Heat equations

The heat equations or dQdQ equations (not to be confused with the heat partial differential equation) are a set of equations for the infinitesimal variation of heat dQdQ in a physical system. They are equations of two thermodynamic variable chosen among pressure PP, volume VV and temperature TT, with the third constrained by the system's equation of state.

The equations are

dQ(P,V)=(∂U∂P)VdP+[(∂U∂V)P+P]dVdQ(P,T)=[(∂U∂P)T+P(∂V∂P)T]dP+(∂H∂T)PdTdQ(V,T)=[(∂U∂V)T+P]dV+(∂U∂T)VdT\begin{align} dQ(P,V)&=\left( \frac{ \partial U }{ \partial P } \right)_{V}dP+\left[ \left( \frac{ \partial U }{ \partial V } \right)_{P}+P \right]dV \\ dQ(P,T)&=\left[ \left( \frac{ \partial U }{ \partial P } \right)_{T}+P\left( \frac{ \partial V }{ \partial P } \right)_{T} \right]dP+\left( \frac{ \partial H }{ \partial T } \right)_{P}dT \\ dQ(V,T)&=\left[ \left( \frac{ \partial U }{ \partial V } \right)_{T}+P \right]dV+\left( \frac{ \partial U }{ \partial T } \right)_{V}dT \end{align}

where UU is the internal energy of the system and HH is enthalpy. The subscripts mean taking a derivative while holding that variable constant. These derivatives are coefficients that are measured experimentally. Specifically, we have

CV=(∂U∂T)VCP=(∂H∂T)PC_{V}=\left( \frac{ \partial U }{ \partial T } \right)_{V}\qquad C_{P}=\left( \frac{ \partial H }{ \partial T } \right)_{P}

as the heat capacity for constant-volume and isobaric transformations.

Derivation

The heat equations are directly derived from the first law of thermodynamics. The internal energy must be

dU=dQ−dW=dQ−PdVdU=dQ-dW=dQ-PdV

with dQdQ heat and dW=PdVdW=PdV work. The equation of state allows us to lock one the three variables PP, VV or TT down, giving us three ways to express dUdU

dU(P,V)=(∂U∂P)VdP+(∂U∂V)PdVdU(P,T)=(∂U∂P)TdP+(∂U∂T)PdTdU(V,T)=(∂U∂V)TdV+(∂U∂T)VdT\begin{align} dU(P,V)&=\left( \frac{ \partial U }{ \partial P } \right)_{V}dP+ \left( \frac{ \partial U }{ \partial V } \right)_{P}dV \\ dU(P,T)&=\left( \frac{ \partial U }{ \partial P } \right)_{T}dP+\left( \frac{ \partial U }{ \partial T } \right)_{P}dT \\ dU(V,T)&=\left( \frac{ \partial U }{ \partial V } \right)_{T}dV+\left( \frac{ \partial U }{ \partial T } \right)_{V}dT \end{align}

for all permutations of the variables. Combining these with the first law dQ=dU+PdVdQ=dU+PdV gives us

dQ(P,V)=(∂U∂P)VdP+[(∂U∂V)P+P]dVdQ(P,T)=(∂U∂P)TdP+(∂U∂T)PdT+PdVdQ(V,T)=[(∂U∂V)T+P]dV+(∂U∂T)VdT\begin{align} dQ(P,V)&=\left( \frac{ \partial U }{ \partial P } \right)_{V}dP+\left[ \left( \frac{ \partial U }{ \partial V } \right)_{P}+P \right]dV \\ dQ(P,T)&=\left( \frac{ \partial U }{ \partial P } \right)_{T}dP+\left( \frac{ \partial U }{ \partial T } \right)_{P}dT+PdV \\ dQ(V,T)&=\left[ \left( \frac{ \partial U }{ \partial V } \right)_{T}+P \right]dV+\left( \frac{ \partial U }{ \partial T } \right)_{V}dT \end{align}

The first and third equations are done, but the second equation can't be used as is, because dVdV is still leftover from the work. We can get rid of it by expressing VV in function of the other two:

dV=(∂V∂P)TdP+(∂V∂T)PdTdV=\left( \frac{ \partial V }{ \partial P } \right)_{T}dP+\left( \frac{ \partial V }{ \partial T } \right)_{P}dT

and so

dQ(P,T)=[(∂U∂P)T+P(∂V∂P)T]dP+(∂(U+PV)∂T)PdTdQ(P,T)=\left[ \left( \frac{ \partial U }{ \partial P } \right)_{T}+P\left( \frac{ \partial V }{ \partial P } \right)_{T} \right]dP+\left( \frac{ \partial (U+PV) }{ \partial T } \right)_{P}dT

For convenience, we can use the enthalpy H=U+PVH=U+PV to simplify things:

dQ(P,T)=[(∂U∂P)T+P(∂V∂P)T]dP+(∂H∂T)PdTdQ(P,T)=\left[ \left( \frac{ \partial U }{ \partial P } \right)_{T}+P\left( \frac{ \partial V }{ \partial P } \right)_{T} \right]dP+\left( \frac{ \partial H }{ \partial T } \right)_{P}dT