Question

Gold-198 has a half-life of 2.7 days. How much of a 96g sample of gold-198 will be left after 8.1 days?

Answer 1

Answer: 12 grams

Explanation: Radioactive decay follows first order kinetics.

Half-life $t_{\frac{1}{2}$ of gold-198 = 2.7 days

$\lambda =\frac{0.693}{t_{\frac{1}{2}}}=\frac{0.693}{2.7}= 0.256days^{-1}$

$N=N_o\times e^{-\lambda t}$

N = amount left after time t= ?

$N_0$ = initial amount  = 96 g

$\lambda$ = rate constant= 0.26

t= time  = 8.1 days

$N=96\times e^{- 0.256days^{-1}\times 8.1days}$

$N=12g$

Thus the amount left after 8.1 days is 12 grams.

Answer 2

Answer:

The amount of gold-198 left after 8.1 days will be 11.99 g.

Explanation:

Half life of the gold-198 sample = $t_{\frac{1}{2}}=2.7 days$

$\lambda =\frac{0.693}{t_{\frac{1}{2}}}=\frac{0.693}{2.7 days}= 0.2566 days^{-1}$

$N=N_o\times e^{-\lambda t}$

N = amount left after time t

$N_0$ = initial amount  = 96 grams

$\lambda$ = rate constant

t= time = 8.1 days

$\log[N]=\log[N_o]-\frac{\lambda t}{2.303}$

$\log[N]=\log[96 g]-\frac{0.2566 day^{-1}\times 8.1 days}{2.303}$

N = 11.99 g

The amount of gold-198 left after 8.1 days will be 11.99 g.