Question

If a 10 gram sample of a radioactive substance has a half-life of 6000 years, how much would be present after 8000 years?

Answer 1

Ans :

Half life formula is given as :

Nt = No (1/2)^{t / h}

Nt is amount of sample left

No is initial amount of sample = 10 g

t = time lapsed = 8000 y

h = half life = 6000 y

Nt = 10 g (1/2)^{8000 y / 6000 y}

= 3.97 g

The amount of sample left will be 3.97 g.

Answer 2

Step 1: Understand the Half-Life Formula

The amount of substance remaining after a given time can be calculated using the formula:

$$N=N_0\times {\left(\frac{1}{2}\right)}^{\frac{t}{t_{1\mathrm{/}2}}}$$

where:

• $N$ = remaining quantity after time $t$,

• $N_0$ = initial quantity,

• $t$ = elapsed time,

• $t_{1\mathrm{/}2}$ = half-life of the substance.

Step 2: Plug in the Given Values

• Initial quantity, $N_0=10$ grams,

• Half-life, $t_{1\mathrm{/}2}=6000$ years,

• Elapsed time, $t=8000$ years.

$$N=10\times {\left(\frac{1}{2}\right)}^{\frac{8000}{6000}}$$

Step 3: Simplify the Exponent

$$\frac{8000}{6000}=\frac{4}{3}\approx 1.3333$$

Step 4: Calculate the Remaining Amount

$$N=10\times {\left(\frac{1}{2}\right)}^{1.3333}$$$$N\approx 10\times 0.3969$$$$N\approx 3.969\text{ grams}$$

Answer is :-

After 8000 years, approximately 3.97 grams of the radioactive substance would remain