Using Kirchhoff’s law to determine the standard enthalpy change of reaction at 298K

The experimental result of the standard change in enthalpy for the combustion reaction $CO(g)+\frac{1}{2}O_2(g)\rightarrow CO_2(g)$ is 281.93 kJmol^-1 at 1200K. Find its standard change in enthalpy at 298.15K given:

According to Hess’ law,

$\Delta H_r^{\circ}=\sum_Pv_P(\Delta H_f^{\circ})_P-\sum_Rv_R(\Delta H_f^{\circ})_R\;\;\;\;\;\;\;\;(69)$

At 1200K,

$281.93=\Delta H_{f,1200K}^{\circ}[CO_2(g)]-\Delta H_{f,1200K}^{\circ}[CO(g)]-\frac{1}{2}\Delta H_{f,1200K}^{\circ}[O_2(g)]\;\;\;\;\;\;\;\;(70)$

At 298.15K,

$\Delta H_{r,298.15K}^{\circ}=\Delta H_{f,298.15K}^{\circ}[CO_2(g)]-\Delta H_{f,298.15K}^{\circ}[CO(g)]-\frac{1}{2}\Delta H_{f,298.15K}^{\circ}[O_2(g)]\;\;\;\;\;\;\;\;(71)$

Subtracting eq71 from eq70,

$281.93-\Delta H_{r,298.15K}^{\circ}=\Delta H_{f,1200K}^{\circ}[CO_2(g)]-\Delta H_{f,298.15K}^{\circ}[CO_2(g)]-\{\Delta H_{f,1200K}^{\circ}[CO(g)]-\Delta H_{f,298.15K}^{\circ}[CO(g)]\}-\frac{1}{2}\{\Delta H_{f,1200K}^{\circ}[O_2(g)]-\Delta H_{f,298.15K}^{\circ}[O_2(g)]\}\;\;\;\;\;\;\;\;(72)$

Since all species in the reaction are in the gaseous state from 298.15K to 1200K, we can substitute eq66 in eq72,

$281.93-\Delta H_{r,298.15K}^{\circ}=\int_{1200}^{298.15}C_{p,CO_2}(T)dT-\int_{1200}^{298.15}C_{p,CO}(T)dT-\frac{1}{2}\int_{1200}^{298.15}C_{p,O_2}(T)dT\;\;\;\;\;\;\;\;(73)$

Substituting eq68 in the 1^st integral on the RHS of eq73,

$\int_{1200}^{298.15}C_{p,CO_2}(T)dT=\int_{1200}^{298.15}(a+bT+cT^2+dT^3)dT\;\;\;\;\;\;\;\;(74)$

Substituting the relevant values from the table above in eq73 and evaluating the integral (using the form expressed by eq74 for each integral in eq73),

$\int_{1200}^{298.15}C_{p,CO_2}(T)dT=-44.467\;kJmol^{-1}\;\;\;\;\;\;\;\;(75)$

$\int_{1200}^{298.15}C_{p,CO}(T)dT=-28.428\;kJmol^{-1}\;\;\;\;\;\;\;\;(76)$

$\frac{1}{2}\int_{1200}^{298.15}C_{p,O_2}(T)dT=-14.981\;kJmol^{-1}\;\;\;\;\;\;\;\;(77)$

and

$\Delta H_{r,298.15K}^{\circ}=+282.99\;kJmol^{-1}\;\;\;\;\;\;\;\;(78)$

Combining eq66 and eq69, for a reaction $A+B+\cdots\rightarrow M+N+\cdots$

$\Delta H_{r}^{\circ}(T_2)-\Delta H_{r}^{\circ}(T_1)=\sum_Pv_P\left[\int_{T_1}^{T_2}C_P(T)dT\right]_P-\sum_Rv_R\left[\int_{T_1}^{T_2}C_P(T)dT\right]_R\;\;\;\;\;\;\;\;(79)$

Using Kirchhoff’s law to determine the standard enthalpy change of reaction at 298K

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