Chapter 24: Q24.41 P (page 1104)

The isotope ${^{212}}_{83} Bi$ has a half-life of $1.01$ yr. What mass (in mg) of a $2.00$ -mg sample will remain after ${3.75×10}^{3} h$ ?

Short Answer

Expert verified

The amount of ${^{212}}_{83} Bi$ that remains is $1.49 mg$

Step by step solution

Radioactive decay

The radioactive decay follows the first order kinetics. The relation between the rate constant and the half-life of the reaction is as shown below:

$k = \frac{0.693}{t_{1 / 2}}$

Calculation

- The half-life of $_{83}^{212} {Biist}_{1/2} =1.01yr$

- The initial amount of $_{83}^{212} {BiwasN}_{0} =2.00mg$

We have to find the mass of $_{83}^{212} Bi$ after time_{$t = 3.75 \cdot 10^{3} h$}

First, let us convert the time elapsed from hours to years

_{$t = 3.75 \times 10^{3} h \times \frac{1 d}{24 h} \times \frac{1 yr}{365 d} = 0.428 yr$}

Now, let us calculate the decay constant of_{$_{83}^{212} Bi$}

_{$k= \frac{ln(2)}{t_{1/2}} = \frac{0.693}{1.01yr} {=0.686yr}^{-1}$}

Therefore, the amount of ${^{212}}_{83} Bi$ that remains after time t is

_{$\begin{matrix} ln (\frac{N_{t}}{N_{0}}) =-kt \\ ln (\frac{N_{t}}{2.00mg}) {=-0.686yr}^{-1} ×0.428yr \\ =-0.2936 \end{matrix}$}

_{\ $\begin{matrix} \frac{N_{t}}{2.00mg} {=e}^{-0.2936} \\ \frac{N_{t}}{2.00mg} =0.7456 \\ N_{t} =1.49mg \end{matrix}$}

Hence,the amount of ${^{212}}_{83} Bi$ that remains is $1.49 mg$ _.

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The isotope ${^{212}}_{83} Bi$ has a half-life of $1.01$ yr. What mass (in mg) of a $2.00$ -mg sample will remain after ${3.75×10}^{3} h$ ?

Short Answer

Step by step solution

Radioactive decay

Calculation

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The isotope 21283Bihas a half-life of1.01 yr. What mass (in mg) of a2.00 -mg sample will remain after3.75×103h ?

Short Answer

Step by step solution

Radioactive decay

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Chemistry Textbooks

Kinetics

Organic Chemistry

Chemistry Branches

Inorganic Chemistry

The Earths Atmosphere

Making Measurements

Study anywhere. Anytime. Across all devices.

The isotope ${^{212}}_{83} Bi$ has a half-life of $1.01$ yr. What mass (in mg) of a $2.00$ -mg sample will remain after ${3.75×10}^{3} h$ ?