Chapter 2, Problem 2.8PR

(a)

Interpretation Introduction

Interpretation:

A version of Kirchhoff’s law has to be determined to study the temperature-true-dependence of the internal energy of the reaction.

Concept Introduction:

Kirchhoff’s law:

This law states that the variation of change of enthalpy of a reaction with temperature at constant pressure is equal to the change in specific heat capacity at constant temperature of the system.

Internal energy:

Internal energy of a system is the total energy contained in the system.  It keeps an account for the loss and gain of energy of the system due to changes in internal state.  It is dependent on temperature and pressure.

It is denoted as $\text{U}$.

From 1^st law of thermodynamics,

$\text{ΔU}\text{=}\text{W}\text{+Q}$

Where,

$\text{ΔU}\text{=}$ Change in internal energy of the system

W is the energy transferred as the form of work to the system.

Q is the energy transferred as the form of heat to the system

Heat capacity at constant volume:

Specific heat capacity at constant volume is defined as the amount of heat required to increase the temperature by ${\text{1}}^{\text{ο}}\text{C}$ keeping the system at constant volume.

It is denoted as ${\text{C}}_{\text{v}}$

  ${\text{C}}_{\text{v}}\text{=}{\left(\frac{\partial \text{U}}{\partial \text{T}}\right)}_{\text{v}}$

$\begin{array}{l}\text{U}\text{=}\text{internal}\text{energy}\text{of}\text{the}\text{system}\\ \\ \text{T}\text{=}\text{temperature}\text{of}\text{the}\text{system}\end{array}$

Enthalpy:

Enthalpy is a property of a thermodynamic system that is equal to the sum of the internal energy of the system and the product of pressure and volume.  For a closed system where transfer of matter between system and surroundings is prohibited, for the processes that occur at constant pressure, the heat absorbed or released equals to the change in enthalpy.

It is denoted as $\text{H}$. The molar enthalpy is defined as the enthalpy per mole. It is denoted as ${\text{H}}_{\text{m}}$.

From thermodynamics,

$\text{H}\text{=}\text{U}\text{+}\text{PV}$

Where,

$\text{U}\text{=}$ Internal energy of the system

H is the enthalpy energy

P is the Pressure

V is the volume.

Specific heat capacity at constant pressure:

Specific heat capacity at constant pressure can be defined as the amount of energy required to increase the temperature of a substance by ${\text{1}}^{\text{ο}}\text{C}$.

It is denoted as ${\text{C}}_{\text{p}}$. The molar specific heat capacity is defined as the specific heat capacity per mole and it is denoted by ${\text{C}}_{\text{p,m}}$.

  ${\text{C}}_{\text{p}}={\left(\frac{\partial \text{H}}{\partial \text{T}}\right)}_{\text{p}}$

Explanation of Solution

The enthalpy and internal energy both are dependent on temperature. This can be as follows,

The enthalpy of the following reaction,

$\text{aA}\text{+bB}\underset{}{\overset{}{\rightleftarrows }}\text{cC}\text{+}\text{dD}$

is given by,

  $\begin{array}{l}\text{ΔH}\text{=}{\text{ΣH}}_{\text{products}}\text{-}{\text{ΣH}}_{\text{reactants}}\\ \\ \text{ΔH}\text{=}\left({\text{cH}}_{\text{C}}\text{+}{\text{dH}}_{\text{D}}\right)\text{-}\left({\text{aH}}_{\text{A}}\text{+}{\text{bH}}_{\text{B}}\right)\end{array}$

Now, differentiating the above equation with temperature at constant pressure,

  $\begin{array}{l}{\left[\frac{\partial \left(\text{ΔH}\right)}{\partial \text{T}}\right]}_{\text{P}}\text{=}\left(\text{c}{\left[\frac{\partial {\text{H}}_{\text{C}}}{\partial \text{T}}\right]}_{\text{P}}\text{+}\text{d}{\left[\frac{\partial {\text{H}}_{\text{D}}}{\partial \text{T}}\right]}_{\text{P}}\right)\text{-}\left(\text{a}{\left[\frac{\partial {\text{H}}_{\text{A}}}{\partial \text{T}}\right]}_{\text{P}}\text{+}\text{b}{\left[\frac{\partial {\text{H}}_{\text{B}}}{\partial \text{T}}\right]}_{\text{P}}\right)\\ \\ {\text{ΔC}}_{\text{P}}\text{=}\left(\text{c}{\left({\text{C}}_{\text{P}}\right)}_{\text{C}}\text{+}\text{d}{\left({\text{C}}_{\text{P}}\right)}_{\text{D}}\right)\text{-}\left(\text{a}{\left({\text{C}}_{\text{P}}\right)}_{\text{A}}\text{+}\text{b}{\left({\text{C}}_{\text{P}}\right)}_{\text{B}}\right)\end{array}$

Now from thermodynamics,

  ${\text{C}}_{\text{p}}={\left(\frac{\partial \text{H}}{\partial \text{T}}\right)}_{\text{p}}$

Here, ${\text{ΔC}}_{\text{P}}\text{=}{\left[\frac{\partial \left(\text{ΔH}\right)}{\partial \text{T}}\right]}_{\text{P}}$

This is the Kirchhoff’s law that concludes,

$\text{d}{\left(\text{ΔH}\right)}_{\text{P}}\text{=}{\text{ΔC}}_{\text{P}}\text{dT}$

Similarly,

For internal energy change of the same reaction can be given as,

  $\begin{array}{l}\text{ΔU}\text{=}{\text{ΣU}}_{\text{products}}\text{-}{\text{ΣU}}_{\text{reactants}}\\ \\ \text{ΔU}\text{=}\left({\text{cU}}_{\text{C}}\text{+}{\text{dU}}_{\text{D}}\right)\text{-}\left({\text{aU}}_{\text{A}}\text{+}{\text{bU}}_{\text{B}}\right)\end{array}$

Now, differentiating the above equation with temperature at constant volume,

  $\begin{array}{l}{\left[\frac{\partial \left(\text{ΔU}\right)}{\partial \text{T}}\right]}_{\text{P}}\text{=}\left(\text{c}{\left[\frac{\partial {\text{U}}_{\text{C}}}{\partial \text{T}}\right]}_{\text{P}}\text{+}\text{d}{\left[\frac{\partial {\text{U}}_{\text{D}}}{\partial \text{T}}\right]}_{\text{P}}\right)\text{-}\left(\text{a}{\left[\frac{\partial {\text{U}}_{\text{A}}}{\partial \text{T}}\right]}_{\text{P}}\text{+}\text{b}{\left[\frac{\partial {\text{U}}_{\text{B}}}{\partial \text{T}}\right]}_{\text{P}}\right)\\ \\ {\text{ΔC}}_{\text{v}}\text{=}\left(\text{c}{\left({\text{C}}_{\text{v}}\right)}_{\text{C}}\text{+}\text{d}{\left({\text{C}}_{\text{v}}\right)}_{\text{D}}\right)\text{-}\left(\text{a}{\left({\text{C}}_{\text{v}}\right)}_{\text{A}}\text{+}\text{b}{\left({\text{C}}_{\text{v}}\right)}_{\text{B}}\right)\end{array}$

Now from thermodynamics,

  ${\text{C}}_{\text{v}}\text{=}{\left(\frac{\partial \text{U}}{\partial \text{T}}\right)}_{\text{v}}$

Here, ${\text{ΔC}}_{\text{v}}\text{=}{\left[\frac{\partial \left(\text{ΔU}\right)}{\partial \text{T}}\right]}_{\text{P}}$

Thus the form of Kirchhoff’s law that shows the true temperature dependence of the internal energy of the reaction is,

  $\text{d}{\left(\text{ΔU}\right)}_{\text{v}}\text{=}{\text{ΔC}}_{\text{v}}\text{dT}$

(b)

Interpretation Introduction

Interpretation:

A more accurate form of Kirchhoff’s law has to be derived when ${\text{Δ}}_{\text{r}}{\text{C}}_{\text{P}}^{\text{θ}}\text{=}{\text{Δ}}_{\text{r}}\text{a}\text{+}{\text{Δ}}_{\text{r}}\text{bT}\text{+}\frac{{\text{Δ}}_{\text{r}}\text{c}}{{\text{T}}^{\text{2}}}$ is given and also in between two temperature range it has to be derived.

Concept Introduction:

Kirchhoff’s law:

This law states that the variation of change of enthalpy of a reaction with temperature at constant pressure is equal to the change in specific heat capacity at constant temperature of the system.

  $\text{d}{\left(\text{ΔH}\right)}_{\text{P}}\text{=}{\text{ΔC}}_{\text{P}}\text{dT}$

Enthalpy:

Enthalpy is a property of a thermodynamic system that is equal to the sum of the internal energy of the system and the product of pressure and volume.  For a closed system where transfer of matter between system and surroundings is prohibited, for the processes that occur at constant pressure, the heat absorbed or released equals to the change in enthalpy.

It is denoted as $\text{H}$. The molar enthalpy is defined as the enthalpy per mole. It is denoted as ${\text{H}}_{\text{m}}$.

From thermodynamics,

$\text{H}\text{=}\text{U}\text{+}\text{PV}$

Where,

$\text{U}\text{=}$ Internal energy of the system

H is the enthalpy energy

P is the Pressure

V is the volume.

Specific heat capacity at constant pressure:

Specific heat capacity at constant pressure can be defined as the amount of energy required to increase the temperature of a substance by ${\text{1}}^{\text{ο}}\text{C}$.

It is denoted as ${\text{C}}_{\text{p}}$. The molar specific heat capacity is defined as the specific heat capacity per mole and it is denoted by ${\text{C}}_{\text{p,m}}$.

  ${\text{C}}_{\text{p}}={\left(\frac{\partial \text{H}}{\partial \text{T}}\right)}_{\text{p}}$

Explanation of Solution

According to the given condition,

  ${\text{Δ}}_{\text{r}}{\text{C}}_{\text{P}}^{\text{θ}}\text{=}{\text{Δ}}_{\text{r}}\text{a}\text{+}{\text{Δ}}_{\text{r}}\text{bT}\text{+}\frac{{\text{Δ}}_{\text{r}}\text{c}}{{\text{T}}^{\text{2}}}$

According to Kirchhoff’s law,

  $\text{d}{\left(\text{ΔH}\right)}_{\text{P}}\text{=}{\text{ΔC}}_{\text{P}}\text{dT}$

Now applying integration,

${\displaystyle \int \text{d}{\left({\text{Δ}}_r\text{H}\right)}_{\text{P}}}\text{=}{\displaystyle \int \left({\text{Δ}}_{\text{r}}\text{a}\text{+}{\text{Δ}}_{\text{r}}\text{bT}\text{+}\frac{{\text{Δ}}_{\text{r}}\text{c}}{{\text{T}}^{\text{2}}}\right)\text{dT}}$

$\begin{array}{l}{\displaystyle \int \text{d}{\left({\text{Δ}}_{\text{r}}\text{H}\right)}_{\text{P}}}\text{=}{\displaystyle \int \left({\text{Δ}}_{\text{r}}\text{a}\text{+}{\text{Δ}}_{\text{r}}\text{bT}\text{+}\frac{{\text{Δ}}_{\text{r}}\text{c}}{{\text{T}}^{\text{2}}}\right)\text{dT}}\\ \\ {\displaystyle \int {\left({\text{Δ}}_{\text{r}}\text{H}\right)}_{\text{P}}}\text{=}{\text{Δ}}_{\text{r}}\text{a}{\displaystyle \int \text{dT}}\text{+}{\text{Δ}}_{\text{r}}\text{b}{\displaystyle \int \text{TdT}}\text{+}{\text{Δ}}_{\text{r}}\text{c}{\displaystyle \int \frac{\text{dT}}{{\text{T}}^{\text{2}}}}\\ \\ {\left({\text{Δ}}_{\text{r}}\text{H}\right)}_{\text{P}}\text{=}{\text{Δ}}_{\text{r}}\text{aT}\text{+}\frac{{\text{Δ}}_{\text{r}}\text{b}}{\text{2}}{\text{T}}^{\text{2}}\text{-}\frac{{\text{Δ}}_{\text{r}}\text{c}}{\text{T}}\end{array}$

Now taking the change between the temperature range ${\text{T}}_{\text{2}}$ to ${\text{T}}_1$ where ${\text{T}}_2>{\text{T}}_1$ and the change in enthalpy in these temperatures are respectively ${\text{Δ}}_r{\text{H}}_1$ and ${\text{Δ}}_r{\text{H}}_2$,

  $\begin{array}{l}{\displaystyle \underset{{\text{Δ}}_{\text{r}}{\text{H}}_{\text{1}}}{\overset{{\text{Δ}}_{\text{r}}{\text{H}}_{\text{2}}}{\int }}\text{d}{\left({\text{Δ}}_{\text{r}}\text{H}\right)}_{\text{P}}}\text{=}{\displaystyle \underset{{\text{T}}_{\text{1}}}{\overset{{\text{T}}_{\text{2}}}{\int }}\left({\text{Δ}}_{\text{r}}\text{a}\text{+}{\text{Δ}}_{\text{r}}\text{bT}\text{+}\frac{{\text{Δ}}_{\text{r}}\text{c}}{{\text{T}}^{\text{2}}}\right)\text{dT}}\\ \\ {\displaystyle \underset{{\text{Δ}}_{\text{r}}{\text{H}}_{\text{1}}}{\overset{{\text{Δ}}_{\text{r}}{\text{H}}_{\text{2}}}{\int }}\text{d}{\left({\text{Δ}}_{\text{r}}\text{H}\right)}_{\text{P}}}\text{=}{\text{Δ}}_{\text{r}}\text{a}{\displaystyle \underset{{\text{T}}_1}{\overset{{\text{T}}_2}{\int }}\text{dT}}+{\text{Δ}}_{\text{r}}\text{b}{\displaystyle \underset{{\text{T}}_1}{\overset{{\text{T}}_2}{\int }}\text{TdT}}+{\text{Δ}}_{\text{r}}\text{c}{\displaystyle \underset{{\text{T}}_1}{\overset{{\text{T}}_2}{\int }}\frac{\text{dT}}{{\text{T}}^{\text{2}}}}\\ \\ \left({\text{Δ}}_{\text{r}}{\text{H}}_{\text{2}}-{\text{Δ}}_{\text{r}}{\text{H}}_{\text{1}}\right)={\text{Δ}}_{\text{r}}\text{a}\left({\text{T}}_2-{\text{T}}_1\right)+\frac{{\text{Δ}}_{\text{r}}\text{b}}{2}\left({\text{T}}_2^2-{\text{T}}_1^2\right)-{\text{Δ}}_{\text{r}}\text{c}\left(\frac{1}{{\text{T}}_2}-\frac{1}{{\text{T}}_1}\right)\end{array}$

Thus, the more accurate form of Kirchhoff’s law in the given condition ${\text{Δ}}_{\text{r}}{\text{C}}_{\text{P}}^{\text{θ}}\text{=}{\text{Δ}}_{\text{r}}\text{a}\text{+}{\text{Δ}}_{\text{r}}\text{bT}\text{+}\frac{{\text{Δ}}_{\text{r}}\text{c}}{{\text{T}}^{\text{2}}}$ is ${\left({\text{Δ}}_{\text{r}}\text{H}\right)}_{\text{P}}\text{=}{\text{Δ}}_{\text{r}}\text{aT}\text{+}\frac{{\text{Δ}}_{\text{r}}\text{b}}{\text{2}}{\text{T}}^{\text{2}}\text{-}\frac{{\text{Δ}}_{\text{r}}\text{c}}{\text{T}}$.

In the temperature range of ${\text{T}}_{\text{2}}$ to ${\text{T}}_1$ the modified Kirchhoff’s law is,

  $\left({\text{Δ}}_{\text{r}}{\text{H}}_{\text{2}}-{\text{Δ}}_{\text{r}}{\text{H}}_{\text{1}}\right)={\text{Δ}}_{\text{r}}\text{a}\left({\text{T}}_2-{\text{T}}_1\right)+\frac{{\text{Δ}}_{\text{r}}\text{b}}{2}\left({\text{T}}_2^2-{\text{T}}_1^2\right)-{\text{Δ}}_{\text{r}}\text{c}\left(\frac{1}{{\text{T}}_2}-\frac{1}{{\text{T}}_1}\right)$