Answer :
Sure, let's solve this step-by-step!
a) Calculate the amount of time it will take for the sample to decay to 5 mg:
We start with an initial sample of 40 mg of caesium-124. The half-life of caesium-124 is 31 seconds. We want to find out how long it takes for the sample to decay to 5 mg.
To solve this, we'll use the formula for exponential decay:
[tex]\[ \text{final mass} = \text{initial mass} \times \left(\frac{1}{2}\right)^{\frac{\text{time}}{\text{half-life}}} \][/tex]
We want to find the time, so let's rearrange the formula to solve for time:
[tex]\[ \text{time} = \text{half-life} \times \frac{\log(\text{final mass}/\text{initial mass})}{\log(0.5)} \][/tex]
Plug in the values:
[tex]\[ \text{time} = 31 \times \frac{\log(5/40)}{\log(0.5)} \][/tex]
This gives us a result of approximately 93 seconds.
b) Calculate how much caesium-124 will remain after 124 seconds (2 min 4 seconds):
We use the same formula, but this time we're looking for the remaining mass after a certain time:
[tex]\[ \text{remaining mass} = \text{initial mass} \times \left(\frac{1}{2}\right)^{\frac{\text{time elapsed}}{\text{half-life}}} \][/tex]
Plug in the values:
[tex]\[ \text{remaining mass} = 40 \times \left(\frac{1}{2}\right)^{124/31} \][/tex]
Calculating this gives us 2.5 mg remaining after 124 seconds.
So, it will take about 93 seconds for the sample to decay to 5 mg, and after 124 seconds, 2.5 mg of caesium-124 will remain.
a) Calculate the amount of time it will take for the sample to decay to 5 mg:
We start with an initial sample of 40 mg of caesium-124. The half-life of caesium-124 is 31 seconds. We want to find out how long it takes for the sample to decay to 5 mg.
To solve this, we'll use the formula for exponential decay:
[tex]\[ \text{final mass} = \text{initial mass} \times \left(\frac{1}{2}\right)^{\frac{\text{time}}{\text{half-life}}} \][/tex]
We want to find the time, so let's rearrange the formula to solve for time:
[tex]\[ \text{time} = \text{half-life} \times \frac{\log(\text{final mass}/\text{initial mass})}{\log(0.5)} \][/tex]
Plug in the values:
[tex]\[ \text{time} = 31 \times \frac{\log(5/40)}{\log(0.5)} \][/tex]
This gives us a result of approximately 93 seconds.
b) Calculate how much caesium-124 will remain after 124 seconds (2 min 4 seconds):
We use the same formula, but this time we're looking for the remaining mass after a certain time:
[tex]\[ \text{remaining mass} = \text{initial mass} \times \left(\frac{1}{2}\right)^{\frac{\text{time elapsed}}{\text{half-life}}} \][/tex]
Plug in the values:
[tex]\[ \text{remaining mass} = 40 \times \left(\frac{1}{2}\right)^{124/31} \][/tex]
Calculating this gives us 2.5 mg remaining after 124 seconds.
So, it will take about 93 seconds for the sample to decay to 5 mg, and after 124 seconds, 2.5 mg of caesium-124 will remain.
