Chapter 20, Problem 20.97P

(a)

Interpretation Introduction

Interpretation:

For the following dehydrogenation reaction, the $\Delta {{\rm H}}^{\text{o}},\Delta {\text{G}}_{\text{rxn}}^{\text{o}}\text{and}\Delta {\text{S}}_{\text{rxn}}^{\text{o}}$ values have to be founded at $298K$, using the given data’s.

Concept introduction:

Enthalpy is the amount energy absorbed or released in a process.

The enthalpy change in a system $\text{(Δ}{{\rm H}}_{\text{sys}}\text{)}$ can be calculated by the following equation.

  ${\text{ΔH}}_{\text{rxn}}\text{ = }{\displaystyle \sum {\text{ΔH}}^{\text{°}}{}_{\text{produdcts}}}\text{-}{\displaystyle \sum {\text{ΔH}}^{\text{°}}{}_{\text{reactants}}}$

Where,

  ${\text{ΔH}}_{\text{f}}^{\text{o}}{}_{\left(\text{reactants}\right)}$ is the standard enthalpy of the reactants

  ${\text{ΔH}}_{\text{f}}^{\text{o}}{}_{\left(\text{produdcts}\right)}$ is the standard enthalpy of the products

Entropy is the measure of randomness in the system.  Standard entropy change in a reaction is the difference in entropy of the products and reactants.  ${\text{(ΔS}}^{\text{°}}{}_{\text{rxn}}\text{)}$ can be calculated by the following equation.

  ${\text{ΔS}}^{\text{°}}{}_{\text{rxn}}\text{ = }{\displaystyle \sum \text{m }}{\text{S}}^{\text{°}}{}_{\text{Products}}-{\displaystyle \sum \text{n}}{\text{S}}^{\text{°}}{}_{\text{reactants}}$

Where,

  ${\text{S}}^{\text{°}}{}_{\text{reactants}}$ is the standard entropy of the reactants

  ${\text{S}}^{\text{°}}{}_{\text{Products}}$ is the standard entropy of the products

Standard entropy change in a reaction and entropy change in the system are same.

Free energy (Gibbs free energy) is the term that is used to explain the total energy content in a thermodynamic system that can be converted into work.  The free energy is represented by the letter $\text{G}$.  All spontaneous process is associated with the decrease of free energy in the system.  The standard free energy change ${\text{(ΔG}}^{\text{°}}{}_{\text{rxn}})$ is the difference in free energy of the reactants and products in their standard state.

${\text{ΔG}}^{\text{°}}{}_{\text{rxn}}\text{=}{\displaystyle \sum {\text{mΔG}}_{\text{f}}{}^{\text{°}}}\text{(Products)-}{\displaystyle \sum {\text{nΔG}}_{\text{f}}{}^{\text{°}}}\text{(Reactants)}$

(b)

Interpretation Introduction

Interpretation:

For the dehydration reaction, in which temperature become a spontaneous and temperature should be calculated.

Concept introduction:

Free energy (or) entropy change is the term that is used to explain the total energy content in a thermodynamic system that can be converted into work.  The free energy is represented by the letter G.  All spontaneous process is associated with the decrease of free energy in the system.  The equation given below helps us to calculate the change in free energy in a system.

  ${\text{ΔG}}^{\text{o}}\text{ = }\Delta {{\rm H}}^{\text{o}}\text{- T}\Delta {\text{S}}^{\text{o}}$

Entropy is a thermodynamic quantity, which is the measure of randomness in a system.  The term entropy is useful in explaining the spontaneity of a process. For all spontaneous process in an isolated system there will be an increase in entropy. Entropy is represented by the letter ‘S’.  It is a state function.  The change in entropy gives information about the magnitude and direction of a process.  The entropy changes associated with a phase transition reaction can be found by the following equation.

  $\text{Τ }\text{}\text{=}\text{}\text{}\frac{{\text{ΔΗ}}^{\text{o}}}{{\text{ΔS}}^{\text{o}}}$

Where,

  $\Delta {{\rm H}}^{\text{o}}$ is the change in enthalpy of the system

  ${\text{ΔG}}^{\text{o}}$ is the standard change in free energy of the system

  $\text{T}$ is the absolute value of the temperature

(c)

Interpretation Introduction

Interpretation:

For the dehydration reaction, the $\Delta G_{\text{rxn}}^{\text{o}}$ and equilibrium $K$ values have to be calculated at ${600}^{o}C$.

Concept introduction:

Free energy (or) entropy change is the term that is used to explain the total energy content in a thermodynamic system that can be converted into work.  The free energy is represented by the letter G.  All spontaneous process is associated with the decrease of free energy in the system.  The equation given below helps us to calculate the change in free energy in a system.

  ${\text{ΔG}}^{\text{o}}\text{ = }\Delta {{\rm H}}^{\text{o}}\text{- T}\Delta {\text{S}}^{\text{o}}$

Where,

  $\Delta {{\rm H}}^{\text{o}}$ is the change in enthalpy of the system

  ${\text{ΔG}}^{\text{o}}$ is the standard change in free energy of the system

  $\text{T}$ is the absolute value of the temperature

(d)

Interpretation Introduction

Interpretation:

$\Delta G$ at $50\%$  conversion has to be determined where $50\%$ of ethylbenzene is reacted.

Concept introduction:

Reaction quotient$\left(Q\right)$: Reaction quotient and equilibrium constant has same expression.  Reaction quotient is also the ratio between the concentrations of the reactant to product, but these concentrations are not necessarily the equilibrium concentrations.

        $\text{Q=}\frac{\left[\text{Product}\right]}{\left[\text{Reactant}\right]}$

The reaction quotient $\left(Q\right)$ is helpful in predicting the direction of the reaction.

• When $\text{Q}\text{>}{\text{K}}_{\text{c}}$, the reaction proceeds towards left to increase the concentration of the reactants.
• When $\text{Q}<{\text{K}}_{\text{c}}$, the reaction proceeds towards right to increase the concentration of the products.
• When $\text{Q}={\text{K}}_{\text{c}}$, the reaction is at equilibrium.

Free energy change$\text{ΔG}$: change in the free energy takes place while reactants convert to product where both are in standard state. It depends on the equilibrium constant $\text{K}$

  $\begin{array}{l}\text{ΔG =}{\text{ΔG}}^{\text{o}}\text{+}\text{RT}\ln \left(\text{K}\right)\\ {\text{ΔG}}^{\text{o}}\text{=}{\text{ΔH}}^{\text{o}}-{\text{TΔS}}^{\text{o}}\end{array}$

  Where,

  $\text{T}$ is the temperature

  $\text{ΔG}$ is the free energy

  ${\text{ΔG}}^{\text{o}}$ is standard free energy.

Mole fraction: Mole fraction ($X$) of any compound in a solution is the number of moles of the component divided by total number of moles making up a solution. It is denoted by $X$

  $Molefraction(X)=\frac{Molesofcomponent}{\begin{array}{l}Totalnumberofmoles\\ makingupthesolutions\end{array}}$