The half-life of an element is $4.5 \times 10 ^ { 11 } \text { yr. }$
How long does it take a sample of the element to decay to $\frac { 2 } { 3 }$ of its original mass? Use $\mathrm { A } = \mathrm { A } _ { 0 } ( 0.5 ) ^ { ( \mathrm { t } / \mathrm { T } ) }$ where $\mathrm { A } 0$ is the initial amount, T is the half-life, and t is the time. (Express results in scientific notation, rounded to the nearest hundredth.)
A) $2.63 \times 10 ^ { 11 } \mathrm { yr }$
B) $5.83 \times 10 ^ { 11 } \mathrm { yr }$
C) $1.29 \times 10 ^ { 11 } \mathrm { yr }$
D) $1.26 \times 10 ^ { 11 } \mathrm { yr }$

Question

The half-life of an element is $4.5 \times 10 ^ { 11 } \text { yr. }$
How long does it take a sample of the element to decay to $\frac { 2 } { 3 }$ of its original mass? Use $\mathrm { A } = \mathrm { A } _ { 0 } ( 0.5 ) ^ { ( \mathrm { t } / \mathrm { T } ) }$ where $\mathrm { A } 0$ is the initial amount,  T  is the half-life, and  t  is the time. (Express results in scientific notation, rounded to the nearest hundredth.)
A) $2.63 \times 10 ^ { 11 } \mathrm { yr }$
B) $5.83 \times 10 ^ { 11 } \mathrm { yr }$
C) $1.29 \times 10 ^ { 11 } \mathrm { yr }$
D) $1.26 \times 10 ^ { 11 } \mathrm { yr }$

Quizplus · Accepted Answer

To find the time it takes for a sample to decay to $\frac{2}{3}$ of its original mass, we use the given formula with $A/A_0 = \frac{2}{3}$ and $T = 4.5 \times 10^{11}$ years. Solving for $t$, we get $t = T \times \frac{\log(2/3)}{\log(0.5)} \approx 2.63 \times 10^{11}$ years.

The Half-Life of an Element Is 4.5×1011 yr. 4.5 \times 10 ^ { 11 } \text { yr. }4.5×1011 yr.

The Half-Life of an Element Is $4.5 \times 10 ^ { 11 } \text { yr. }$