Skip to main content
Chemistry LibreTexts

4.6: Useful Definitions and Relationships

  • Page ID
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    In this chapter (and in the previous chapter), several useful definitions have been stated.

    Toolbox of useful Relationships

    The following “measurable quantities” have been defined:

    • Heat Capacities: \[ C_V \equiv \left( \dfrac{\partial U}{\partial T} \right)_V \] and\[ C_p \equiv \left( \dfrac{\partial H}{\partial T} \right)_p \]
    • Coefficient of Thermal Expansion: \[ \alpha \equiv \left( \dfrac{\partial V}{\partial T} \right)_p \] or \[ \left( \dfrac{\partial V}{\partial T} \right)_p = V \alpha\]
    • Isothermal Compressibility: \[ \kappa_T \equiv - \dfrac{1}{V} \left( \dfrac{\partial V}{\partial p} \right)_T \] or \[ \left( \dfrac{\partial V}{\partial p} \right)_T = -V \kappa _T\]

    The following relation has been derived:

    \[ \dfrac{ \alpha}{\kappa_T} = \left( \dfrac{\partial p}{\partial T} \right)_V \]

    And the following relationships were given without proof (yet!):

    \[\left( \dfrac{\partial U}{\partial V} \right)_T = T \left( \dfrac{\partial p}{\partial T} \right)_V - p\]


    \[\left( \dfrac{\partial H}{\partial p} \right)_T = - T \left( \dfrac{\partial V}{\partial T} \right)_p - p\]

    Together, these relationships and definitions make a powerful set of tools that can be used to derive a number of very useful expressions.

    Example \(\PageIndex{1}\): Expanding Thermodynamic Function

    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 \(H(p, T)\):

    \[ dH = \left( \dfrac{\partial H}{\partial p} \right)_T dp + \left( \dfrac{\partial H}{\partial T} \right)_p dT\]

    Divide by \(dV\) and constrain to constant \(T\) (to generate the partial of interest on the left):

    \[\left.\dfrac{dH}{dV} \right\rvert_{T}= \left( \dfrac{\partial H}{\partial p} \right)_T \left.\dfrac{dp}{dV} \right\rvert_{T} + \cancelto{0}{\left( \dfrac{\partial H}{\partial T} \right)_p \left.\dfrac{dT}{dV} \right\rvert_{T}}\]

    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”:

    \[ \left(\dfrac{\partial H}{\partial V} \right)_{T} = \left[ -T \left(\dfrac{\partial V}{\partial T} \right)_{p} +V \right] \left(\dfrac{\partial p}{\partial V} \right)_{T}\]

    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

    \[ \left(\dfrac{\partial H}{\partial V} \right)_{T} = T \left(\dfrac{\partial p}{\partial T} \right)_{V} + V \left(\dfrac{\partial p}{\partial V} \right)_{T}\]

    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”.

    \[ \left(\dfrac{\partial H}{\partial V} \right)_{T} = T \dfrac{\alpha}{\kappa_T} + V \dfrac{1}{-V \kappa_T}\]


    \[ \left(\dfrac{\partial H}{\partial V} \right)_{T} = \dfrac{1}{\kappa_T} ( T \alpha -1)\]

    Example \(\PageIndex{2}\): Isothermal Compression

    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 \(\kappa_T = 7.9 \times 10^{-5} atm^{-1}\)).


    Integrating the total differential of \(H\) at constant temperature results in

    \[ \Delta H = \left(\dfrac{\partial H}{\partial V} \right)_{T} \Delta V\]

    From Example \(\PageIndex{1}\), we know that

    \[ \left(\dfrac{\partial H}{\partial V} \right)_{T} = \dfrac{1}{\kappa_T} ( T \alpha -1)\]


    \[ \Delta H = \left [ \dfrac{1}{ 7.9 \times 10^{-5} atm^{-1}} \left( (300 \,K) (1.1 \times 10^{-3 }K^{-1}) -1 \right) \right] ( - 0.010\, L/mol ) \]

    \[ \Delta H = \left( 84.81 \, \dfrac{\cancel{atm\,L}}{mol}\right) \underbrace{\left(\dfrac{8.314\,J}{0.8206\, \cancel{atm\,L}}\right)}_{\text{conversion factor}} = 9590 \, J/mol\]


    • Patrick E. Fleming (Department of Chemistry and Biochemistry; California State University, East Bay)

    4.6: Useful Definitions and Relationships is shared under a not declared license and was authored, remixed, and/or curated by Patrick Fleming.