Chapter 19: Problem 33

The number of radioactive nuclides in a sample decays from $1.00 \times 10^{20}$ to $2.50 \times 10^{19}$ in 10.0 minutes. What is the half-life of this radioactive species?

Short Answer

Expert verified

The half-life of this radioactive species is approximately 10.0 minutes.

Step by step solution

Understand the radioactive decay formula

The radioactive decay formula is given by: \[N(t) = N_0 \cdot e^{-\lambda t}\] where $N(t)$ is the quantity of radioactive nuclides remaining after time $t$, $N_0$ is the initial number of radioactive nuclides, $\lambda$ is the decay constant, and $t$ is time.

Use the given data to find the decay constant

We have been given the initial number of radioactive nuclides $N_0 = 1.00 \times 10^{20}$, the final number of nuclides after the decay $N(t) = 2.50 \times 10^{19}$, and the time interval during which this decay occurs $t = 10.0 \, \text{minutes}$. Plugging these values into the radioactive decay formula, we get: \[2.50 \times 10^{19} = (1.00 \times 10^{20}) \cdot e^{-\lambda (10.0 \, \text{min})}\] Let's isolate $\lambda$ on one side of the equation.

Isolate the decay constant

To do this, first divide both sides of the equation by $1.00 \times 10^{20}$: \[\frac{2.50 \times 10^{19}}{1.00 \times 10^{20}} = e^{-\lambda (10.0 \, \text{min})}\] Next, take the natural logarithm of both sides. \[\ln{\left(\frac{2.50 \times 10^{19}}{1.00 \times 10^{20}}\right)} = -\lambda (10.0 \, \text{min})\] Now, divide by $-10.0 \, \text{min}$ to isolate $\lambda$: \[\lambda = -\frac{\ln{\left(\frac{2.50 \times 10^{19}}{1.00 \times 10^{20}}\right)}}{10.0 \, \text{min}} \approx 0.0693 \, \text{min}^{-1}\] We found the decay constant, $\lambda \approx 0.0693 \, \text{min}^{-1}$.

Find half-life using the decay constant

The half-life of a radioactive species is related to the decay constant by the formula: \[T_{1/2} = \frac{\ln{2}}{\lambda}\] We can now substitute the value of $\lambda$ that we found in the previous step and compute the half-life: \[T_{1/2} = \frac{\ln{2}}{0.0693 \, \text{min}^{-1}} \approx 10.0 \, \text{minutes}\] Therefore, the half-life of this radioactive species is approximately 10.0 minutes.

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The number of radioactive nuclides in a sample decays from $1.00 \times 10^{20}\( to \)2.50 \times 10^{19}$ in 10.0 minutes. What is the half-life of this radioactive species?

Short Answer

Step by step solution

Understand the radioactive decay formula

Use the given data to find the decay constant

Isolate the decay constant

Find half-life using the decay constant

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