Problem 36 - Chemistry The Central Science In Si Units

Question

It takes 180 minutes for a 200 -mg sample of an unknown radioactive substance to decay to \(112 \mathrm{mg}\). What is the halflife of this substance?

Step-by-Step Solution

Answer

The half-life of the substance is approximately 183 minutes.

Step 1: Understanding the Problem

We need to find the half-life of a radioactive substance. We know the initial mass, the final mass after 180 minutes, and need to determine the time it takes for the mass to decay to half its initial value.

Step 2: Using the Exponential Decay Formula

The formula for exponential decay is given by \[N(t) = N_0 imes e^{-kt}\]Where \(N(t)\) is the final amount of the substance, \(N_0\) is the initial amount, \(k\) is the decay constant, and \(t\) is the time elapsed.

Step 3: Substituting Known Values

Substitute \(N_0 = 200 \text{ mg}\), \(N(t) = 112 \text{ mg}\), and \(t = 180 \text{ minutes}\) into the formula, \[112 = 200 imes e^{-180k}\]

Step 4: Solving for the Decay Constant

Rearrange the equation to solve for \(k\): \[\frac{112}{200} = e^{-180k}\]Take the natural logarithm of both sides: \[\ln\left(\frac{112}{200}\right) = -180k\]Simplifying gives:\[k = -\frac{1}{180} \ln\left(\frac{112}{200}\right)\]

Step 5: Calculating the Decay Constant

Compute \(k\):\[k \approx -\frac{1}{180} \times \ln(0.56)\]Calculating this yields approximately \(k \approx 0.0038 \text{ per minute}\).

Step 6: Finding the Half-Life using the Decay Constant

The half-life \(T_{1/2}\) can be found using the formula:\[T_{1/2} = \frac{\ln(2)}{k}\]Substitute the calculated \(k\) into the formula.

Step 7: Calculating the Half-Life

Plug in the value of \(k\) to find the half-life:\[T_{1/2} \approx \frac{\ln(2)}{0.0038}\]Computing this gives approximately \(T_{1/2} \approx 182.98 \text{ minutes}\).

Key Concepts

Half-Life Calculation

When talking about radioactive decay, the half-life of a substance is extremely important. It tells us how long it takes for half of the substance to decay away. In practical terms, if you start with 200 mg of a substance, after one half-life, only 100 mg will be left.

Calculating the half-life involves using the decay constant, which links how quickly a substance decays over time. Remember, each different substance can have a very different half-life. Some can be over in seconds, while others take years. For calculation, use the formula:

• \[ T_{1/2} = \frac{\ln(2)}{k} \]
• Here, \( T_{1/2} \) is the half-life, and \( k \) is the decay constant.

In our earlier problem, we discovered a half-life of about 183 minutes for the substance, meaning every 183 minutes, the sample decreases by half. This predictable pattern is what makes the half-life such a powerful concept.

Exponential Decay Formula

Radioactive decay describes a process where the quantity of a substance decreases over time. This pattern is not linear but exponential, which means the rate of decay is proportional to the amount present.

The exponential nature of decay can be described using a specific mathematical formula:

• \[ N(t) = N_0 \times e^{-kt} \]
• Where \( N(t) \) is the amount left after time \( t \), \( N_0 \) is the initial amount, and \( k \) is the decay constant.

This formula helps us calculate how much of a sample remains after any given amount of time. For example, if 112 mg of a 200 mg sample of a radioactive substance remains after 180 minutes, this formula was used to calculate that. Understanding and using the exponential decay formula is key to mastering tasks involving radioactivity.

Decay Constant

The decay constant, often denoted by \( k \), represents how quickly a substance undergoes radioactive decay. A larger decay constant means the substance decays faster.

In our example, we calculated \( k \) using the exponential decay formula. First, by rearranging and substituting known values:

• \[ \ln\left(\frac{112}{200}\right) = -180k \]
• Giving us the decay constant:
• \[ k \approx -\frac{1}{180} \times \ln(0.56) \approx 0.0038 \text{ per minute} \]

This means every minute, the substance decays at a rate of 0.0038 times its current amount. Understanding the decay constant helps us predict how substances will behave over time, which is essential for calculating both the remaining quantity and the half-life. By grasping how the decay constant influences decay processes, you gain useful insight into the nature of radioactive substances.