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A half-life is the amount of time that it takes for a radioactive substance to decay to one half of its original quantity.
Suppose radioactive decay causes 300 grams of a substance to decrease to 300 × 2⁻³ grams after 3 half-lives. Evaluate 300 × 2⁻³ to determine how many grams of the substance are left. Explain why the expression 300 × 2⁻ⁿ can be used to find the amount of the substance that remains after n half-lives.

Solution:

Given, a half-life is the amount of time that it takes for a radioactive substance to decay to one half of its original quantity.

Radioactive decay causes 300 grams of a substance to decrease to 300 × 2⁻³ grams after 3 half-lives.

We have to find the amount of substance left after decay.

According to the question,

After one half-life, 300 g decrease to 300/2¹ gm

Amount of substance left = original weight - decayed weight

= 300 - 300/2

= 300(2 - 1)/2

= 300/2 gm

= 300 × 2⁻¹ gm

In 2 half-life, the initial weight of the substance is 300/2 gm

So, 1/2 (300/2) = 300/2² gm

= 300 × 2⁻² gm

Amount of substance left = 300/2 - 300/4

= 300(2 - 1)/4

= 300/4 gm

= 300 × 2⁻² gm

In 3 half-life, the initial weight of the substance is 300/4 gm

So, 1/2 (300/4) = 300/2³ gm

= 300 × 2⁻³ gm

Amount of substance left = 300/4 - 300/8

= 300(2 - 1)/8

= 300/8 gm

= 37.5 gm

Therefore, the amount of substance left is 37.5 gm.

From the above explanation, it is clear that 300 × 2⁻ⁿ can be used to find the amount of the substance that remains after n half-lives.

✦ Try This: Consider a quantity of a radioactive substance. The fraction of this quantity that remains after t half - lives can be found by using the expression 3^-t. What fraction of substance remains after 7 half-lives?

☛ Also Check: NCERT Solutions for Class 8 Maths Chapter 12

NCERT Exemplar Class 8 Maths Chapter 8 Problem 139

A half-life is the amount of time that it takes for a radioactive substance to decay to one half of its original quantity. Suppose radioactive decay causes 300 grams of a substance to decrease to 300 × 2⁻³ grams after 3 half-lives. Evaluate 300 × 2⁻³ to determine how many grams of the substance are left. Explain why the expression 300 × 2⁻ⁿ can be used to find the amount of the substance that remains after n half-lives

Summary:

A half-life is the amount of time that it takes for a radioactive substance to decay to one half of its original quantity. Suppose radioactive decay causes 300 grams of a substance to decrease to 300 × 2⁻³ grams after 3 half-lives. 37.5 grams of the substance are left.

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