4.6: Useful Definitions and Relationships
- Page ID
- 84313
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In this chapter (and in the previous chapter), several useful definitions have been stated.
The following “measurable quantities” have been defined:
- Heat Capacities: \[ C_V \equiv \left( \dfrac{\partial U}{\partial T} \right)_V \nonumber \] and\[ C_p \equiv \left( \dfrac{\partial H}{\partial T} \right)_p \nonumber \]
- Coefficient of Thermal Expansion: \[ \alpha \equiv \left( \dfrac{\partial V}{\partial T} \right)_p \nonumber \] or \[ \left( \dfrac{\partial V}{\partial T} \right)_p = V \alpha \nonumber \]
- Isothermal Compressibility: \[ \kappa_T \equiv - \dfrac{1}{V} \left( \dfrac{\partial V}{\partial p} \right)_T \nonumber \] or \[ \left( \dfrac{\partial V}{\partial p} \right)_T = -V \kappa _T \nonumber \]
The following relation has been derived:
\[ \dfrac{ \alpha}{\kappa_T} = \left( \dfrac{\partial p}{\partial T} \right)_V \nonumber \]
And the following relationships were given without proof (yet!):
\[\left( \dfrac{\partial U}{\partial V} \right)_T = T \left( \dfrac{\partial p}{\partial T} \right)_V - p \nonumber \]
and
\[\left( \dfrac{\partial H}{\partial p} \right)_T = - T \left( \dfrac{\partial V}{\partial T} \right)_p - p \nonumber \]
Together, these relationships and definitions make a powerful set of tools that can be used to derive a number of very useful expressions.
Derive an expression for \(\left( \dfrac{\partial H}{\partial V} \right)_T\) in terms of measurable quantities.
Solution 1:
Begin by using the total differential of \(H(p, T)\):
\[ dH = \left( \dfrac{\partial H}{\partial p} \right)_T dp + \left( \dfrac{\partial H}{\partial T} \right)_p dT \nonumber \]
Divide by \(dV\) and constrain to constant \(T\) (to generate the partial of interest on the left):
\[\left.\dfrac{dH}{dV} \right\rvert_{T}= \left( \dfrac{\partial H}{\partial p} \right)_T \left.\dfrac{dp}{dV} \right\rvert_{T} + \cancelto{0}{\left( \dfrac{\partial H}{\partial T} \right)_p \left.\dfrac{dT}{dV} \right\rvert_{T}} \nonumber \]
The last term on the right will vanish (since \(dT = 0\) for constant \(T\)). After converting to partial derivatives
\[ \left(\dfrac{\partial H}{\partial V} \right)_{T} = \left( \dfrac{\partial H}{\partial p} \right)_T \left(\dfrac{\partial p}{\partial V} \right)_{T} \label{eq5} \]
This result is simply a demonstration of the “chain rule” on partial derivatives! But now we are getting somewhere. We can now substitute for \(\left(\dfrac{\partial H}{\partial V} \right)_{T}\) using our “toolbox of useful relationships”:
\[ \left(\dfrac{\partial H}{\partial V} \right)_{T} = \left[ -T \left(\dfrac{\partial V}{\partial T} \right)_{p} +V \right] \left(\dfrac{\partial p}{\partial V} \right)_{T} \nonumber \]
Using the distributive property of multiplication, this expression becomes
\[ \left(\dfrac{\partial H}{\partial V} \right)_{T} = -T \left(\dfrac{\partial V}{\partial T} \right)_{p}\left(\dfrac{\partial p}{\partial V} \right)_{T} + V \left(\dfrac{\partial p}{\partial V} \right)_{T} \label{eq7} \]
Using the cyclic permutation rule (Transformation Type II), the middle term of Equation \ref{eq7} can be simplified
\[ \left(\dfrac{\partial H}{\partial V} \right)_{T} = T \left(\dfrac{\partial p}{\partial T} \right)_{V} + V \left(\dfrac{\partial p}{\partial V} \right)_{T} \nonumber \]
And now all of the partial derivatives on the right can be expressed in terms of \(\alpha\) and \(\kappa_T\) (along with \(T\) and \(V\), which are also “measurable properties”.
\[ \left(\dfrac{\partial H}{\partial V} \right)_{T} = T \dfrac{\alpha}{\kappa_T} + V \dfrac{1}{-V \kappa_T} \nonumber \]
or
\[ \left(\dfrac{\partial H}{\partial V} \right)_{T} = \dfrac{1}{\kappa_T} ( T \alpha -1) \nonumber \]
Calculate \(\Delta H\) for the isothermal compression of ethanol which will decrease the molar volume by \(0.010\, L/mol\) at 300 K. (For ethanol, \(\alpha = 1.1 \times 10^{-3 }K^{-1}\) and \(\kappa_T = 7.9 \times 10^{-5} atm^{-1}\)).
Solution
Integrating the total differential of \(H\) at constant temperature results in
\[ \Delta H = \left(\dfrac{\partial H}{\partial V} \right)_{T} \Delta V \nonumber \]
From Example \(\PageIndex{1}\), we know that
\[ \left(\dfrac{\partial H}{\partial V} \right)_{T} = \dfrac{1}{\kappa_T} ( T \alpha -1) \nonumber \]
so
\[ \Delta H = \left [ \dfrac{1}{ 7.9 \times 10^{-5} atm^{-1}} \left( (300 \,K) (1.1 \times 10^{-3 }K^{-1}) -1 \right) \right] ( - 0.010\, L/mol ) \nonumber \]
\[ \Delta H = \left( 84.81 \, \dfrac{\cancel{atm\,L}}{mol}\right) \underbrace{\left(\dfrac{8.314\,J}{0.8206\, \cancel{atm\,L}}\right)}_{\text{conversion factor}} = 9590 \, J/mol \nonumber \]
Contributors
Patrick E. Fleming (Department of Chemistry and Biochemistry; California State University, East Bay)
cyclic permutation rule? Where is that on the Libraries?