Understanding Radioactive Decay and Disintegration Constants

Radioactive decay is a spontaneous process where an unstable atomic nucleus loses energy by emitting radiation. This process happens at a specific rate for each radioactive substance. This rate is characterized by either the half-life or the disintegration constant.

Key Concepts: Half-life and Disintegration Constant

Let's define the two key terms involved in this problem:

• Half-life (\(T\)): The time required for half of the radioactive nuclei in a sample to undergo radioactive decay. It is a measure of how long a radioactive substance remains active.
• Disintegration Constant (\(\lambda\)): Also known as the decay constant, it is the probability per unit time that a nucleus will decay. It represents the rate of decay of a radioactive substance. A larger disintegration constant means faster decay.

Half-life and disintegration constant are inversely related. This means if a substance has a shorter half-life, it decays faster and thus has a larger disintegration constant.

Relating Half-life and Disintegration Constant

The mathematical relationship between the half-life (\(T\)) and the disintegration constant (\(\lambda\)) is given by the formula:

\( T = \frac{\ln 2}{\lambda} \)

Here, \( \ln 2 \) is the natural logarithm of 2, which is approximately 0.693.

From this formula, we can express the disintegration constant in terms of the half-life:

\( \lambda = \frac{\ln 2}{T} \)

Calculating the Ratio of Disintegration Constants

We are given two radioactive specimens with different half-lives. Let's denote the half-life and disintegration constant of the first specimen as \( T_1 \) and \( \lambda_1 \), and for the second specimen as \( T_2 \) and \( \lambda_2 \).

The given values are:

Using the relationship \( \lambda = \frac{\ln 2}{T} \), we can write the disintegration constants for the two specimens:

For specimen 1: \( \lambda_1 = \frac{\ln 2}{T_1} \)

For specimen 2: \( \lambda_2 = \frac{\ln 2}{T_2} \)

We need to find the ratio of their disintegration constants, \( \frac{\lambda_1}{\lambda_2} \). We can calculate this ratio by dividing the expression for \( \lambda_1 \) by the expression for \( \lambda_2 \):

\( \frac{\lambda_1}{\lambda_2} = \frac{\frac{\ln 2}{T_1}}{\frac{\ln 2}{T_2}} \)

To simplify this complex fraction, we can invert the denominator and multiply:

\( \frac{\lambda_1}{\lambda_2} = \frac{\ln 2}{T_1} \times \frac{T_2}{\ln 2} \)

The \( \ln 2 \) terms cancel out, leaving:

\( \frac{\lambda_1}{\lambda_2} = \frac{T_2}{T_1} \)

Now, substitute the given numerical values for \( T_1 \) and \( T_2 \):

\( \frac{\lambda_1}{\lambda_2} = \frac{7.5 \text{ years}}{5 \text{ years}} \)

To calculate the ratio, we can simplify the fraction:

\( \frac{\lambda_1}{\lambda_2} = \frac{7.5}{5} = \frac{75/10}{5} = \frac{75}{50} \)

Divide the numerator and denominator by their greatest common divisor, which is 25:

\( \frac{75}{50} = \frac{75 \div 25}{50 \div 25} = \frac{3}{2} \)

Thus, the ratio of the disintegration constants \( \lambda_1 \) and \( \lambda_2 \) is \( \frac{3}{2} \).

Conclusion: Ratio of Disintegration Constants

Based on the half-lives provided, the ratio of the disintegration constants \( \lambda_1 \) to \( \lambda_2 \) is found to be \( \frac{3}{2} \).

Revision Table: Radioactive Decay Formulas

Additional Information: Radioactive Decay Law

The number of radioactive nuclei \( N(t) \) remaining at time \( t \) from an initial number \( N_0 \) at \( t=0 \) is given by the radioactive decay law:

\( N(t) = N_0 e^{-\lambda t} \)

where \( e \) is the base of the natural logarithm and \( \lambda \) is the disintegration constant. The activity of a radioactive sample, which is the rate of decay, is proportional to the number of nuclei present and the disintegration constant (\( A = \lambda N \)). This fundamental law governs how radioactive substances decrease over time.