Chapter 11, Problem 12Q

A.

Summary Introduction

To determine: Whether the given numbers in the question are in agreement or not.

Introduction: Lipid molecules are made up of hydrocarbons that help in different functions of a living cell. The plasma membrane is composed of a lipid bilayer.

Explanation of Solution

$\begin{array}{l}\begin{array}{l}\text{Given,}\hfill \\ \text{Diameter of lipid head is = 0}\text{.5 nm}\hfill \\ {\text{Time = 10}}^{\text{-7}}\hfill \end{array}\\ \text{Size}\text{of}\text{the}\text{bacterial}\text{cell}\text{=}\text{2}\text{μm}\end{array}$

$\text{Approximate}\text{time}\text{(t)}\text{is}\text{calculated}\text{as follows:}$

$\begin{array}{l}\text{t}\text{=}{\text{x}}^{\text{2}}\text{/2D}\mathrm{.....}\text{Equation}\text{(1)}\\ \text{Here,}\\ \text{t}\text{=}\text{Time}\text{taken}\\ \text{x}\text{=}\text{Average}\text{distance}\\ \text{D}\text{=}\text{Diffusion}\text{constant}\end{array}$$\begin{array}{l}\text{The total number of lipid molecules present in the bacterial cell}\text{=}\text{2}\text{μm/0}\text{.5}\text{nm}\\ \text{=4000}\text{lipid}\text{molecules}\end{array}$

    $\begin{array}{c}\text{Time required for a lipid molecule to move in a straight line to reach another end is = 4000}\text{×}{\text{10}}^{\text{-7}}\\ \text{=}\text{4}\text{×}{\text{10}}^{\text{-4}}\text{sec}\end{array}$

Generally, the lipid molecule considerably takes more time if it moves in a random motion.

Substituting the values of x and t in Equation (1), the diffusion constant (D) can be calculated as follows:

$\begin{array}{l}\text{D =}{\text{x}}^{\text{2}}\text{/2t}\\ \text{=(0}{\text{.5)}}^{\text{2}}\text{/2×}{\text{10}}^{\text{-7}}\\ \text{=}\text{1}\text{.25}\text{×}{\text{10}}^{\text{-8}}{\text{cm}}^{\text{2}}\text{/sec}\end{array}$

Substituting the obtained value of D in Equation (1) for a distance of 2 μm, the approximate time (t) is calculated as follows:

$\begin{array}{ccc}\text{t}& =& {(2\times {10}^{-4}\text{c}\text{m})}^2/2\times 1.25\times {10}^{-8}\\ & =& 1.6\text{seconds}\end{array}$

Conclusion

The approximate time required for a lipid molecule to move from one end to the other end in a 2-µm long bacterial cell is 1.6 seconds rather than 0.2 seconds. Thus, these two numbers are not in agreement.

Since the lipid molecules exchange their position in every 10^-7 seconds, it would appear that the distance must be exchanged more quickly than one second. However, the lipid exchange can occur in either direction or in opposite direction or it would slow down the overall production.

B1.

Summary Introduction

To calculate: The speed of a lipid molecule.

Introduction: Lipids are heterogeneous groups of compounds that are insoluble in water and soluble in ether, alcohol, and chloroform. Fats, phospholipids, and steroids are major groups of lipids.

Explanation of Solution

$\begin{array}{l}\text{Given,}\\ \text{Length}\text{of}\text{ping-pong}\text{ball}\text{=}\text{4}\text{cm =}\frac{4}{{10}^5}\text{km}\\ \text{Time}(\text{t})\text{=}{\text{10}}^{\text{-7}}\text{sec =}\frac{{10}^{-7}}{3600}\text{hours}\\ \text{ Distance}(\text{x})\text{=}\text{6}\text{m = 600cm}\end{array}$

$\begin{array}{c}\text{Speed}\text{=}\text{Distance/time}\\ \text{=}\frac{4}{{10}^5}\text{km/}\frac{{10}^{-7}}{3600}\text{hr}\\ =\frac{4}{{10}^5}\text{km}\times \frac{3600}{{10}^{-7}}\text{hr}\\ \text{=}\text{1,440,000 km/hr}\end{array}$

Conclusion

The speed of the lipid molecule is 1,440,000.

B2.

Summary Introduction

To calculate: The time required to move a ball from one side to the opposite side.

Introduction: Lipid molecules are made up of hydrocarbons that help in different functions of a living cell. The plasma membrane is composed of a lipid bilayer.

Explanation of Solution

Substituting the values of x and t in Equation (1), the diffusion constant (D) is calculated as follows:

$\begin{array}{l}\text{D =}{\text{x}}^{\text{2}}\text{/2t}\\ {\text{=(4)}}^{\text{2}}{\text{/2×10}}^{\text{-7}}\\ \text{=}8\text{×}{\text{10}}^{\text{7}}{\text{cm}}^{\text{2}}\text{/sec}\end{array}$

The time required to travel 6 m is given below:

$\begin{array}{l}\text{ t =}{\text{x}}^{\text{2}}\text{/2D}\\ \text{=}{(\text{600 cm)}}^{\text{2}}\text{/2}\times \text{(8}\text{×}{\text{10}}^7{\text{) cm}}^2/\sec \\ \text{ =}\text{0}\text{.00225}\text{seconds}\end{array}$

Conclusion

The time required for the lipid molecule to travel 6 m distance is 0.00225 seconds.