Give the definition of enthalpy
Write an expression for the differential of the enthalpy
Explain which thermodynamic variables are related to the derivatives of the enthalpy
Give the Maxwell relation that can be derived based on the definition of the enthalpy
Define the Helmholtz free energy
Write the differential of the Helmholtz free energy
Which thermodynamic variables are related to the derivatives of the Helmholtz free energy
Give the Maxwell relation that can be derived based on the definition of F
Define the Gibbs free energy, G
Write the differential of the Gibbs free energy
Which thermodynamic variables are related to the derivatives of the G
Give the Maxwell relation that can be derived based on the definition G
The enthalpy is defined as:
<aside> 💡 $H = E + PV$
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We can thus write the differential of the enthalpy as:
$\begin{aligned}\textrm{d}H & = \textrm{d}E + P\textrm{d}V + V\textrm{d}P = T\textrm{d}S - P\textrm{d}V + \mu \textrm{d}N + P\textrm{d}V + V\textrm{d}P \\& = T\textrm{d}S + \mu \textrm{d}N + V\textrm{d}P\end{aligned}$In the second step above we insert the result for dE that we obtained by combining the first and second laws.
Enthalpy is an exact differential so we can write:
$\textrm{d}H =\left( \frac{ \partial H }{\partial S} \right){P,N} \textrm{d}S + \left( \frac{ \partial H }{\partial P} \right){S,N} \textrm{d}P + \left( \frac{ \partial H }{\partial N} \right)_{P,S} \textrm{d}N$
Equating coefficients in these two expressions gives:
$T = \left( \frac{ \partial H}{\partial S} \right){P,N} \qquad V = \left( \frac{ \partial H}{\partial P} \right){S,N} \qquad \mu = \left( \frac{ \partial H }{\partial N} \right)_{P,S}$
Equality of second, crossed derivatives gives:
$\left(\frac{\partial T}{\partial P} \right){S,N} = \left(\frac{\partial V}{\partial S} \right){P,N}$
The Helmholtz free energy is defined as:
<aside> 💡 $F= E - TS$
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We can thus write the differential of the Helmholtz free energy as:
$\begin{aligned}\textrm{d}F & = \textrm{d}E - T\textrm{d}S - S\textrm{d}T = T\textrm{d}S - P\textrm{d}V + \mu \textrm{d}N - T\textrm{d}S - S\textrm{d}T \\& = - P\textrm{d}V + \mu \textrm{d}N - S\textrm{d}T\end{aligned}$In the second step above we insert the result for dE that we obtained by combining the first and second laws.
Helmholtz free energy is an exact differential so we can write:
$\textrm{d}F =\left( \frac{ \partial F }{\partial T} \right){V,N} \textrm{d}T + \left( \frac{ \partial F }{\partial V} \right){T,N} \textrm{d}V + \left( \frac{ \partial F }{\partial N} \right)_{V,T} \textrm{d}N$
Equating coefficients in these two expressions gives:
$S = -\left( \frac{ \partial F}{\partial T} \right){V,N} \qquad P = -\left( \frac{ \partial F}{\partial V} \right){T,N} \qquad \mu = \left( \frac{ \partial F }{\partial N} \right)_{V,T}$
Equality of second, crossed derivatives gives:
$\left(\frac{\partial P}{\partial T} \right){V,N} = \left(\frac{\partial S}{\partial V} \right){T,N}$
The Gibbs free energy is defined as:
<aside> 💡 $G = H - TS$
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We can thus write the differential of the Gibbs free energy as:
$\begin{aligned}\textrm{d}G & = \textrm{d}H - T\textrm{d}S - S\textrm{d}T = T\textrm{d}S + V\textrm{d}P + \mu \textrm{d}N - T\textrm{d}S - S\textrm{d}T \\& = V\textrm{d}P + \mu \textrm{d}N - S\textrm{d}T\end{aligned}$In the second step above we insert the result for dH that we obtained when we considered the entropy
Gibbs free energy is an exact differential so we can write:
$\textrm{d}G =\left( \frac{ \partial G }{\partial T} \right){P,N} \textrm{d}T + \left( \frac{ \partial G }{\partial P} \right){T,N} \textrm{d}V + \left( \frac{ \partial G }{\partial N} \right)_{P,T} \textrm{d}N$
Equating coefficients in these two expressions gives:
$S = -\left( \frac{ \partial G}{\partial T} \right){P,N} \qquad V = \left( \frac{ \partial G}{\partial P} \right){T,N} \qquad \mu = \left( \frac{ \partial G }{\partial N} \right)_{P,T}$
Equality of second, crossed derivatives gives:
$\left(\frac{\partial V}{\partial T} \right){P,N} = -\left(\frac{\partial S}{\partial P} \right){T,N}$
<aside> 📌 SUMMARY: We introduce reservoirs so that we can understand the behaviours of systems at constant pressure and constant temperature. New extensive thermodynamic variables that are known as thermodynamic potentials are required to understand systems in contact with reservoirs.
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