Problem 56 - College Physics

Question

The nucleus \(_{8}^{15} \mathrm{O}\) has a half-life of 2.0 min. \(_{8}^{19}\mathrm{O}\) has a half-life of about 0.5 min. (a) If, at some, time, a sample contains equal amounts of \(_{8}^{15} \mathrm{O}\) and \(_{8}^{19} \mathrm{O},\) what is the ratio of \(_{8}^{15} \mathrm{O}\) to \(_{8}^{19} \mathrm{O}\) after 2.0 min? (b) After 10.0 min?

Step-by-Step Solution

Answer

(a) 8:1 after 2 min; (b) 32768:1 after 10 min.

Step 1: Understand Half-life

The half-life of a radioactive substance is the time it takes for half of the material to decay. In this problem, 1. For _{8}^{15} \( \mathrm{O} \), the half-life is 2.0 minutes,2. For \( _{8}^{19} \mathrm{O} \), the half-life is 0.5 minutes.We need to find how much of each substance remains after given time intervals.

Step 2: Formulate General Decay Formula

The decay of a radioactive element can be described by the formula:\( N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}}\)where \(N(t)\) represents the amount remaining at time \(t\), \(N_0\) represents the initial amount, and \(T_{1/2}\) is the half-life.

Step 3: Calculate Remaining \( _{8}^{15} O \) after 2 Min

Since the half-life of \( _{8}^{15} \mathrm{O} \) is 2.0 minutes, after 2 minutes, half of it will remain:\( N_{\,_{8}^{15}O}(2) = \frac{N_0}{2}\)Therefore, quantity remaining = \(\frac{N_0}{2}\).

Step 4: Calculate Remaining \( _{8}^{19} O \) after 2 Min

Since the half-life of \( _{8}^{19} \mathrm{O} \) is 0.5 minutes, after 2 minutes, we calculate the decay through two half-lives:\( N_{\,_{8}^{19}O}(2) = N_0 \left( \frac{1}{2} \right)^{\frac{2}{0.5}} = N_0 \left( \frac{1}{2} \right)^4 = \frac{N_0}{16}\)

Step 5: Calculate Ratio After 2 Min

The ratio of \( _{8}^{15} \mathrm{O} \) to \( _{8}^{19} \mathrm{O} \) after 2 minutes is:\( \frac{N_{\,_{8}^{15}O}(2)}{N_{\,_{8}^{19}O}(2)} = \frac{\frac{N_0}{2}}{\frac{N_0}{16}} = 8\)So, the ratio is 8:1.

Step 6: Calculate Remaining \( _{8}^{15} O \) after 10 Min

After 10 minutes, we calculate using \(T_{1/2} = 2\) minutes:\( N_{\,_{8}^{15}O}(10) = N_0 \left( \frac{1}{2} \right)^{\frac{10}{2}} = \frac{N_0}{32}\)

Step 7: Calculate Remaining \( _{8}^{19} O \) after 10 Min

After 10 minutes, we calculate using \(T_{1/2} = 0.5\) minutes:\( N_{\,_{8}^{19}O}(10) = N_0 \left( \frac{1}{2} \right)^{\frac{10}{0.5}} = N_0 \left( \frac{1}{2} \right)^{20} = \frac{N_0}{1048576}\)

Step 8: Calculate Ratio After 10 Min

The ratio of \( _{8}^{15} \mathrm{O} \) to \( _{8}^{19} \mathrm{O} \) after 10 minutes is:\( \frac{N_{\,_{8}^{15}O}(10)}{N_{\,_{8}^{19}O}(10)} = \frac{\frac{N_0}{32}}{\frac{N_0}{1048576}} = \frac{1048576}{32} = 32768\)So, the ratio is 32768:1.

Key Concepts

Half-life

In radioactive decay, understanding the concept of half-life is crucial. It refers to the period needed for half of a sample of a radioactive substance to decay into another substance. This measure provides insight into how long an isotope remains active before significant decay occurs.
For example, in our exercise, the half-life of the isotope \(_{8}^{15} \mathrm{O}\) is 2.0 minutes, meaning that in 2 minutes, half of its sample would have decayed into a different element. Similarly, \(_{8}^{19} \mathrm{O}\) has a half-life of 0.5 minutes, indicating that in only 30 seconds, half of it transforms into another substance.
Understanding half-life helps us calculate how long it will take for a radioactive substance to reduce to a particular amount. It's a pivotal concept in radiochemistry and nuclear physics because it lays the foundation for safely handling radioactive materials and determining timelines for their natural decay.

Decay Formula

The decay formula is essential for calculating how quickly a radioactive isotope decays over time. This mathematical equation is expressed as: \[ N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \]Here, \(N(t)\) stands for the amount of substance remaining at time \(t\), \(N_0\) is the initial quantity of the substance, and \(T_{1/2}\) is the half-life of the substance.
Using this formula, you can predict how much of an isotope will remain after a certain time has passed. For instance, with \(_{8}^{15} \mathrm{O}\), after 2 minutes, its amount reduces to half because its half-life is 2 minutes. For \(_{8}^{19} \mathrm{O}\), with a half-life of 0.5 minutes, the calculations after 2 minutes show significant decay via four half-lives.
The decay formula's elegance lies in its ability to model exponential decay, making it an essential tool in sciences that deal with radioactivity. It helps predict decay rates accurately, which is vital for various practical applications such as nuclear medicine and radioactive dating.

Isotope Calculation

Calculating isotopic decay and ratios requires a firm grip on the half-life and decay formula concepts. In the given exercise, equal initial amounts of \(_{8}^{15} \mathrm{O}\) and \(_{8}^{19} \mathrm{O}\) are used. Over time, as these isotopes decay, their remains can be calculated using the decay formula.
For \(_{8}^{15} \mathrm{O}\), after 2 minutes, it decays through one half-life, leaving half its initial amount. Meanwhile, \(_{8}^{19} \mathrm{O}\) undergoes four half-lives in the same period, leaving just \( \frac{1}{16} \) of its initial amount. The ratio between them at this time is 8:1.
When extended to 10 minutes, calculations show \(_{8}^{15} \mathrm{O}\) reduces through five half-lives to \( \frac{1}{32} \) of its initial amount. \(_{8}^{19} \mathrm{O}\) undergoes twenty half-lives, shrinking to \( \frac{1}{1048576} \). The resulting dramatic ratio is 32768:1.
Accurate isotope calculation helps us discern how different isotopes behave under identical timeframes, guiding important decisions in fields such as radiotherapy and archaeology.