Chapter 4, Problem 4.66P

Interpretation Introduction

(a)

Interpretation:

The formula for $K_{\text{e}}\left(T\right)$ is to be derived and $K_{\text{e}}\left(450\text{ K}\right)=0.0548{\text{ atm}}^{-2}$ is to be shown.

Concept introduction:

The Arrhenius equation for the equilibrium rate constant as a function of the absolute temperature is:

$K_{\text{e}}\left(T\right)=A_{\text{0}}\exp \left(-\frac{E_{\text{a}}}{RT\left(\text{K}\right)}\right)$ ...... (1)

Here, $E_{\text{a}}$ is the activation energy of the reaction, $R$ is the universal gas constant with the value $8.314\text{ J/mol}·\text{K}$ and $A_{\text{0}}$ is an Arrhenius constant.

The equation of a straight line on semi-log plot is, $\ln y=\text{m}x+\ln \text{b}$.

Given two points $\left(x_1,y_1\right)\text{ and }\left(x_2,y_2\right)$, the slope $\left(\text{m}\right)$ for this equation is calculated as:

$\text{m}=\frac{\ln y_2-\ln y_1}{x_2-x_1}$ ...... (2)

The value of intercept can be calculated using this slope and any one of the given points.

Interpretation Introduction

(b)

Interpretation:

The expressions for $n_{\text{A}},n_{\text{B}},\text{ and }{n}_{\text{C}}$ and then for $y_{\text{A}},y_{\text{B}},\text{ and }{y}_{\text{C}}$ in terms of $n_{\text{A0}},n_{\text{B0}},n_{\text{C0}}\text{ and }\xi$ is to be written. Also, an expression involving only $n_{\text{A0}},n_{\text{B0}},n_{\text{C0}}\text{,}P,T,\text{ and }{\xi }_{\text{e}}$ is to be derived.

Concept introduction:

For a single reaction system, the extent of reaction for each of the components present in the products can be estimated by the equation:

$n_i=n_{io}+v_i\xi$ ...... (5)

Here, $n_i$ is the final moles of the component $i$, $n_{io}$ is the initial moles of the component $i$, $v_i$ is the stoichiometric coefficient of the component $i$ in the reaction and $\xi$ is the extent of the reaction. $v_i$ is taken as negative for reactants and positive for products.

Mole fraction $\left(y_i\right)$ of any component is given by:

$y_i=\frac{n_i}{\sum n_i}$ ...... (6)

Here, $n_i$ is the moles of component $i$ and $\sum n_i$ is the total moles of all the components present in the mixture.

Interpretation Introduction

(c)

Interpretation:

The molar composition of the product stream and fractional conversion of $\text{CO}$ is to be calculated.

Concept introduction:

The formula for the fractional conversion $\left(f\right)$ is:

$f=\frac{\text{moles reacted}}{\text{moles fed}}$ ...... (13)

Interpretation Introduction

(d)

Interpretation:

Any physical constraints on ${\xi }_{\text{e}}$ associated with $n_{\text{A0}}=n_{\text{B0}}=1\text{ mol}$ is to be determined. Also, fractional conversion of $\text{CO}$ and total gram moles of methanol produced in the modified process is to be calculated.

Concept introduction:

The formula for the fractional conversion $\left(f\right)$ is:

$f=\frac{\text{moles reacted}}{\text{moles fed}}$ ...... (13)

Interpretation Introduction

(e)

Interpretation:

The calculations done in part (d) are to be repeated for $n_{\text{B0}}=2\text{ mol}$. Also, the subsequent increase in the fractional conversion of $\text{CO}$ is to be explained.

Concept introduction:

The formula for the fractional conversion $\left(f\right)$ is:

$f=\frac{\text{moles reacted}}{\text{moles fed}}$ ...... (13)

Interpretation Introduction

(f)

Interpretation:

The equations for $y_{\text{A}},y_{\text{B}},y_{\text{C}},\text{ and }{f}_{\text{A}}$ is to be written in terms of $y_{\text{A0}},y_{\text{B0}},P\text{ and }T$ in an excel spreadsheet and checked for the values given in part (c). Also, the effect of increase in the moles of ${\text{CH}}_3\text{OH}$ in the feed, temperature and pressure on $f_{\text{A}}$ is to be determined.

Concept introduction:

Mole fraction $\left(y_i\right)$ of any component is given by:

$y_i=\frac{n_i}{\sum n_i}$ ...... (6)

Here, $n_i$ is the moles of component $i$ and $\sum n_i$ is the total moles of all the components present in the mixture.

The formula for the fractional conversion $\left(f\right)$ is:

$f=\frac{\text{moles reacted}}{\text{moles fed}}$ ...... (13)