Problem 56
Question
Two radioactive elements \(\mathrm{A}\) and \(\mathrm{B}\) have decay constant \(\lambda\) and \(10 \lambda\) respectively. If the decay begins with the same number of atoms of the \(\mathrm{n}\), the ratio of atoms of \(\mathrm{A}\) to those of \(\mathrm{B}\) after time \(1 / 9 \lambda\) will be (a) \(\mathrm{e}^{-3}\) (b) \(\mathrm{e}^{2}\) (c) \(\mathrm{e}\) (d) \(\mathrm{e}^{-1}\)
Step-by-Step Solution
Verified Answer
The ratio is \( e \), choice (c).
1Step 1: Understand the Decay Formula
The number of atoms remaining after a certain time for an element with a decay constant \( \lambda \) is given by the formula \( N(t) = N_0 e^{-\lambda t} \), where \( N_0 \) is the initial number of atoms.
2Step 2: Apply Decay Formula to Element A
For element \( \mathrm{A} \), which has a decay constant \( \lambda \), the number of atoms after time \( t = \frac{1}{9 \lambda} \) is \( N_A(t) = n e^{-\lambda \cdot \frac{1}{9 \lambda}} = n e^{-\frac{1}{9}} \).
3Step 3: Apply Decay Formula to Element B
For element \( \mathrm{B} \), which has a decay constant \( 10\lambda \), the number of atoms after the same time \( t = \frac{1}{9 \lambda} \) is \( N_B(t) = n e^{-10\lambda \cdot \frac{1}{9 \lambda}} = n e^{-\frac{10}{9}} \).
4Step 4: Find the Ratio of Atoms
The ratio of remaining atoms of \( \mathrm{A} \) to \( \mathrm{B} \) after time \( t = \frac{1}{9 \lambda} \) is calculated by dividing \( N_A(t) \) by \( N_B(t) \). Hence, the ratio is \( \frac{n e^{-\frac{1}{9}}}{n e^{-\frac{10}{9}}} = e^{\frac{10}{9} - \frac{1}{9}} = e^{\frac{9}{9}} = e^{1} = e \).
5Step 5: Verify Answer
After confirming each step, verify that the calculations follow from the decay formula. The correct choice is (c) \( e \), indicating the provided steps match the potential solutions.
Key Concepts
Decay ConstantExponential Decay FormulaHalf-life
Decay Constant
In the world of radioactive decay, the decay constant (often represented by \( \lambda \)) is a crucial parameter that reflects how quickly a radioactive element disintegrates. It is a characteristic of each radioactive isotope and varies from one element to another. Think of the decay constant like a "rate of decay," indicating the likelihood of a single atom decaying in a given time period.
The decay constant is directly tied to the stability of the nucleus: the larger the value of \( \lambda \), the faster the decay process. Conversely, a smaller decay constant means a slower rate of decay. For example, in the exercise given, element \( \mathrm{A} \) has a decay constant of \( \lambda \), while element \( \mathrm{B} \) has a decay constant of \( 10\lambda \). This means element \( \mathrm{B} \) decays much faster than element \( \mathrm{A} \).
Radioactive decay is random at the level of individual atoms, but with a large collection of atoms, the decay process follows a predictable pattern. By understanding decay constants, scientists can make precise predictions about how much of a substance will remain after a certain time.
The decay constant is directly tied to the stability of the nucleus: the larger the value of \( \lambda \), the faster the decay process. Conversely, a smaller decay constant means a slower rate of decay. For example, in the exercise given, element \( \mathrm{A} \) has a decay constant of \( \lambda \), while element \( \mathrm{B} \) has a decay constant of \( 10\lambda \). This means element \( \mathrm{B} \) decays much faster than element \( \mathrm{A} \).
Radioactive decay is random at the level of individual atoms, but with a large collection of atoms, the decay process follows a predictable pattern. By understanding decay constants, scientists can make precise predictions about how much of a substance will remain after a certain time.
Exponential Decay Formula
The exponential decay formula is key to quantifying how radioactive substances transform over time. It mathematically encapsulates the way substances decrease in quantity as they undergo radioactive decay.
The formula is given by \( N(t) = N_0 e^{-\lambda t} \), where:
By utilizing this formula, one can determine how many atoms remain after a specified time interval. In practice, given an initial quantity and a time period, you can apply the exponential decay formula to quantify the remaining substance, as shown in the solution for both elements \( \mathrm{A} \) and \( \mathrm{B} \) in the exercise.
The formula is given by \( N(t) = N_0 e^{-\lambda t} \), where:
- \( N(t) \) is the number of atoms remaining at time \( t \).
- \( N_0 \) is the initial number of atoms.
- \( \lambda \) is the decay constant.
By utilizing this formula, one can determine how many atoms remain after a specified time interval. In practice, given an initial quantity and a time period, you can apply the exponential decay formula to quantify the remaining substance, as shown in the solution for both elements \( \mathrm{A} \) and \( \mathrm{B} \) in the exercise.
Half-life
The half-life of a substance is an important concept that complements the decay constant and exponential decay formula. It refers to the amount of time required for half of the radioactive atoms in a given sample to decay.
The relationship between half-life and the decay constant is given by the formula:
\[ T_{1/2} = \frac{\ln(2)}{\lambda} \]
Here, \( T_{1/2} \) is the half-life, and \( \ln(2) \) (the natural logarithm of 2) is approximately equal to 0.693.
This concept allows us to estimate how long a radionuclide will remain active. A substance with a short half-life decays quickly and becomes stable sooner, while a substance with a long half-life persists longer. Understanding half-lives helps in managing radioactive materials and planning for their safe use and disposal.
Even though half-life wasn't directly used in solving the exercise, it provides additional insight into how decay constants impact the longevity of radioactive elements.
The relationship between half-life and the decay constant is given by the formula:
\[ T_{1/2} = \frac{\ln(2)}{\lambda} \]
Here, \( T_{1/2} \) is the half-life, and \( \ln(2) \) (the natural logarithm of 2) is approximately equal to 0.693.
This concept allows us to estimate how long a radionuclide will remain active. A substance with a short half-life decays quickly and becomes stable sooner, while a substance with a long half-life persists longer. Understanding half-lives helps in managing radioactive materials and planning for their safe use and disposal.
Even though half-life wasn't directly used in solving the exercise, it provides additional insight into how decay constants impact the longevity of radioactive elements.
Other exercises in this chapter
Problem 54
A radioactive form of phosphorus undergoes Y-decay. What will the radioactivity level (R) versus time graph for the decay process appear?
View solution Problem 55
Nuclear reactors are usually surrounded by lead and concrete. Which of the following is this safety precaution particularly for? (a) neutron (b) \(\alpha\) part
View solution Problem 58
During a K-electron capture (a) X-rays are emitted (b) neutrous are emitted (c) \(\alpha\) particles are emitted (d) \(\gamma\) rays are emitted
View solution Problem 60
\({ }_{7} \mathrm{~N}^{13}\) changes to \({ }_{6} \mathrm{C}^{13}\) by the emission of (a) proton (b) electron (c) neutron (d) positron
View solution