Problem 127 - A Complete Resource Book in Chemistry for JEE Main

Question

How much time is required for a \(5.75-\mathrm{mg}\) sample of \({ }^{51} \mathrm{Cr}\) to decay to \(1.50 \mathrm{mg}\) if it has a half-life of \(27.8\) days? (a) \(5.39\) days (b) \(2.69\) days (c) \(53.9\) days (d) \(5.49\) days

Step-by-Step Solution

Answer

(c) 53.9 days.

Step 1: Identify Known Values

We know the initial mass \( m_0 \) of \( {}^{51}\mathrm{Cr} \) is \( 5.75 \ \mathrm{mg} \) and the remaining mass \( m \) is \( 1.50 \ \mathrm{mg} \). The half-life \( t_{1/2} \) of \( {}^{51}\mathrm{Cr} \) is \( 27.8 \) days.

Step 2: Use the Decay Formula

The formula for radioactive decay is given by \( m = m_0 \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \), where \( m \) is the remaining mass, \( m_0 \) is the initial mass, \( t \) is time, and \( t_{1/2} \) is the half-life.

Step 3: Rearrange the Formula to Solve for Time

We want to find \( t \). Rearrange the decay formula to find \( t \):\[ t = t_{1/2} \times \frac{\log\left( \frac{m}{m_0} \right)}{\log\left( \frac{1}{2} \right)} \]

Step 4: Calculate the Logarithms

First, calculate the values of the logarithms needed:- \( \log\left( \frac{m}{m_0} \right) = \log\left( \frac{1.50}{5.75} \right) \approx -0.585 \)- \( \log\left( \frac{1}{2} \right) \approx -0.301 \)

Step 5: Solve for Time \( t \)

Substitute the logarithms and known half-life into the rearranged formula:\[ t = 27.8 \times \frac{-0.585}{-0.301} \approx 53.9 \text{ days} \]

Step 6: Compare with Given Options

From the calculation, \( t \) is approximately \( 53.9 \) days, which matches option (c).

Key Concepts

Understanding Half-Life Calculation

In radioactive decay, half-life is a crucial term. It refers to the time needed for half of the radioactive isotopes in a sample to decay. Understanding this concept is vital for solving decay-related problems.

When you begin with a certain amount of a radioactive material, say a sample of chromium-51, over time, its quantity decreases as it decays. This rate of decrease can be measured by the half-life.

• Initial mass: The starting amount of the radioactive substance.
• Remaining mass: The amount left after a certain period.
• Half-life (\(t_{1/2}\)): The time it takes for half of the sample to decay.

For chromium-51 (\({}^{51}\mathrm{Cr}\)), knowing that its half-life is 27.8 days helps us determine how long it takes for the isotope quantity to reduce from any initial mass to a desired value.

Given a problem, you can apply the decay formula:
\[ m = m_0 \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \]
Here, \(m_0\) denotes the original amount, \(m\) is the remaining mass, and \(t\) is the desired time, while \(t_{1/2}\) represents the half-life.

Logarithmic Functions in Decay

Logarithms are highly useful in calculations involving exponential decay. The decay formula can require the rearrangement and use of logarithms when determining the time it takes for a radioactive sample to decay to a particular mass.

In this case, we take the decay formula and rearrange it into:
\[ t = t_{1/2} \times \frac{\log\left( \frac{m}{m_0} \right)}{\log\left( \frac{1}{2} \right)} \]
This formula simplifies the process by enabling us to solve for \(t\) directly when we know the other variables: initial mass \(m_0\), remaining mass \(m\), and the half-life \(t_{1/2}\).

• Calculate \(\log\left( \frac{m}{m_0} \right)\) to understand how much the sample has decreased relative to its initial amount.
• The constant \(\log\left( \frac{1}{2} \right)\) arises from the half-life concept itself and equates to approximately -0.301.

By inserting the known values and solving, we can find \(t\), the time it takes for the decay to reach a specified remaining mass from the original quantity.

Specific Decay of Chromium-51

Chromium-51 is an isotope used in various scientific applications, including medical diagnostics. Its decay process follows the general rules of radioactive decay, featuring a distinct half-life, which for \({}^{51}\mathrm{Cr}\) is 27.8 days.

Consider a problem where you're given a sample initially weighing 5.75 mg of chromium-51, and you are told to find how long it takes for this sample to decay to 1.50 mg.

• Initial Mass: 5.75 mg
• Remaining Mass: 1.50 mg
• Half-Life: 27.8 days

Using the decay formula, substitute these values to find the time \(t\). Calculate the logarithms as described in the problem:
\[ \log\left( \frac{1.50}{5.75} \right) \approx -0.585 \]\[ \log\left( \frac{1}{2} \right) \approx -0.301 \]
Multiply the half-life by the ratio of the logarithms to find \(t\):
\[ t = 27.8 \times \frac{-0.585}{-0.301} \approx 53.9 \text{ days} \]
The solution reveals that it takes about 53.9 days for the chromium-51 sample to decay from 5.75 mg to 1.50 mg.