Question

These exercises use the radioactive decay model. The half-life of strontium-90 is 28 years. How long will it take a 50-mg sample to decay to a mass of 32 mg?

Answer 1

The radioactive decay of a substance can be modeled using the formula:

\[ N(t) = N_0 \cdot \left( \frac{1}{2} \right)^{\frac{t}{T}} \]

where:
- \( N(t) \) is the remaining quantity after time \( t \),
- \( N_0 \) is the initial quantity,
- \( T \) is the half-life of the substance.

In this case, \( N_0 \) is the initial mass of strontium-90, \( T \) is the half-life (28 years), and \( N(t) \) is the final mass (32 mg). We are asked to find the time \( t \).

\[ 32 = 50 \cdot \left( \frac{1}{2} \right)^{\frac{t}{28}} \]

To solve for \( t \), we can take the logarithm of both sides. Let's use the natural logarithm (ln):

\[ \ln(32) = \ln\left(50 \cdot \left( \frac{1}{2} \right)^{\frac{t}{28}}\right) \]

Using logarithm properties, we can simplify this equation:

\[ \ln(32) = \ln(50) + \frac{t}{28} \cdot \ln\left(\frac{1}{2}\right) \]

Now, isolate \( t \) by subtracting \( \ln(50) \) and solving for \( t \):

\[ \frac{t}{28} \cdot \ln\left(\frac{1}{2}\right) = \ln(32) - \ln(50) \]

\[ t = \frac{28 \cdot (\ln(32) - \ln(50))}{\ln\left(\frac{1}{2}\right)} \]

Now, you can use a calculator to find the numerical value of \( t \). The result will be the time it takes for the strontium-90 sample to decay from 50 mg to 32 mg.

Answer 2

To find the time it takes for a 50-mg sample of strontium-90 to decay to a mass of 32 mg, we can use the radioactive decay model. The half-life of strontium-90 is 28 years.

Step-by-step explanation:

To find the time it takes for a 50-mg sample of strontium-90 to decay to a mass of 32 mg, we can use the radioactive decay model. The half-life of strontium-90 is 28 years.

First, we need to find the number of half-lives it takes for the sample to decay. Since each half-life reduces the mass by half, the number of half-lives can be calculated by dividing the initial mass by the final mass:

Number of half-lives = log2 (initial mass / final mass)

Once we have the number of half-lives, we can multiply it by the half-life of strontium-90 to find the total time:

Total time = number of half-lives × half-life of strontium-90