Uranium-238 ( ) decays to form thorium-234 ( ) with a half-life of 4.5 $\times$ 10 ⁹ years. How many years will it take for 75% of the uranium-238 to decay?

Question

Quizplus · Accepted Answer

We can use the half-life equation to solve for the time it takes for 75% of the uranium-238 to decay:

N/N_0 = (1/2)^(t/t_1/2)

where N/N_0 is the fraction of uranium-238 remaining, t is the time elapsed, and t_1/2 is the half-life. We want N/N_0 to be 0.25, which means that 75% of the uranium-238 has decayed.

0.25 = (1/2)^(t/4.5x10^9)

Taking the natural logarithm of both sides gives:

ln(0.25) = (t/4.5x10^9)ln(1/2)

Solving for t gives:

t = ln(0.25)(4.5x10^9)/ln(1/2)

t ≈ 9.0x10^9 years

Therefore, the answer is D) 9.0x10^9 years.

Uranium-238 ( ) Decays to Form Thorium-234 ×\times× 10 9 Years

Uranium-238 ( ) Decays to Form Thorium-234 $\times$ 10 ⁹ Years