# 19.13: The Temperature Dependence of ΔH

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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

$dH = C_p \,dT \nonumber$

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

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

##### 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*}

19.13: The Temperature Dependence of ΔH is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.