# An original sample of K-40 has a mass of 25.00 grams. After 3.9 xx 10^9 years, 3.125 grams of the original sample remains unchanged. What is the half-life of K-40?

Jun 21, 2017

$1.3 \times {10}^{9} y r s$

#### Explanation:

From ${C}_{\text{final" = C_"initial}} {e}^{- k t}$ solve for rate constant (k) ...

=> $k = \left(\ln \frac{{C}_{f} / {C}_{i}}{-} t\right)$ = $\ln \frac{\frac{3.125}{25}}{- 3.9 \times {10}^{9}}$ = $5.33 \times {10}^{-} 10 y r {s}^{-} 1$

Use the rate constant (k) in the half-life equation ...

=> ${t}_{\frac{1}{2}} = \left(\frac{0.693}{k}\right) = \left(\frac{0.693}{5.33 \times {10}^{-} 10}\right) y e a r s$ $1.3 \times {10}^{9} y e a r s$