6.2: Other Relations
- Page ID
- 32031
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The TdS Equations
The entropy \(S\) is a function of the state of the system. We can take it to be a function of any two of the three variables (\(p\), \(T\), \(V\)). Taking \(S\) to be a function of \(p\) and \(T\), we write
\[T dS = T \biggl( \frac{\partial S}{\partial T} \biggr)_p dT \;+\; T \biggl( \frac{\partial S}{\partial p} \biggr)_T dp \]
For the first term on the right hand side, we can use
\[T \biggl( \frac{\partial S}{\partial T} \biggr)_p dT \;=\; \biggl( \frac{\partial Q}{\partial T} \biggr)_p dT \;=\; C_p \label{6.2.2} \]
where \(C_p\) is the specific heat at constant pressure. Further, using the last of the Maxwell relations, we can now write Equation \ref{6.2.2} as
\[Tds \;=\; C_pdT \;-\;T\biggl( \frac{\partial V}{\partial T} \biggr)_p dp \label{6.2.3}\]
The coefficient of volumetric expansion (due to heating) is defined by
\[ \alpha \;=\; \frac{1}{V} \biggl( \frac{\partial V}{\partial T} \biggr)_p \]
Equation \ref{6.2.3} can thus be rewritten as
\[T dS \;=\; C_p dT \;−\; \alpha T V dp \label{6.2.5}\]
If we take \(S\) to be a function of \(V\) and \(T\),
\[T dS \;=\; T \biggl( \frac{\partial S}{\partial T} \biggr)_V dT \;+\; T \biggl( \frac{\partial S}{\partial V} \biggr)_T dV \]
Again the first term on the right hand side can be expressed in terms of Cv, the specific heat at constant volume, using
\[ T \biggl( \frac{\partial S}{\partial T} \biggr)_V \;=\; C_v \]
Further using the Maxwell relations, we get
\[T dS \;=\; C_vdT \;+\; T \biggl( \frac{\partial p}{\partial T} \biggr)_V dV \label{6.2.8} \]
Equations \ref{6.2.3} (or \ref{6.2.5} and \ref{6.2.8}) are known as the \(T dS\) equations.
Equations for Specific Heats
Equating the two expressions for \(T dS\), we get
\[(C_p \;−\; C_v) dT \;=\; T \biggl[ \biggl( \frac{\partial p}{\partial T} \biggr)_V dV \;+\; \biggl( \frac{\partial V}{\partial T} \biggr)_pdp \biggr] \label{6.2.9}\]
By the equation of state, we can write \(p\) as a function of \(V\) and \(T\), so that
\[dp \;=\; \biggl( \frac{\partial p}{\partial T} \biggr)_V dT \;+\; \biggl( \frac{\partial p}{\partial V} \biggr)_T dV\]
Using this in Equation \ref{6.2.9}, we find
\[(C_p \;−\; C_v) dT \;=\; T \biggl[ \biggl( \frac{\partial p}{\partial T} \biggr)_V \;+\; \biggl( \frac{\partial V}{\partial T} \biggr)_p \biggl( \frac{\partial p}{\partial V} \biggr)_T \biggr]dV \;+\; T \biggl( \frac{\partial V}{\partial T} \biggr)_p \biggl( \frac{\partial p}{\partial T} \biggr)_V dT \label{6.2.11}\]
However, using Equation 6.1.9, taking \(X \;=\; V\), \(Y \;=\; p\) and \(Z \;=\; T\), we have
\[ \biggl( \frac{\partial p}{\partial T} \biggr)_V \;+\; \biggl( \frac{\partial V}{\partial T} \biggr)_p \biggl( \frac{\partial p}{\partial V} \biggr)_T \;=\; 0 \label{6.2.12}\]
Thus the coefficient of \(dV\) in Equation \ref{6.2.11} vanishes and we can simplify it as
\[C_p \;-\; C_v \;=\; T \biggl( \frac{\partial V}{\partial T} \biggr)_p \biggl( \frac{\partial p}{\partial T} \biggr)_V \;=\; -T \biggl[ \biggl( \frac{\partial V}{\partial T} \biggr)_p \biggr]^2 \biggl( \frac{\partial p}{\partial V} \biggr)_T \]
where we have used Equation \ref{6.2.12} again. We have already defined the coefficient of volumetric expansion \( \alpha \). The isothermal compressibility \( \kappa_T\) is defined by
\[ \frac{1}{ \kappa_T} \;=\; -V \biggl( \frac{\partial p}{\partial V} \biggr)_T \]
In terms of these we can express \(C_p \;−\; C_v\) as
\[ C_p \;-\; C_v \;=\; V \frac{ \alpha^2T}{ \kappa_T} \]
This equation is very useful in calculating \(C_v\) from measurements of \(C_p\) and \(\alpha\) and \( \kappa_T\). Further, for all substances, \( \kappa_T \;>\; 0\). Thus, we see from this equation that \(C_p ≥ C_v\). (The result \( \kappa_T \;>\; 0\) can be proved in statistical mechanics.)
In the \(T dS\) equations, if \(dT\), \(dV\) and \(dp\) are related adiabatically, \(dS \;=\; 0\) and we get
\[ C_p \;=\; T \biggl( \frac{\partial V}{\partial T} \biggr)_p \biggl( \frac{\partial p}{\partial T} \biggr)_S,\;\;\;\; C_v \;=\; -T \biggl( \frac{\partial p}{\partial T} \biggr)_V \biggl( \frac{\partial V}{\partial T} \biggr)_S \]
This gives
\[ \frac{C_p}{C_v} \;=\; - \biggl( \frac{\partial V}{\partial T} \biggr)_p \biggl( \frac{\partial p}{\partial T} \biggr)_S \biggl[ \biggl( \frac{\partial p}{\partial T} \biggr)_V \biggl( \frac{\partial V}{\partial T} \biggr)_S \biggr]^{-1} \]
We have the following relations among the terms involved in this expression,
\[ \biggl( \frac{\partial p}{\partial T} \biggr)_S \;=\; \biggl( \frac{\partial V}{\partial T} \biggr)_S \biggl( \frac{\partial p}{\partial V} \biggr)_S \;=\; \biggl( \frac{\partial V}{\partial T} \biggr)_S \frac{1}{V \kappa_S} \\ \biggl( \frac{\partial V}{\partial T} \biggr)_p \;=\; \biggl( \frac{\partial p}{\partial T} \biggr)_V \biggl( \frac{\partial T}{\partial p} \biggr)_V \biggl( \frac{\partial V}{\partial T} \biggr)_p \;=\; -\biggl( \frac{\partial p}{\partial T} \biggr)_V \frac{1}{\bigl( \frac{\partial p}{\partial V} \bigr)_T} \;=\; \biggl( \frac{\partial p}{\partial T} \biggr)_V V \kappa_T \]
Using these we find
\[ \frac{C_p}{C_v} \;=\; \frac{\kappa_T}{\kappa_S}\]
Going back to the Maxwell relations and using the expressions for \(T dS\), we find
\[dU \;=\; C_vdT \;+\; \biggl[T \biggl( \frac{\partial p}{\partial T} \biggr)_V -p \biggr]dV \\ dH \;=\; C_pdT \;+\; \biggl[V- T \biggl( \frac{\partial V}{\partial T} \biggr)_p \biggr]dp \]
these immediately yield the relations
\[ C_v \;=\; \biggl( \frac{\partial U}{\partial T} \biggr)_V,\;\;\;\;C_p \;=\; \biggl( \frac{\partial H}{\partial T} \biggr)_p \\ \biggl( \frac{\partial U}{\partial V} \biggr)_T \;=\; T \biggl( \frac{\partial p}{\partial T} \biggr)_V -p \;\;\;\;\;\;\; \biggl( \frac{\partial H}{\partial p} \biggr)_T \;=\; V- T \biggl( \frac{\partial V}{\partial T} \biggr)_p \]
Gibbs-Helmholtz Relation
Since the Helmholtz free energy is defined as \(F = U − T S\),
\[dF \;=\; dU \;−\; T dS \;−\; SdT \;=\; −S dT \;−\; p dV\]
This gives immediately
\[S \;=\; -\biggl( \frac{\partial F}{\partial T} \biggr)_V,\;\;\;\; p \;=\; -\biggl( \frac{\partial F}{\partial V} \biggr)_T \]
Using this equation for entropy, we find
\[U \;=\; F \;+\; T S \;=\; F \;−\; T \biggl( \frac{\partial F}{\partial T} \biggr)_V \]
This is known as the Gibbs-Helmholtz relation. If \(F\) is known as a function of \(T\) and \(V\), we can use these to obtain \(S\), \(p\) and \(U\). Thus, all thermodynamic variables can be obtained from \(F\) as a function of \(T\) and \(V\).