Inthermodynamics ,Bridgman's thermodynamic equations are a basic set of thermodynamic equations, derived using a method of generating multiple thermodynamic identities involving a number of thermodynamic quantities. The equations are named after the American physicistPercy Williams Bridgman . (See also theexact differential article for general differential relationships).
The extensive variables of the system are fundamental. Only theentropy S , thevolume V and the four most common thermodynamic potentials will be considered. The four most common thermodynamic potentials are:
The first derivatives of the internal energy with respect to its (extensive) natural variablesS andV yields the intensive parameters of the system - Thepressure P and thetemperature T . For a simple system in which theparticle numbers are constant, the second derivatives of the thermodynamic potentials can all be expressed in terms of only threematerial properties
Bridgman's equations are a series of relationships between all of the above quantities.
Many thermodynamic equations are expressed in terms of partial derivatives. For example, the expression for the heat capacity at constant pressure is:
C P = ( ∂ H ∂ T ) P {\displaystyle C_{P}=\left({\frac {\partial H}{\partial T}}\right)_{P}} which is the partial derivative of the enthalpy with respect to temperature while holding pressure constant. We may write this equation as:
C P = ( ∂ H ) P ( ∂ T ) P {\displaystyle C_{P}={\frac {(\partial H)_{P}}{(\partial T)_{P}}}} This method of rewriting the partial derivative was described by Bridgman (and also Lewis & Randall), and allows the use of the following collection of expressions to express many thermodynamic equations. For example from the equations below we have:
( ∂ H ) P = C P {\displaystyle (\partial H)_{P}=C_{P}} and
( ∂ T ) P = 1 {\displaystyle (\partial T)_{P}=1} Dividing, we recover the proper expression for CP .
The following summary restates various partial terms in terms of the thermodynamic potentials, the state parameters S, T, P, V, and the following threematerial properties which are easily measured experimentally.
( ∂ V ∂ T ) P = α V {\displaystyle \left({\frac {\partial V}{\partial T}}\right)_{P}=\alpha V} ( ∂ V ∂ P ) T = − β T V {\displaystyle \left({\frac {\partial V}{\partial P}}\right)_{T}=-\beta _{T}V} ( ∂ H ∂ T ) P = C P = c P N {\displaystyle \left({\frac {\partial H}{\partial T}}\right)_{P}=C_{P}=c_{P}N} [ edit ] Note that Lewis and Randall use F and E for the Gibbs energy and internal energy, respectively, rather than G and U which are used in this article.
( ∂ T ) P = − ( ∂ P ) T = 1 {\displaystyle (\partial T)_{P}=-(\partial P)_{T}=1} ( ∂ V ) P = − ( ∂ P ) V = ( ∂ V ∂ T ) P {\displaystyle (\partial V)_{P}=-(\partial P)_{V}=\left({\frac {\partial V}{\partial T}}\right)_{P}} ( ∂ S ) P = − ( ∂ P ) S = C p T {\displaystyle (\partial S)_{P}=-(\partial P)_{S}={\frac {C_{p}}{T}}} ( ∂ U ) P = − ( ∂ P ) U = C P − P ( ∂ V ∂ T ) P {\displaystyle (\partial U)_{P}=-(\partial P)_{U}=C_{P}-P\left({\frac {\partial V}{\partial T}}\right)_{P}} ( ∂ H ) P = − ( ∂ P ) H = C P {\displaystyle (\partial H)_{P}=-(\partial P)_{H}=C_{P}} ( ∂ G ) P = − ( ∂ P ) G = − S {\displaystyle (\partial G)_{P}=-(\partial P)_{G}=-S} ( ∂ A ) P = − ( ∂ P ) A = − S − P ( ∂ V ∂ T ) P {\displaystyle (\partial A)_{P}=-(\partial P)_{A}=-S-P\left({\frac {\partial V}{\partial T}}\right)_{P}} ( ∂ V ) T = − ( ∂ T ) V = − ( ∂ V ∂ P ) T {\displaystyle (\partial V)_{T}=-(\partial T)_{V}=-\left({\frac {\partial V}{\partial P}}\right)_{T}} ( ∂ S ) T = − ( ∂ T ) S = ( ∂ V ∂ T ) P {\displaystyle (\partial S)_{T}=-(\partial T)_{S}=\left({\frac {\partial V}{\partial T}}\right)_{P}} ( ∂ U ) T = − ( ∂ T ) U = T ( ∂ V ∂ T ) P + P ( ∂ V ∂ P ) T {\displaystyle (\partial U)_{T}=-(\partial T)_{U}=T\left({\frac {\partial V}{\partial T}}\right)_{P}+P\left({\frac {\partial V}{\partial P}}\right)_{T}} ( ∂ H ) T = − ( ∂ T ) H = − V + T ( ∂ V ∂ T ) P {\displaystyle (\partial H)_{T}=-(\partial T)_{H}=-V+T\left({\frac {\partial V}{\partial T}}\right)_{P}} ( ∂ G ) T = − ( ∂ T ) G = − V {\displaystyle (\partial G)_{T}=-(\partial T)_{G}=-V} ( ∂ A ) T = − ( ∂ T ) A = P ( ∂ V ∂ P ) T {\displaystyle (\partial A)_{T}=-(\partial T)_{A}=P\left({\frac {\partial V}{\partial P}}\right)_{T}} ( ∂ S ) V = − ( ∂ V ) S = C P T ( ∂ V ∂ P ) T + ( ∂ V ∂ T ) P 2 {\displaystyle (\partial S)_{V}=-(\partial V)_{S}={\frac {C_{P}}{T}}\left({\frac {\partial V}{\partial P}}\right)_{T}+\left({\frac {\partial V}{\partial T}}\right)_{P}^{2}} ( ∂ U ) V = − ( ∂ V ) U = C P ( ∂ V ∂ P ) T + T ( ∂ V ∂ T ) P 2 {\displaystyle (\partial U)_{V}=-(\partial V)_{U}=C_{P}\left({\frac {\partial V}{\partial P}}\right)_{T}+T\left({\frac {\partial V}{\partial T}}\right)_{P}^{2}} ( ∂ H ) V = − ( ∂ V ) H = C P ( ∂ V ∂ P ) T + T ( ∂ V ∂ T ) P 2 − V ( ∂ V ∂ T ) P {\displaystyle (\partial H)_{V}=-(\partial V)_{H}=C_{P}\left({\frac {\partial V}{\partial P}}\right)_{T}+T\left({\frac {\partial V}{\partial T}}\right)_{P}^{2}-V\left({\frac {\partial V}{\partial T}}\right)_{P}} ( ∂ G ) V = − ( ∂ V ) G = − V ( ∂ V ∂ T ) P − S ( ∂ V ∂ P ) T {\displaystyle (\partial G)_{V}=-(\partial V)_{G}=-V\left({\frac {\partial V}{\partial T}}\right)_{P}-S\left({\frac {\partial V}{\partial P}}\right)_{T}} ( ∂ A ) V = − ( ∂ V ) A = − S ( ∂ V ∂ P ) T {\displaystyle (\partial A)_{V}=-(\partial V)_{A}=-S\left({\frac {\partial V}{\partial P}}\right)_{T}} ( ∂ U ) S = − ( ∂ S ) U = P C P T ( ∂ V ∂ P ) T + P ( ∂ V ∂ T ) P 2 {\displaystyle (\partial U)_{S}=-(\partial S)_{U}={\frac {PC_{P}}{T}}\left({\frac {\partial V}{\partial P}}\right)_{T}+P\left({\frac {\partial V}{\partial T}}\right)_{P}^{2}} ( ∂ H ) S = − ( ∂ S ) H = − V C P T {\displaystyle (\partial H)_{S}=-(\partial S)_{H}=-{\frac {VC_{P}}{T}}} ( ∂ G ) S = − ( ∂ S ) G = − V C P T + S ( ∂ V ∂ T ) P {\displaystyle (\partial G)_{S}=-(\partial S)_{G}=-{\frac {VC_{P}}{T}}+S\left({\frac {\partial V}{\partial T}}\right)_{P}} ( ∂ A ) S = − ( ∂ S ) A = P C P T ( ∂ V ∂ P ) T + P ( ∂ V ∂ T ) P 2 + S ( ∂ V ∂ T ) P {\displaystyle (\partial A)_{S}=-(\partial S)_{A}={\frac {PC_{P}}{T}}\left({\frac {\partial V}{\partial P}}\right)_{T}+P\left({\frac {\partial V}{\partial T}}\right)_{P}^{2}+S\left({\frac {\partial V}{\partial T}}\right)_{P}} ( ∂ H ) U = − ( ∂ U ) H = − V C P + P V ( ∂ V ∂ T ) P − P C P ( ∂ V ∂ P ) T − P T ( ∂ V ∂ T ) P 2 {\displaystyle (\partial H)_{U}=-(\partial U)_{H}=-VC_{P}+PV\left({\frac {\partial V}{\partial T}}\right)_{P}-PC_{P}\left({\frac {\partial V}{\partial P}}\right)_{T}-PT\left({\frac {\partial V}{\partial T}}\right)_{P}^{2}} ( ∂ G ) U = − ( ∂ U ) G = − V C P + P V ( ∂ V ∂ T ) P + S T ( ∂ V ∂ T ) P + S P ( ∂ V ∂ P ) T {\displaystyle (\partial G)_{U}=-(\partial U)_{G}=-VC_{P}+PV\left({\frac {\partial V}{\partial T}}\right)_{P}+ST\left({\frac {\partial V}{\partial T}}\right)_{P}+SP\left({\frac {\partial V}{\partial P}}\right)_{T}} ( ∂ A ) U = − ( ∂ U ) A = P ( C P + S ) ( ∂ V ∂ P ) T + P T ( ∂ V ∂ T ) P 2 + S T ( ∂ V ∂ T ) P {\displaystyle (\partial A)_{U}=-(\partial U)_{A}=P(C_{P}+S)\left({\frac {\partial V}{\partial P}}\right)_{T}+PT\left({\frac {\partial V}{\partial T}}\right)_{P}^{2}+ST\left({\frac {\partial V}{\partial T}}\right)_{P}} ( ∂ G ) H = − ( ∂ H ) G = − V ( C P + S ) + T S ( ∂ V ∂ T ) P {\displaystyle (\partial G)_{H}=-(\partial H)_{G}=-V(C_{P}+S)+TS\left({\frac {\partial V}{\partial T}}\right)_{P}} ( ∂ A ) H = − ( ∂ H ) A = − [ S + P ( ∂ V ∂ T ) P ] [ V − T ( ∂ V ∂ T ) P ] + P C P ( ∂ V ∂ P ) T {\displaystyle (\partial A)_{H}=-(\partial H)_{A}=-\left[S+P\left({\frac {\partial V}{\partial T}}\right)_{P}\right]\left[V-T\left({\frac {\partial V}{\partial T}}\right)_{P}\right]+PC_{P}\left({\frac {\partial V}{\partial P}}\right)_{T}} ( ∂ A ) G = − ( ∂ G ) A = − S [ V + P ( ∂ V ∂ P ) T ] − P V ( ∂ V ∂ T ) P {\displaystyle (\partial A)_{G}=-(\partial G)_{A}=-S\left[V+P\left({\frac {\partial V}{\partial P}}\right)_{T}\right]-PV\left({\frac {\partial V}{\partial T}}\right)_{P}}
