Problem 35 - Chemistry: An Atoms-Focused Approach

Question

What is the half-life of \(a^{199}\) Au if \(16.5 \%\) of it decays in 168 hours?

Step-by-Step Solution

Answer

Answer: The half-life of \(a^{199}\) Au is approximately 407.71 hours.

Step 1: Understanding the half-life formula

The formula we will use is derived from the decay rate equation: \(\displaystyle N\thinspace (\!t\!) =N_{0} e^{ \operatorname{-}\lambda t}\), where \(N(t)\) is the amount of the sample remaining at time \(t\), \(N_{0}\) is the initial amount, \(e\) is the base of the natural logarithm, and \(\lambda\) is the decay rate. We will find the half-life \(T_{\frac{1}{2}}\) by using the formula: \(\displaystyle T_{\frac{1}{2}}=\frac{\ln 2}{\lambda}\).

Step 2: Analyzing the given information

The problem states that 16.5% of the \(a^{199}\) Au decays in 168 hours. Therefore, 83.5% (\(\displaystyle 100\% \operatorname{-}16.5\%\)) remains after 168 hours. We can now use this information to find the decay rate, \(\lambda\).

Step 3: Calculating the decay rate

As we know the initial amount, \(\displaystyle N_{0}\), the remaining amount, \(N(168)\), and \(t = 168\thinspace h\) we can plug these values into the decay equation to solve for the decay constant, \(\lambda\). \(\displaystyle \frac{N\thinspace (\!t\!)}{N_{0}} =\frac{N(\!168\!)}{N_{0}} =0.835= e^{ \operatorname{-}\lambda 168}\). To find the decay rate, \(\lambda\), we can take the natural logarithm of both sides of the equation and solve for \(\lambda\): \(\displaystyle \ln 0.835= \operatorname{-}\lambda 168 \Rightarrow \lambda =\frac{\ln 0.835}{\operatorname{-}168}\)

Step 4: Calculating the half-life

Now that we have the decay rate, we can find the half-life of \(a^{199}\) Au by plugging in the value of \(\lambda\) we just found into the half-life formula: \(\displaystyle T_{\frac{1}{2}}=\frac{\ln 2}{\lambda} =\frac{\ln 2}{\displaystyle \frac{\ln 0.835}{\operatorname{-}168}}\)

Step 5: Simplifying the expression and finding the result

Simplifying the expression, we get: \(\displaystyle T_{\frac{1}{2}}=\frac{\ln 2\thinspace (\!-\!168\!)}{\ln 0.835}\) Now, calculate the numerical value of \(\displaystyle T_{\frac{1}{2}}\): \(\displaystyle T_{\frac{1}{2}}\approx 407.71\thinspace h\) Therefore, the half-life of \(a^{199}\) Au is approximately 407.71 hours.

Key Concepts

Decay Rate

The decay rate, often symbolized by the Greek letter lambda (\(\lambda\)), is a crucial concept in understanding how substances decay over time. In this context, we're dealing with substances that decrease at a rate dependent on their current amount. The decay rate can be thought of as a measure of how quickly this decrease happens.

To calculate the decay rate in our example, we used the exponential decay formula:

• \(N(t) = N_0 e^{-\lambda t}\)
• Where \(N(t)\) is the remaining amount at time \(t\), \(N_0\) is the initial amount, and \(e\) is the base of the natural logarithm.

By knowing the percentage of the substance that remains after a certain amount of time, we can solve the formula for \(\lambda\). In our exercise, we have 83.5% remaining after 168 hours, which leads to the equation \(0.835 = e^{-\lambda 168}\). By taking the natural logarithm of both sides and rearranging, we find the decay rate.

Exponential Decay

Exponential decay is a process where the quantity diminishes at a rate proportional to its current value. This concept is paramount in understanding many natural and scientific phenomena, such as radioactive decay, population decline, and even financial depreciation.

In the realm of exponential decay, the amount of a substance decreases continuously and asymptotically approaches zero over time. The formula representing this decay is:

• \(N(t) = N_0 e^{-\lambda t}\)
• Here, \(N(t)\) represents the amount left after time \(t\), \(N_0\) is the initial quantity, and \(\lambda\) is the decay rate.

An important feature of exponential decay is that the rate of decay slows down as time progresses. For example, once you know the percentage of decrease (like the 16.5% decay after 168 hours from the problem), you can determine how much substance decreases further using the same mathematical principles.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is the logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. The natural logarithm is notably used in calculating exponential growth or decay processes.

In our exercise, the natural logarithm is used to rearrange and solve the exponential decay formula. For instance, when dealing with the equation \(0.835 = e^{-\lambda 168}\), to find the decay rate, \(\lambda\), we take the natural logarithm of both sides, resulting in \(\ln 0.835 = -\lambda 168\). This simplification allows us to isolate \(\lambda\) and thus fully solve the decay rate.

The natural logarithm provides a straightforward way to linearize exponential relationships, making complex exponential equations manageable and solvable.