Question

It \(\mathrm{N}_{0}\) is the original mass of the substance of halt life 5 years, the amount of substance left after 15 years is (A) \(\left(\mathrm{N}_{0} / 16\right)\) (B) \(\left(\mathrm{N}_{0} / 8\right)\) (C) \(\left(\mathrm{N}_{0} / 4\right)\) (D) \(\left(\mathrm{N}_{0} / 2\right)\)

Short answer

The amount of substance left after 15 years is \(\frac{1}{8}\) of its original mass, which corresponds to answer choice (B).

Step-by-step solution

Understand the Half-Life formula

\ To solve this problem, we need to use the half-life formula, which is: \[N(t) = N_0 * (1/2)^{t/T}\] Where: - \(N(t)\) = remaining mass of the substance at time \(t\) - \(N_0\) = original mass of the substance - \(t\) = time in years - \(T\) = half-life of the substance in years In this exercise, \(T = 5\) years and \(t = 15\) years.

Substitute the given values into the formula

\ Now, we can substitute the given values for \(T\) and \(t\) into the half-life formula: \[N(15) = N_0 * (1/2)^{15/5}\]

Simplify the expression

\ Simplify the exponent 15/5: \[N(15) = N_0 * (1/2)^3\] Now raise the fraction to the power of 3: \[N(15) = N_0 * \frac{1}{8}\] We now have the expression for the remaining mass of the substance after 15 years as a fraction of the original mass.

Compare the result with the answer choices

\ Our result, \(N(15) = N_0 * \frac{1}{8}\), matches answer choice (B): (B) \(\left(N_0 / 8\right)\) Thus, after 15 years, the remaining mass of the substance is \(\frac{1}{8}\) of its original mass.