A radioactive sample contains 1020 atoms. Its half-life is 20 seconds. How many disintegrations occur in the…

Question

A radioactive sample contains 1020 atoms. Its half-life is 20 seconds. How many disintegrations occur in the first second?

ScanSolve AI Tutor · Accepted Answer

To find the number of disintegrations in the first second, we need to calculate the initial activity of the radioactive material. The activity is given by $A = \lambda N$ , where $\lambda$ is the decay constant and $N$ is the number of radioactive atoms. The number of disintegrations in a short time interval $\Delta t$ is approximately $A \Delta t$ .

Determine the initial number of atoms ( $N_0$ ).
The question states "1020 atoms". However, given the options are in powers of $10^{18}$ and $10^{20}$ , it is highly probable that "1020" is a typo and should be $10^{20}$ atoms. We will proceed with this assumption.
So, $N_0 = 10^{20}$ atoms.
Calculate the decay constant ( $\lambda$ ).
The half-life ( $T_{1/2}$ ) is given as 20 seconds. The relationship between half-life and decay constant is:
$\lambda = \frac{\ln(2)}{T_{1/2}}$ $\lambda = \frac{0.693}{20 \text{ s}}$ $\lambda = 0.03465 \text{ s}^{-1}$
Calculate the initial activity ( $A_0$ ).
The initial activity is the rate of disintegration at $t=0$ .
$A_0 = \lambda N_0$ $A_0 = (0.03465 \text{ s}^{-1}) \times (10^{20} \text{ atoms})$ $A_0 = 3.465 \times 10^{18} \text{ disintegrations/s}$
Calculate the number of disintegrations in the first second.
Since the time interval is very short ( $\Delta t = 1 \text{ s}$ ), we can approximate the number of disintegrations as $A_0 \Delta t$ .
$\text{Number of disintegrations} = A_0 \times \Delta t$ $\text{Number of disintegrations} = (3.465 \times 10^{18} \text{ disintegrations/s}) \times (1 \text{ s})$ $\text{Number of disintegrations} = 3.465 \times 10^{18}$
Comparing this result with the given options:
A. $3.47 \times 10^{20}$
B. $3.47 \times 10^{18}$
C. $6.93 \times 10^{20}$
D. $6.93 \times 10^{18}$

Our calculated value $3.465 \times 10^{18}$ is approximately $3.47 \times 10^{18}$ , which matches option B.

The final answer is $\boxed{\text{B}}$ .

A radioactive sample contains 1020 atoms. Its half-life is 20 seconds. How many disintegrations occur in the first second?

Step-by-step solution

Determine the initial number of atoms ( $N_0$ ).

Calculate the decay constant ( $\lambda$ ).

Calculate the initial activity ( $A_0$ ).

Calculate the number of disintegrations in the first second.

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More Physics Questions

A radioactive sample contains 1020 atoms. Its half-life is 20 seconds. How many disintegrations occur in the first second?

Step-by-step solution

Determine the initial number of atoms (N0N_0N0​).

Calculate the decay constant (λ\lambdaλ).

Calculate the initial activity (A0A_0A0​).

Calculate the number of disintegrations in the first second.

Still stuck on this one?

More Physics Questions

Determine the initial number of atoms ( $N_0$ ).

Calculate the decay constant ( $\lambda$ ).

Calculate the initial activity ( $A_0$ ).