A half-life is the time it takes an amount of substance to decay to one-half of its original amount. For example, the sample is one-half after the first half-life but one-quarter (i. e., one-half of one-half) after the second half-life. If the half life of uranium-235 is 700 million years, how long does it take for 10.0 grams to decay to 2.5 grams? Express your answer in billions of years.

I'm failing to understand how it isn't 1.4 billion years. From 10g to 5g would be 700 million years. Then, from 5g to 2.5g would be another 700 million years.

2.10 * 10⁹ yr

Step-by-step explanation:

The half-life of U-235 is the time it takes for half the U to decay.

After one half-life, half (50 %) of the original amount will remain.

After a second half-life, half of that amount (25 %) will remain, and so on.

We can construct a table as follows:

No. of Fraction Amount

half-lives t / (yr * 10⁶) remaining remaining/g

1 700 ½ 10.0

2 1400 ¼ 5.00

3 2100 ⅛ 2.50

4 2800 ¹/₁₆ 1.25

We see that 2100 * 10⁶ yr is three half-lives, and the amount of U-235 remaining is 2.50 g.

It takes 2.10 * 10⁹ yr for the U-238 to decay to 2.50 g.