High School

Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ Sodium-25 was to be used in an experiment, but it took 3.0 minutes to get the sodium from the reactor to the laboratory. If 5.0 mg of sodium-25 was removed from the reactor, how many mg of sodium-25 were placed in the reaction vessel 3.0 minutes later, given that the half-life of sodium-25 is 60 seconds?

Answer :

Let's solve the problem about the decay of sodium-25 together!

### Step-by-Step Solution

1. Understanding the Half-life:
- Sodium-25 has a half-life of 60 seconds. This means that every 60 seconds, half of the sodium-25 will have decayed.

2. Time Calculation:
- You need to find out how much sodium-25 remains after 3.0 minutes. First, we convert this time into seconds since the half-life is given in seconds:
[tex]\[
3.0 \text{ minutes} = 3.0 \times 60 = 180 \text{ seconds}
\][/tex]

3. Decay Formula:
- The decay of a radioactive isotope like sodium-25 can be calculated using the formula:
[tex]\[
N(t) = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}
\][/tex]

- Where:
- [tex]\(N(t)\)[/tex] is the remaining amount after time [tex]\(t\)[/tex]
- [tex]\(N_0\)[/tex] is the initial amount (5.0 mg in this case)
- [tex]\(t\)[/tex] is the total time elapsed (180 seconds)
- [tex]\(t_{1/2}\)[/tex] is the half-life of the isotope (60 seconds)

4. Calculating Remaining Sodium-25:
- Substitute the known values into the formula:
[tex]\[
N(180) = 5.0 \times \left(\frac{1}{2}\right)^{\frac{180}{60}}
\][/tex]

- Now, calculate the exponent:
[tex]\[
\frac{180}{60} = 3
\][/tex]

- Therefore, the formula becomes:
[tex]\[
N(180) = 5.0 \times \left(\frac{1}{2}\right)^3 = 5.0 \times \frac{1}{8}
\][/tex]

- Calculating this gives:
[tex]\[
N(180) = 5.0 \times 0.125 = 0.625 \text{ mg}
\][/tex]

So, after 3.0 minutes, 0.625 mg of sodium-25 would remain and be placed in the reaction vessel.