Chapter 23: Q43P (page 631)

23-43. An infinitely sharp zone of solute is placed at the center of a column at time $t = 0$ . After diffusion for time $t_{1}$ , the standard deviation of the Gaussian band is $1.0 mm$ . After $20 \min$ more, at time $t_{2}$ , the standard deviation is $2.0 mm$ . What will be the width after another $20 \min$ , at time $t_{3}$ ?

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The solution is

$p_{3} = 2.646$

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An indefinitely sharp zone of solute was positioned at the center of a column at time $t = 0$ . After diffusion, the Gaussian band's standard deviation wasrole="math" localid="1654840869606" $1.0 mm$ for time $t_{1}$ . The standard deviation was $2.0 mm$ at time after $t_{2}$ after $20 \min$ . At time $t_{3}$ , we'll calculate the width after another $20 \min$ .

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Standard deviation is proportional to $\sqrt{t}$ and it is given:

p_{1} = 1 mm p_{2} = 2 mm t_{2} = t_{1} + 20 t_{3} = t_{1} + 40 t_{1^{'}} p_{3} = ?

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To find the value of $t_{1}$ , we'll first compare the following expressions for $p_{1}$ and $p_{2}$ :

$\frac{p_{1}}{p_{2}} = \sqrt{\frac{t_{1}}{t_{1} + 20}} \frac{1}{2} = \sqrt{\frac{t_{1}}{t_{1} + 20}}$

We'll now subtract the root from the equation and find $t_{1}$

${(\frac{1}{2})}^{2} = \frac{t_{1}}{t_{1} + 20} \frac{1}{4} t_{1} + 5 = t_{1} 5 = t_{1} - \frac{1}{4} t_{1} t_{1} = \frac{20}{3}$

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The value of $p_{3}$ is then calculated by comparing the values of $p_{2}$ and $p_{3}$ :

$\frac{p_{1}}{p_{3}} = \sqrt{\frac{t_{1}}{t_{1} + 40}} \frac{1}{p_{3}} = \sqrt{\frac{\frac{20}{3}}{\frac{20}{3} + 40}} \frac{1}{p_{3}} = 0.378 p_{3} = \frac{1}{0.378} p_{3} = 2.646$

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23-43. An infinitely sharp zone of solute is placed at the center of a column at time t=0. After diffusion for timet1 , the standard deviation of the Gaussian band is 1.0mm. After20min more, at time t2, the standard deviation is2.0mm . What will be the width after another20min , at time t3?

Short Answer

Step by step solution

of 4

of 4

of 4

of 4

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Chemistry Textbooks

Chemical Reactions

Organic Chemistry

Ionic and Molecular Compounds

Kinetics

Chemical Analysis

Nuclear Chemistry

Study anywhere. Anytime. Across all devices.