Chapter 5, Problem 11P

(a)

Summary Introduction

To calculate: The concentration of antigen when Y is 0.2 and ${\text{K}}_{\text{d}}{\text{= 5×10}}^{\text{-8}}$.

Introduction:

An antigen is any foreign body that evades the immune system of an individual. An antibody is a protein called immunoglobulin, which is induced by the antigen. These antibodies used by the individual’s immune system to neutralize the antigen.

Explanation of Solution

Calculation:

$\begin{array}{l}\text{According to the formula}\\ L=\frac{\theta K_d}{1-\theta }\\ \end{array}$

$\text{Y=θ = 0}{\text{.2 and K}}_{\text{d}}{\text{= 5×10}}^{\text{-8}}\text{M}$

$\begin{array}{l}\text{Substituting the formula:}\\ \text{= }\frac{\text{0}{\text{.2×10}}^{\text{-8}}}{\text{1-0}\text{.2}}\\ \text{= }\frac{\text{0}\text{.00000001}}{\text{0}\text{.8}}\\ =0.0000000125\text{= }1\times 10^{-8}\end{array}$

Conclusion

The concentration of antigen at which Y= 0.2 is $1\times 10^{-8}\text{M}$.

(b)

Summary Introduction

To calculate: The concentration of antigen when Y is 0.5 and ${\text{K}}_{\text{d}}{\text{= 5×10}}^{\text{-8}}\text{M}$.

Introduction:

An antigen is any foreign body that evades the immune system of an individual. An antibody is a protein called immunoglobulin, which is induced by the antigen. These antibodies used by the individual’s immune system to neutralize the antigen.

Explanation of Solution

Calculation:

$\begin{array}{l}\text{According to the formula}\\ L=\frac{\theta K_d}{1-\theta }\\ \end{array}$

$\text{Y=θ = 0}{\text{.5 and K}}_{\text{d}}{\text{= 5×10}}^{\text{-8}}\text{M}$

$\begin{array}{l}\text{Substituting the formula:}\\ \text{= }\frac{\text{0}{\text{.5×10}}^{\text{-8}}}{\text{1-0}\text{.5}}\\ \text{= }\frac{\text{0}\text{.000000025}}{\text{0}\text{.5}}\\ \text{= }5\times 10^{-8}\text{M}\end{array}$

Conclusion

The concentration of antigen at which Y= 0.5 is $5\times 10^{-8}\text{M}$.

Summary Introduction

(c)

To calculate: The concentration of antigen when Y is 0.6 and ${\text{K}}_{\text{d}}{\text{= 5×10}}^{\text{-8}}$.

Introduction:

An antigen is any foreign body that evades the immune system of an individual. An antibody is a protein called immunoglobulin, which is induced by the antigen. These antibodies used by the individual’s immune system to neutralize the antigen.

Explanation of Solution

Calculation:

$\begin{array}{l}\text{According to the formula}\\ L=\frac{\theta K_d}{1-\theta }\\ \end{array}$

$\text{Y=θ = 0}{\text{.6 and K}}_{\text{d}}{\text{= 5×10}}^{\text{-8}}\text{M}$

$\begin{array}{l}\text{Substituting the formula:}\\ \text{= }\frac{\text{0}{\text{.6×10}}^{\text{-8}}}{\text{1-0}\text{.6}}\\ \text{= }\frac{\text{0}\text{.00000003}}{\text{0}\text{.4}}\\ =0.00000008\text{= }8\times 10^{-8}\text{M}\end{array}$

Conclusion

The concentration of antigen at which Y= 0.6 is $8\times 10^{-8}\text{M}$/

Summary Introduction

(d)

To calculate: The concentration of antigen when Y is 0.8 and ${\text{K}}_{\text{d}}{\text{= 5×10}}^{\text{-8}}\text{M}$.

Introduction:

An antigen is any foreign body that evades the immune system of an individual. An antibody is a protein called immunoglobulin, which is induced by the antigen. These antibodies used by the individual’s immune system to neutralize the antigen.

Explanation of Solution

Calculation:

$\begin{array}{l}\text{According to the formula}\\ L=\frac{\theta K_d}{1-\theta }\\ \end{array}$

$\text{Y=θ = 0}{\text{.8 and K}}_{\text{d}}{\text{= 5×10}}^{\text{-8}}\text{M}$

$\begin{array}{l}\text{Substituting the formula:}\\ \text{= }\frac{\text{0}{\text{.8×10}}^{\text{-8}}}{\text{1-0}\text{.8}}\\ \text{= }\frac{\text{0}\text{.00000004}}{\text{0}\text{.2}}\\ =0.0000002\text{= }2\times 10^{-7}\end{array}$

Conclusion

The concentration of antigen at which Y= 0.8 is $2\times 10^{-7}\text{M}$.