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Chapter 5: Q. 21 (page 351)

A 50-mg sample of a radioactive substance decays to 34 mg after 30 days. How long will it take for there to be 2 mg remaining?

Short Answer

Expert verified

The time taken is approximately 250.5 days.

Step by step solution

01

Given Information 

It is given that:

Initial amount is 50mg.

Amount after 30 days is 34mg.

Amount to be determined when it will be 2mg.

02

Initial equation and assumption

It is assumed that the radioactive substance decay is exponential, therefore the equation can be expressed as:

At=A0ebt

where At is the amount after time t, A0 is initial amount and b is time factor.

03

Determining time factor

As per the question,

A0 = 50mg, A30 = 34mg

Using the formula,

A30=A0eb(30)34=50e30b3450=e30b

Taking log both sides

30b=ln1725b=130ln1725b=-0.01285

04

Calculating the time required

Let the time to be determined is t days

Given At = 2mg

Putting it in the equation we get,

At=A0ebt2=50e-0.01285脳t250=e-0.01285t

Taking log on both sides

-0.01285t=ln125t=-10.01285脳-3.21887t鈮�250.5days

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