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A certain sample of radioactive material decays at a rate of 500 counts per second at a certain time. The count rate falls to 200 counts per second after 50 minutes.

(a) What is the decay constant of the sample?

Answer :

Final Answer:

The decay constant (λ) is approximately 7.22 x 10^(-5) per second. This constant represents the probability of decay per unit time. The sample's half-life can be calculated using the formula T(1/2) = ln(2) / λ, which is about 9593 seconds or approximately 2.66 hours.

Explanation:

We have the initial count rate (\(N_0\)) as 500 particles per second, and after 50 minutes, which is equivalent to 3000 seconds, the count rate (\(N(3000)\)) falls to 200 particles per second.

Using the radioactive decay formula, we can write two equations:

1. [tex]\(500 = N_0 e^{-λ \cdot 0}\[/tex]) (at the initial time, \(t = 0\))

2. [tex]\(200 = 500 e^{-λ \cdot 3000}\)[/tex](after 50 minutes, \(t = 3000\) seconds)

Simplifying equation 1:

\[e^{0} = 1\]

So, equation 1 becomes:

\[500 = N_0\]

Substituting this value into equation 2:

\[200 = 500 e^{-3000λ}\]

Now, we can solve for λ:

[tex]\[e^{-3000λ} = \frac{200}{500} = 0.4\][/tex]

Taking the natural logarithm on both sides:

\[-3000λ = \ln(0.4)\]

Now, solving for λ:

[tex]\[λ = -\frac{\ln(0.4)}{3000}\][/tex]

Calculating this value will give us the decay constant of the sample.

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