Chapter 2, Problem 2.10PR

(a)

Interpretation Introduction

Interpretation:

An expression for the difference between the change in enthalpy and the change in internal energy for a gas phase process has to be determined assuming all the species behave as perfect gas.

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 the internal energy of system.

$\text{W}\text{=}$ Energy transferred as the form of work to system

$\text{Q}\text{=}$ Energy transferred as the form of heat to system

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,

$\text{H}\text{=}$ enthalpy of system

$\text{U}\text{=}$ internal energy of system

$\text{P}\text{=}$ pressure of system

$\text{V}\text{=}$ volume of system

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 ${\text{C}}_{\text{p,m}}$.

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

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 ${\text{C}}_{\text{v}}$

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

$\text{U}\text{=}$ internal energy of system

$\text{T}\text{=}$ temperature

Explanation of Solution

Given that for perfect gas ${\text{C}}_{\text{p}}\text{=}\frac{\text{5}}{\text{2}}\text{R}$.

Applying the Kirchhoff’s law,

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

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

Thus by integration of both,

  $\begin{array}{l}\text{ΔH}=\Delta {\text{C}}_{\text{p}}\text{T}\\ \\ \text{ΔU}=\Delta {\text{C}}_{\text{v}}\text{T}\end{array}$

Thus the difference is,

  $\begin{array}{l}\text{ΔH}\text{-}\text{ΔU}\text{=}\left(\Delta {\text{C}}_{\text{p}}\text{-}\Delta {\text{C}}_{\text{v}}\right)\text{T}\\ \\ \text{ΔH}\text{-}\text{ΔU}\text{=}\left(\frac{\text{5}}{\text{2}}\text{-}\frac{\text{3}}{\text{2}}\right)\text{RT}\left[{\text{C}}_{\text{p}}\text{-}{\text{C}}_{\text{v}}\text{=}\text{R}\right]\\ \\ \text{ΔH}\text{-}\text{ΔU}\text{=}\text{RT}\end{array}$

The difference between the change in enthalpy and the change in internal energy for a gas phase process is $\text{RT}$.

(b)

Interpretation Introduction

Interpretation:

For ionization ${\text{ΔH}}_{\text{ion}}\text{-}{\text{ΔU}}_{\text{ion}}\text{=}\frac{\text{5}}{\text{2}}\text{RT}$ that has to be established.

Concept introduction:

Ionization enthalpy:

Ionization enthalpy is the energy required to remove an electron from a gaseous atom or ion

Ionization energy:

Ionization energy is defined as the minimum amount of energy required to remove the most loosely bound valence electron of an isolated neutral gas molecule.

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.

Explanation of Solution

According to Kirchhoff’s law,

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

Applying integration,

$\begin{array}{l}{\displaystyle \underset{\text{H(0)}}{\overset{\text{H(T)}}{\int }}{\left(\text{ΔH}\right)}_{\text{P}}}\text{=}{\text{ΔC}}_{\text{P}}{\displaystyle \underset{\text{0}}{\overset{\text{T}}{\int }}\text{dT}}\\ \\ \text{ΔH(T)}\text{-}\text{ΔH(0)}\text{=}{\text{ΔC}}_{\text{P}}\text{T}\\ \\ {\text{ΔH}}_{\text{ion}}\text{-}{\text{ΔU}}_{\text{ion}}\text{=}\frac{\text{5}}{\text{2}}\text{RT}\left[\begin{array}{l}{\text{ΔC}}_{\text{P}}\text{=}\frac{\text{5}}{\text{2}}\text{R}\\ {\text{ΔU}}_{\text{ion}}\text{=}\text{ionisation}\text{energy}\end{array}\right]\end{array}$

Thus it has been established that, ${\text{ΔH}}_{\text{ion}}\text{-}{\text{ΔU}}_{\text{ion}}\text{=}\frac{\text{5}}{\text{2}}\text{RT}$.

(c)

Interpretation Introduction

Interpretation:

Difference between the standard enthalpy of ionization of $\text{Ca(g)}$ to ${\text{Ca}}^{2+}\text{(g)}$ and the accompanying change in internal energy at ${\text{25}}^{\text{ο}}\text{C}$ has to be calculated.

Concept introduction:

Ionization enthalpy:

Ionization enthalpy is the energy required to remove an electron from a gaseous atom or ion

Answer to Problem 2.10PR

The difference between the standard enthalpy of ionization of $\text{Ca(g)}$ to ${\text{Ca}}^{2+}\text{(g)}$ and the accompanying change in internal energy at ${\text{25}}^{\text{ο}}\text{C}$ is $6193.93\text{J}{\text{mol}}^{-1}$.

Explanation of Solution

From part (b) it can be concluded that,

${\text{ΔH}}_{\text{ion}}\text{-}{\text{ΔU}}_{\text{ion}}\text{=}\frac{\text{5}}{\text{2}}\text{RT}$

According to the question, difference between the standard enthalpy of ionization of $\text{Ca(g)}$ to ${\text{Ca}}^{2+}\text{(g)}$ and the accompanying change in internal energy at ${\text{25}}^{\text{ο}}\text{C}$ has to be calculated.

Given that,

  $\begin{array}{l}\text{T}\text{=}\text{(25+273)}\text{K}\\ \\ \text{T}\text{=}298\text{K}\end{array}$

Hence,

  $\begin{array}{l}{\text{ΔH}}_{\text{ion}}\text{-}{\text{ΔU}}_{\text{ion}}\text{=}\frac{\text{5}}{\text{2}}\text{RT}\\ \\ {\text{ΔH}}_{\text{ion}}\text{-}{\text{ΔU}}_{\text{ion}}\text{=}\frac{\text{5}}{\text{2}}\text{×8}\text{.314}\text{J}{\text{mol}}^{-1}{\text{K}}^{-1}\text{×298}\text{K}\\ \\ {\text{ΔH}}_{\text{ion}}\text{-}{\text{ΔU}}_{\text{ion}}\text{=}6193.93\text{J}{\text{mol}}^{-1}\end{array}$

Hence the difference between the standard enthalpy of ionization of $\text{Ca(g)}$ to ${\text{Ca}}^{2+}\text{(g)}$ and the accompanying change in internal energy at ${\text{25}}^{\text{ο}}\text{C}$ is $6193.93\text{J}{\text{mol}}^{-1}$.

(d)

Interpretation Introduction

Interpretation:

For electron gain ${\text{ΔH}}_{\text{eg}}\text{-}{\text{ΔU}}_{\text{eg}}\text{=}-\frac{\text{5}}{\text{2}}\text{RT}$ has to be established.

Concept introduction:

Electron gain enthalpy:

Electron gain enthalpy can be defined as the amount of energy released when an electron is added to an isolated gaseous atom for electron gain enthalpy the sign convention is negative.

Electron affinity:

Electron affinity can be defined as the change in energy of a neutral atom in the gaseous phase when an electron is added to the atom to form the negative ion.

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.

Explanation of Solution

According to Kirchhoff’s law,

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

Applying integration,

  $\begin{array}{l}{\displaystyle \underset{\text{H(0)}}{\overset{\text{H(T)}}{\int }}{\left(\text{ΔH}\right)}_{\text{P}}}\text{=}{\text{ΔC}}_{\text{P}}{\displaystyle \underset{\text{0}}{\overset{\text{T}}{\int }}\text{dT}}\\ \\ \text{ΔH(T)}\text{-}\text{ΔH(0)}\text{=}{\text{ΔC}}_{\text{P}}\text{T}\\ \\ {\text{ΔH}}_{\text{eg}}\text{-}{\text{ΔU}}_{\text{eg}}\text{=}-\frac{\text{5}}{\text{2}}\text{RT}\left[\begin{array}{l}{\text{ΔC}}_{\text{P}}\text{=}-\frac{\text{5}}{\text{2}}\text{R due to electron gain}\\ {\text{ΔU}}_{\text{eg}}\text{=}\text{electron}\text{affinity}\end{array}\right]\end{array}$

Thus it has been established that, ${\text{ΔH}}_{\text{eg}}\text{-}{\text{ΔU}}_{\text{eg}}\text{=}-\frac{\text{5}}{\text{2}}\text{RT}$.

(e)

Interpretation Introduction

Interpretation:

Difference between the standard electron gain enthalpy of $\text{Br(g)}$ and the corresponding change in the internal energy at ${\text{25}}^{\text{ο}}\text{C}$ has to be calculated.

Concept introduction:

Electron gain enthalpy:

Electron gain enthalpy can be defined as the amount of energy released when an electron is added to an isolated gaseous atom for electron gain enthalpy the sign convention is negative.

Answer to Problem 2.10PR

The difference between the standard electron gain enthalpy of $\text{Br(g)}$ and the corresponding change in the internal energy at ${\text{25}}^{\text{ο}}\text{C}$ is $-6193.93\text{J}{\text{mol}}^{\text{-1}}$.

Explanation of Solution

From the part (d) it can be concluded that,

  ${\text{ΔH}}_{\text{eg}}\text{-}{\text{ΔU}}_{\text{eg}}\text{=}-\frac{\text{5}}{\text{2}}\text{RT}$

Now according to the question the difference between the standard electron gain enthalpy of $\text{Br(g)}$ and the corresponding change in the internal energy at ${\text{25}}^{\text{ο}}\text{C}$ has to be calculated.

Given that,

  $\begin{array}{l}\text{T}\text{=}\text{(25+273)}\text{K}\\ \\ \text{T}\text{=}298\text{K}\end{array}$

Hence,

  $\begin{array}{l}{\text{ΔH}}_{\text{eg}}\text{-}{\text{ΔU}}_{\text{eg}}\text{=}-\frac{\text{5}}{\text{2}}\text{RT}\\ \\ {\text{ΔH}}_{\text{eg}}\text{-}{\text{ΔU}}_{\text{eg}}\text{=}-\frac{\text{5}}{\text{2}}\text{×8}\text{.314}\text{J}{\text{mol}}^{\text{-1}}{\text{K}}^{\text{-1}}\text{×298}\text{K}\\ \\ {\text{ΔH}}_{\text{eg}}\text{-}{\text{ΔU}}_{\text{eg}}\text{=}-6193.93\text{J}{\text{mol}}^{\text{-1}}\end{array}$

Hence the difference between the standard electron gain enthalpy of $\text{Br(g)}$ and the corresponding change in the internal energy at ${\text{25}}^{\text{ο}}\text{C}$ is $-6193.93\text{J}{\text{mol}}^{\text{-1}}$.