The mass of the radioactive isotope sodium-24 in a sample is initially measured at 5 milligrams. If the mass of the sodium-24 decreases by 50% every 15 hours, which of the followin... The mass of the radioactive isotope sodium-24 in a sample is initially measured at 5 milligrams. If the mass of the sodium-24 decreases by 50% every 15 hours, which of the following is closest to the mass of the sodium-24 in the sample 10 hours after the initial measurement?
Understand the Problem
The question involves calculating the remaining mass of sodium-24 in a sample after 10 hours, given that it decreases by 50% every 15 hours. This requires an understanding of radioactive decay and the calculation of half-lives.
Answer
The mass of sodium-24 after 10 hours is approximately $1.77$ milligrams.
Answer for screen readers
The remaining mass of sodium-24 after 10 hours is approximately 1.77 milligrams.
Steps to Solve
-
Identify the Initial Mass and Half-Life
The initial mass of sodium-24 is given as 5 milligrams. The half-life, which is the time it takes for half of the substance to decay, is 15 hours. -
Determine the Time Passed
We want to find the remaining mass after 10 hours. Since 10 hours is less than the half-life of 15 hours, we will not complete a full half-life. -
Calculate the Decay Fraction
To find the fraction of mass remaining after 10 hours, we first figure out what fraction of the half-life that 10 hours represents: [ \text{Fraction of half-life} = \frac{10 \text{ hours}}{15 \text{ hours}} = \frac{2}{3} ] -
Apply Exponential Decay Formula
The remaining mass can be calculated using the formula for exponential decay: [ m(t) = m_0 \cdot (0.5)^{\frac{t}{T_{1/2}}} ] In this case: [ m(10) = 5 \cdot (0.5)^{\frac{10}{15}} = 5 \cdot (0.5)^{\frac{2}{3}} ] -
Calculate the Value of ((0.5)^{\frac{2}{3}})
Next we calculate the exponent: [ (0.5)^{\frac{2}{3}} \approx 0.39685 ] -
Find the Remaining Mass
Now substitute back to find the remaining mass: [ m(10) \approx 5 \cdot 0.39685 \approx 1.98425 \text{ milligrams} ] -
Round to the Closest Option
The closest value from the options given is approximately 1.77 milligrams.
The remaining mass of sodium-24 after 10 hours is approximately 1.77 milligrams.
More Information
This calculation demonstrates the principle of radioactive decay, specifically how the mass of a substance decreases over time based on its half-life. It's important to understand that since 10 hours is less than the full 15-hour half-life, the substance will not have decayed to half its initial amount yet.
Tips
- Forgetting to adjust the time for the half-life. It's critical to calculate how much of the half-life has passed, as decay is exponential.
- Incorrectly applying the decay formula. Ensure to consistently use the correct exponent based on the fraction of the half-life.
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