Chapter 8: Q8-3-19P (page 404)

If $λ_{1} = λ_{2} = λ$ in (3.10), then $\int e^{(λ_{2} - λ_{1}) t} d t = \int d t$ . Find $N_{2}$ for this case.

Short Answer

Expert verified

The value of the given expression is, $N_{2} = N_{0} λ t e^{- λ t}$ .

Step by step solution

Given Information.

The given two equations are,

$\int e^{(λ_{2} - λ_{1}) t} d t = \int d t λ_{1} = λ_{2} = λ$ .

The formula that is used.

The below equation defines the process of decaying is,

$N_{2} e^{λ_{2} t} = \int λ_{1} N_{0} e^{- λ_{1} t} e^{λ_{2} t} d t + c$ .

Here, $N_{0} =$ number of radium atoms at $t = 0$ ,

$N_{1 =}$ Number of radium atoms at time,

$N_{2 =}$ Number of radon atoms at time,

$λ_{1}$ and $λ_{2}$ decay constants for $R_{a}$ and $R_{n}$ .

Find the value of N2.

Substitute the given information in the formula.

$N_{2} e^{λ t} = λ N_{0} \int d t = λ N_{0} t + C N_{2} = (λ N_{0} t + C) e^{- λ t}$

Use the given initial condition in order to find the value of integration constant, that is, put $N_{2} = 0, t = 0$ which gives $C = 0$ .

Thus, the final answer is,

$N_{2} = 0, t = 0$ .

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If $λ_{1} = λ_{2} = λ$ in (3.10), then $\int e^{(λ_{2} - λ_{1}) t} d t = \int d t$ . Find $N_{2}$ for this case.

Short Answer

Step by step solution

Given Information.

The formula that is used.

Find the value of N2.

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If λ1=λ2=λin (3.10), then∫e(λ2-λ1)tdt=∫dt. FindN2for this case.

Short Answer

Step by step solution

Given Information.

The formula that is used.

Find the value of N2.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Linear Momentum

Engineering Physics

Energy Physics

Medical Physics

Electromagnetism

Measurements

Study anywhere. Anytime. Across all devices.

If $λ_{1} = λ_{2} = λ$ in (3.10), then $\int e^{(λ_{2} - λ_{1}) t} d t = \int d t$ . Find $N_{2}$ for this case.