Question

n isotope of sodium has a half-life of 20 hours. Suppose an initial sample of this isotope has mass 10 grams. The amount of the isotope (in grams) remaining after t hours is given by:

Answer 1

The amount of a sodium isotope remaining after a set amount of time can be calculated using the half-life. The formula used is the initial mass multiplied by (1/2) raised to the power of the number of half-lives that have passed.

Step-by-step explanation:

Calculating Remaining Isotope Mass After Time

The amount of a radioactive isotope remaining after a given time can be calculated using the half-life of the isotope. Given a sodium isotope with a half-life of 20 hours and an initial mass of 10 grams, we can determine the amount remaining after any multiple of the half-life. If we let 'n' represent the number of half-lives that have passed, the remaining mass can be determined using the formula:

amount remaining = initial mass × (1/2)^n

For example, if we wanted to find out how much of the sodium isotope would remain after 80 hours, we would note that 80 hours is exactly 4 half-lives (since 80/20 = 4). Then using the equation provided:

amount remaining = 10 grams × (1/2)^4 = 10 grams × 1/16 = 0.625 grams

This calculation can be used for any radioactive isotope to find the mass remaining after a certain period. It is crucial to understand that this method can also be applied when the time is not an exact multiple of the half-life by using the appropriate fraction of n.

Answer 2

In this question the half-life (t½) of the isotope is 20 years, which means that after 20 years half of the isotope would have decayed and half would be left.

Since half of the sample remains, we can say that 10g/2 = 5g

After 20 years, 5 grams of the sample would remain.

If you wanted to know the amount after two half-lives, then after 40 years 2.5 grams of the isotope would exist.