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Chapter 18: Problem 2533

A radioactive substance decays to \((1 / 16)\) th of its initial activity in 40 days. the half life of the radioactive substance expressed in day is (A) 20 (B) 5 (C) 10 (D) 4

Short Answer

Expert verified
Given that a radioactive substance decays to \(\dfrac{1}{16}\) of its initial activity in 40 days, we first set up the radioactive decay equation: \[\dfrac{1}{16} A_{0} = A_{0}e^{-λ(40)}\] Solving for the decay constant (λ): \[λ = \dfrac{\ln \dfrac{1}{16}}{-40}\] Finally, calculating the half-life using the decay constant: \[T_{\frac{1}{2}} = \dfrac{\ln 2}{\dfrac{\ln \dfrac{1}{16}}{-40}} \] Simplifying gives us the half-life to be \(T_{\frac{1}{2}} = 10\) days. Thus, the correct answer is (C) 10.

Step by step solution

01

Given that the initial activity decays to \((1/16)\) after 40 days, the radioactive decay equation can be written as: \[ \dfrac{1}{16} A_{0} = A_{0}e^{-λ(40)}\] #Step 2: Solve the equation for the decay constant (λ)#

Divide both sides of the equation by \(A_{0}\) to get: \[ \dfrac{1}{16} = e^{-λ(40)}\] Take the natural logarithm of both sides of the equation: \[ \ln \dfrac{1}{16} = -λ(40)\] Now, we solve for \(λ\): \[ λ = \dfrac{\ln \dfrac{1}{16}}{-40} \] #Step 3: Find the half-life using the decay constant (λ)#
02

The half-life is given by the formula: \[ T_{\frac{1}{2}} = \dfrac{\ln 2}{λ} \] Use the decay constant found in Step 2 to calculate the half-life: \[ T_{\frac{1}{2}} = \dfrac{\ln 2}{\dfrac{\ln \dfrac{1}{16}}{-40}} \] #Step 4: Simplify and find the answer#

Simplify to get the half-life, and compare your answer with the given options: \[ T_{\frac{1}{2}} = 10\] So, the half-life of the radioactive substance expressed in days is \(10\). Therefore, the correct answer is (C) 10.

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