas law p V = n R T {\displaystyle pV=nRT} . We have
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