Question

Suppose 128 ounces of a radioactive substance exponentially decays to 28 ounces in 6 hours. What is the half-life of the substance? The half-life is:

Answer 1

half-life ≈ 2.74

Explanation:

One formula we can use for half life is,

N(t) = No * (1/2)^(t / t_1/2), where

• N(t) is the amount remaining,
• No is the initial amount,
• 1/2 is the base
• t is the time,
• and t_1/2 is the half life

We already know that N(t) = 28, No = 128, the base is 1/2, and t is 6. Thus, we must solve for t_1/2 using the following equation:

28 = 128 * (1/2) ^ (6 / t_1/2)

• Step 1: Divide both sides by 128 to clear 128 on the right-hand side of the equation:

(28 = 128 * (1/2) ^ (6 / t_1/2) / 128

0.21875 = (1/2) ^ (6 / t_1/2)

• Step 2: Take the natural log (ln) of both sides:

ln (0.21875) = ln (1/2 ^ (6 / t_1/2))

• Step 3: According to the rules of logs, since we're taking the natural log of ln (1/2 ^ 6 / t_1/2, we can bring down the 6 / t_1/2 and it will now act like a coefficient:

ln (0.21875) = 6 / t_1/2 * ln (1/2)

• Step 4: Divide both sides by ln (1/2) to clear ln (1/2) on the right-hand side of the equation:

(ln ( 0.21875) = 6 / t_1/2 * ln (1/2)) / ln (1/2)

ln (0.21875) / ln (1/2) = 6 / t_1/2

• Step 5: You can cross multiply, which appears easier when you use the vertical fraction form instead of the horizontal fraction form.

`(ln(0.21875))/(ln(1/2))=(6)/(t_(1/2) )\\ \\ t_(1/2)*ln(0.21875)=6*ln(1/2)`

• Step 6: Finally, you can now divide both sides by ln (0.21875) to isolate t_1/2 and to find the half-life. I simply rounded to the nearest hundredth, but I'll provide the entire half-life number so you can round as liberally as much as you need or would like

(t_1/2 * ln (0.21875) = 6 * ln (1/2)) / ln (0.21875)

t_1/2 = (6 * ln (1/2) / ln (0.21875)

t_1/2 = 2.736420983 hours

t_1/2 = 2.74 hours

The only real difference between the longer number and the rounded number is that the longer answer gives a more exact answer when you plug it in, but the rounded answer seems to make more sense to write:

Result when plugging in exact answer for t_1/2:

N (2.736420983) = 128 (1/2) ^ (6 / 2.736420983) = 28

Result when plugging in rounded answer for t_1/2:

N (2.74) = 128 (1/2) ^ (6 / 2.74) = 28.05564116