Chapter 16, Problem 16.122P

(a)

Interpretation Introduction

Interpretation:

A Plot of $\mu vs.S$ for $S$ between $0.0and1.0kg/m^3$ has to be drawn.

Concept Introduction:

The effect of a substrate concentration on first-order growth rate of a microbial population follows the Monod equation:

$\mu =\frac{{\mu }_{\max }S}{K_s+S}$

Where,

$\begin{array}{l}\mu ={\text{first-order growth rate (s}}^{\text{-1}}\text{)}\\ \\ {\mu }_{\max }=Maximumgrowthrate\\ \\ S=Substrateconcentration\\ \\ K_s=\text{the value of S that gives one-half of the maximum growth rate}\\ {\text{(in kg/m}}^{\text{3}}\text{)}\end{array}$

(b)

Interpretation Introduction

Interpretation:

The density after one hour if the initial $S$ is $0.30kg/m^3$ with given initial population density of $5.0\times {10}^{3}cells/m^3$  has to be determined.

Concept Introduction:

The effect of a substrate concentration on first-order growth rate of a microbial population follows the Monod equation:

$\mu =\frac{{\mu }_{\max }S}{K_s+S}$

Where,

$\begin{array}{l}\mu ={\text{first-order growth rate (s}}^{\text{-1}}\text{)}\\ \\ {\mu }_{\max }=Maximumgrowthrate\\ \\ S=Substrateconcentration\\ \\ K_s=\text{the value of S that gives one-half of the maximum growth rate}\\ {\text{(in kg/m}}^{\text{3}}\text{)}\end{array}$

The first-order reaction kinetics,

$\ln {\left[A\right]}_t=\ln {\left[A\right]}_0+k_{0}t$

Where,

$\begin{array}{l}{\left[\text{A}\right]}_{\text{t}}\text{=}\text{Concentration}\text{of}\text{reactant}\text{at}\text{time}\text{t}\\ \\ {\left[\text{A}\right]}_{\text{0}}\text{=}\text{Initial}\text{Concentration}\text{of}\text{reactant}\\ \\ {\text{k}}_{\text{0}}\text{=}\text{First-order}\text{rate}\text{constant}\end{array}$

(c)

Interpretation Introduction

Interpretation:

The density after one hour if the initial $S$ is $0.70kg/m^3$ has to be determined.

Concept Introduction:

The effect of a substrate concentration on first-order growth rate of a microbial population follows the Monod equation:

$\mu =\frac{{\mu }_{\max }S}{K_s+S}$

Where,

$\begin{array}{l}\mu ={\text{first-order growth rate (s}}^{\text{-1}}\text{)}\\ \\ {\mu }_{\max }=Maximumgrowthrate\\ \\ S=Substrateconcentration\\ \\ K_s=\text{the value of S that gives one-half of the maximum growth rate}\\ {\text{(in kg/m}}^{\text{3}}\text{)}\end{array}$

The first-order reaction kinetics,

$\ln {\left[A\right]}_t=\ln {\left[A\right]}_0+k_{0}t$

Where,

$\begin{array}{l}{\left[A\right]}_t=\text{Concentration}\text{of}\text{reactant}\text{at}\text{time}\text{t}\\ \\ {\left[A\right]}_0=\text{Initial}\text{Concentration}\text{of}\text{reactant}\\ \\ k_0=First-\text{order}\text{rate}\text{constant}\end{array}$