Problem 103 - A Complete Resource Book in Chemistry for JEE Main

Question

A sample of radioactive substance gave 630 counts per minute and 610 counts per minute at times differing by 1 hour. The decay constant \((\lambda)\) in \(\min ^{-1}\) is given by (a) \(\lambda=\frac{630}{610} \times 60\) (b) \(\mathrm{e}^{60 \mathrm{k}}=\frac{630}{610}\) (c) \(\lambda=\frac{2.303}{60} \log \frac{610}{630}\) (d) \(\lambda=\frac{2.303}{60} \times \frac{630}{610}\)

Step-by-Step Solution

Answer

(c) \(\lambda = \frac{2.303}{60} \log \frac{610}{630}\).

Step 1: Understanding the Radioactive Decay

Radioactive decay follows the formula \( N(t) = N_0 e^{-\lambda t} \), where \( N_0 \) is the initial quantity, \( N(t) \) is the quantity at time \( t \), and \( \lambda \) is the decay constant.

Step 2: Given Values

At time \( t_1 \), the count is 630 counts per minute, and at \( t_2 = t_1 + 60 \text{ minutes} \), the count is 610 counts per minute.

Step 3: Using the Exponential Decay Formula

From \( 610 = 630 e^{-\lambda \cdot 60} \), we know that \( \frac{610}{630} = e^{-60\lambda} \).

Step 4: Solving the Exponential Equation

Taking natural logarithms on both sides, \( \ln \left(\frac{610}{630}\right) = -60\lambda \). Thus, \( \lambda = -\frac{\ln \left(\frac{610}{630}\right)}{60} \).

Step 5: Approximation Using Logarithm

By converting the natural logarithm to a common logarithm: \( \ln \left(\frac{610}{630}\right) = 2.303 \log \left(\frac{610}{630}\right) \). Thus, \( \lambda = \frac{2.303}{60} \log \left(\frac{610}{630}\right) \).

Key Concepts

Decay Constant

The decay constant, often symbolized by \( \lambda \), represents the probability of a radioactive atom decaying per unit time. It's a vital parameter in understanding how substances diminish over time due to radioactive decay. A higher decay constant means the sample will decay more quickly.

When considering radioactive materials, the decay constant plays a critical role in predicting how long it will take for the sample to lose its radioactivity, which is often useful in fields like nuclear physics, medicine, and archaeology.

The relationship between the decay constant and the decay process can be expressed through the formula:

• \( N(t) = N_0 e^{-\lambda t} \)

where:

• \( N(t) \) is the quantity remaining at time \( t \)
• \( N_0 \) is the initial quantity
• \( e \) is the base of natural logarithms approximately equal to 2.71828
• \( \lambda \) is the decay constant

This formula helps track the rate of radioactive decay, essential for scientific calculations involving radioactive substances.

Exponential Decay Formula

The exponential decay formula describes the process through which radioactive materials decrease over time. The formula \( N(t) = N_0 e^{-\lambda t} \) represents how the material diminishes exponentially.

Understanding this formula is crucial in solving problems involving radioactive substances. In our context, at time \( t_1 \), the initial count of radioactive substance is 630 counts per minute, and at \( t_2 = t_1 + 60 \) minutes, it reduces to 610 counts.

We use the exponential decay formula to write:

• \( 610 = 630 e^{-\lambda \cdot 60} \)

Simplifying further, we divide both sides by 630 to get:

• \( \frac{610}{630} = e^{-60\lambda} \)

This equation shows how the quantity of substance decreases with time when impacted by a decay constant. This formula is utilized to set up and solve equations in order to determine the unknown decay constant, \( \lambda \), thus predicting how the substance's radioactivity changes over time.

Natural Logarithms

Natural logarithms help transform exponential equations into linear ones, making them easier to solve. They use the base \( e \), which is approximately equal to 2.71828. In the context of radioactive decay, taking the natural logarithm of both sides of the equation \( \frac{610}{630} = e^{-60\lambda} \) assists in finding the decay constant:

• \( \ln \left( \frac{610}{630} \right) = -60\lambda \)

This transformation allows us to isolate the decay constant \( \lambda \) and solve:

• \( \lambda = -\frac{\ln \left( \frac{610}{630} \right)}{60} \)

For easier computation with decimal logarithms, the natural logarithm \( \ln \) can be converted to a common logarithm \( \log \) using the conversion factor 2.303. Thus:

• \( \ln \left( \frac{610}{630} \right) = 2.303 \log \left( \frac{610}{630} \right) \)

This assists in practical scenarios where calculators typically use the common logarithm. In sum, natural logarithms simplify complex exponential decay equations and are essential in calculating parameters involved in natural processes like radioactive decay.