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Chapter 19: Problem 131

If \(N_{0}\) is the original mass of the substance of half-life period \(t_{1 / 2}=5\) years, then the amount of substance left after 15 years is (A) \(N_{0} / 8\) (B) \(N_{0} / 16\) (C) \(N_{0} / 2\) (D) \(N_{0} / 4\)

Short Answer

Expert verified
The amount of substance left after 15 years is \(N_{0} / 8\).

Step by step solution

01

Understand the half-life formula

The formula for the half-life of a substance is given by: \[ N(t) = N_0 \cdot 2^{-\frac{t}{t_{1/2}}} \] Here, \(N(t)\) is the mass of the substance left after time \(t\), \(N_0\) is the initial mass of the substance, and \(t_{1/2}\) is the half-life period.
02

Plug in the given values

In the given problem, we have \(N_0\) as the initial mass, \(t_{1/2} = 5\) years, and the time elapsed is \(t = 15\) years. We will substitute these values into the half-life formula: \[ N(15) = N_0 \cdot 2^{-\frac{15}{5}} \]
03

Simplify the formula

Now let's simplify the formula: \[ N(15) = N_0 \cdot 2^{-\frac{15}{5}} = N_0 \cdot 2^{-3} \]
04

Calculate the remaining mass

To find the remaining mass, finish the calculation by simplifying the expression: \[ N(15) = N_0 \cdot 2^{-3} = N_0 \cdot \frac{1}{2^3} = \frac{N_0}{8} \]
05

Identify the correct answer

Based on our calculation, the amount of substance left after 15 years is \(N_{0}/8\). Therefore, the correct answer is: (A) \(N_{0} / 8\)

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