Question

Uranium-238 forms thorium-234 after radioactive decay and has a half-life of 4.5 x 109 years. How many years will it take to decay 75% of the initial amount? (a) 4.5 x 109 years (b) 9 x 1010 years (c) 4.5 x 1010 years (d) 9 x 109 years​

Answer 1

Answer:

Your answer is (a)

$4.5 \times 109$

Each decay has its own characteristic half-life. In the first step, uranium-238 decays by alpha emission to thorium-234 with a half-life of 4.5×109 years. This decreases its atomic number by two. The thorium-234 rapidly decays by beta emission to protactinium-234 (t1/2= 24.1 days).

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Answer 2

Given:

A Uranium- 238 forms Thorium-234 after a radioactive decay with a half-life of 4.5 x 109 years.

To Find:

The time that is taken by the sample to decay by 75% of the initial amount.

Solution:

1. It is given that the half-life of the sample is 4.5 x 109 years.

• The half-life of a radioactive is defined as the time taken by the sample to decay by 50% of the initial value.
• Radioactive decays are always First order reactions.
• Let x grams of a sample have a half-life of t years, The time taken by the sample to decay by 50% is t years. Now, x/2 grams of the sample is present. The time that is taken by the sample to decay 50% of the present value(75% of the initial value) is again t years because the reaction is of order 1.

2. It is given that the reaction decays 75% of the initial amount. Therefore, the time taken will be:

• The time that is taken to reach 50% of the initial value + time taken to decay 50% after the first decay. (50% +25%)

=> Time taken = 4.5 x 109 + 4.5 x 109 years,

=> Time taken = 2 x 4.5 x 109 years,

=> Time taken = 9 x 109 years.

Therefore, the time taken by the sample to decay by 75 % of the initial sample is 9 x 109 years. Option D is the correct answer.