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Chemistry LibreTexts

19.13: The Temperature Dependence of ΔH

  • Page ID
    13711
  • It is often required to know thermodynamic functions (such as enthalpy) at temperatures other than those available from tabulated data. Fortunately, the conversion to other temperatures is not difficult.

    At constant pressure

    \[ dH = C_p \,dT\]

    And so for a temperature change from \(T_1\) to \(T_2\)

    \[ \Delta H = \int_{T_2}^{T_2} C_p\, dT \label{EQ1}\]

    Equation \ref{EQ1} is often referred to as Kirchhoff's Law. If \(C_p\) is independent of temperature, then

    \[\Delta H = C_p \,\Delta T \label{intH}\]

    If the temperature dependence of the heat capacity is known, it can be incorporated into the integral in Equation \ref{EQ1}. A common empirical model used to fit heat capacities over broad temperature ranges is

    \[C_p(T) = a+ bT + \dfrac{c}{T^2} \label{EQ15}\]

    After combining Equations \ref{EQ15} and \ref{EQ1}, the enthalpy change for the temperature change can be found obtained by a simple integration

    \[ \Delta H = \int_{T_1}^{T_2} \left(a+ bT + \dfrac{c}{T^2} \right) dT \label{EQ2}\]

    Solving the definite integral yields

    \[ \begin{align} \Delta H &= \left[ aT + \dfrac{b}{2} T^2 - \dfrac{c}{T} \right]_{T_1}^{T_2} \\ &= a(T_2-T_1) + \dfrac{b}{2}(T_2^2-T_1^2) - c \left( \dfrac{1}{T_2} - \dfrac{1}{T_1} \right) \label{ineq} \end{align} \]

    This expression can then be used with experimentally determined values of \(a\), \(b\), and \(c\), some of which are shown in the following table.

    Table \(\PageIndex{1}\): Empirical Parameters for the temperature dependence of \(C_p\)
    Substance a (J mol-1 K-1) b (J mol-1 K-2) c (J mol-1 K)
    C(gr) 16.86 4.77 x 10-3 -8.54 x 105
    CO2(g) 44.22 8.79 x 10-3 -8.62 x 105
    H2O(l) 75.29 0 0
    N2(g) 28.58 3.77 x 10-3 -5.0 x 104
    Pb(s) 22.13 1.172 x 10-2 9.6 x 104

    Example \(\PageIndex{1}\): Heating Lead

    What is the molar enthalpy change for a temperature increase from 273 K to 353 K for Pb(s)?

    Solution:

    The enthalpy change is given by Equation \ref{EQ1} with a temperature dependence \(C_p\) given by Equation \ref{EQ1} using the parameters in Table \(\PageIndex{1}\). This results in the integral form (Equation \ref{ineq}):

    \[ \Delta H = a(T_2-T_1) + \dfrac{b}{2}(T_2^2-T_1^2) - c \left( \dfrac{1}{T_2} - \dfrac{1}{T_1} \right) \nonumber\]

    when substituted with the relevant parameters of Pb(s) from Table \(\PageIndex{1}\).

    \[ \begin{align*} \Delta H = \,& (22.14\, \dfrac{J}{mol\,K} ( 353\,K - 273\,K) \\ & + \dfrac{1.172 \times 10^{-2} \frac{J}{mol\,K^2}}{2} \left( (353\,K)^2 - (273\,K)^2 \right) \\ &- 9.6 \times 10^4 \dfrac{J\,K}{mol} \left( \dfrac{1}{(353\,K)} - \dfrac{1}{(273\,K)} \right) \\ \Delta H = \, & 1770.4 \, \dfrac{J}{mol}+ 295.5\, \dfrac{J}{mol}+ 470.5 \, \dfrac{J}{mol} \\ = & 2534.4 \,\dfrac{J}{mol} \end {align*}\]

    For chemical reactions, the reaction enthalpy at differing temperatures can be calculated from

    \[\Delta H_{rxn}(T_2) = \Delta H_{rxn}(T_1) + \int_{T_1}^{T_2} \Delta C_p \Delta T\]

    Example \(\PageIndex{2}\): Enthalpy of Formation

    The enthalpy of formation of NH3(g) is -46.11 kJ/mol at 25 oC. Calculate the enthalpy of formation at 100 oC.

    Solution:

    \[\ce{N2(g) + 3 H2(g) \rightleftharpoons 2 NH3(g)} \nonumber \]

    with \(\Delta H \,(298\, K) = -46.11\, kJ/mol\)

    Compound Cp (J mol-1 K-1)
    N2(g) 29.12
    H2(g) 28.82
    NH3(g) 35.06

    \[ \begin{align*} \Delta H (373\,K) & = \Delta H (298\,K) + \Delta C_p\Delta T \\ & = -46110 \dfrac{J}{mol} \left[ 2 \left(35.06 \dfrac{J}{mol\,K}\right) - \left(29.12\, \dfrac{J}{mol\,K}\right) - 3\left(28.82\, \dfrac{J}{mol\,K}\right) \right] (373\,K -298\,K) \\ & = -49.5\, \dfrac{kJ}{mol} \end{align*}\]

    Contributors and Attributions

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