Half-Life: Comprehensive NEET Physics Notes

1. Half-Life in Radioactive Decay

1.1 Definition of Half-Life

The half-life of a radioactive substance is the time required for half of the radioactive nuclei in a sample to decay. It is a characteristic property of each radioactive isotope and remains constant, irrespective of the initial amount of the substance.

• Mathematical Expression: If $N_0$ is the initial quantity of radioactive nuclei, after one half-life, the quantity remaining is:
  $N=\frac{N_0}{2}$
  After two half-lives:
  $N=\frac{N_0}{4}$
  This relationship continues for each successive half-life.

1.2 Formula for Half-Life

For a first-order decay process like radioactive decay, the relationship between the decay constant $\lambda$ (which represents the probability of decay per unit time) and the half-life $t_{1/2}$ is expressed by the formula:

$t_{1/2}=\frac{0.693}{\lambda }$

Where:

• $t_{1/2}$ is the half-life,
• $\lambda$ is the decay constant.

A larger value of $\lambda$ indicates a faster decay, resulting in a shorter half-life.

NEET Tip:

For NEET problems, always remember the relation between half-life and the decay constant. This formula is often used in solving numerical problems related to radioactive decay.

1.3 Application of Half-Life in Radioactive Decay

Radioactive isotopes decay at predictable rates, making half-life a valuable tool in several applications:

• Carbon Dating: Used to determine the age of ancient objects by measuring the remaining carbon-14 content.
• Medical Uses: Radioisotopes like iodine-131, with a half-life of approximately 8 days, are utilized in cancer treatment and diagnostic imaging.

Real-Life Application:

Carbon dating helps archaeologists determine the age of fossils and artifacts by measuring the remaining carbon-14, which decays over time.

2. Detailed Derivation of the Half-Life Equation

2.1 First-Order Reaction in Radioactive Decay

Radioactive decay follows first-order kinetics, meaning the rate of decay depends on the number of undecayed nuclei. The general form of the first-order reaction rate equation is:

$\frac{dN}{dt}=-\lambda N$

Where:

• $N$ is the number of undecayed nuclei at time $t$,
• $\lambda$ is the decay constant.

By integrating this equation, we derive the following:

$N=N_0{e}^{-\lambda t}$

Where $N_0$ is the initial number of radioactive nuclei at time $t=0$.

To find the half-life, set $N=\frac{N_0}{2}$ at time $t=t_{1/2}$, which leads to:

$t_{1/2}=\frac{0.693}{\lambda }$

NEET Problem-Solving Strategy:

Use the half-life formula to solve questions quickly, especially when given the decay constant $\lambda$. Most NEET questions will provide the decay constant or half-life to calculate one or the other.

3. Visual Representation of Half-Life

3.1 Decay Curve

A decay curve is a graphical representation of the decrease in the quantity of a radioactive substance over time. It shows an exponential decay, where the number of undecayed nuclei decreases by half in each successive half-life period.

To visualize this:

• The y-axis represents the number of undecayed nuclei.
• The x-axis represents time.

At each half-life interval, the number of remaining nuclei halves, showing a characteristic exponential curve.

4. Examples and Problem-Solving

4.1 Example 1: Calculating Half-Life

A radioactive isotope has a decay constant $\lambda =0.01{\text{s}}^{-1}$. Calculate its half-life.

Solution: Using the formula:

$t_{1/2}=\frac{0.693}{\lambda }$

Substitute the given value:

$t_{1/2}=\frac{0.693}{0.01}=69.3,\text{seconds}$

Thus, the half-life is 69.3 seconds.

4.2 Example 2: Radioactive Decay Over Multiple Half-Lives

A radioactive sample initially contains 1000 nuclei. How many nuclei remain after 3 half-lives?

Solution: After 1 half-life:

$N_1=\frac{1000}{2}=500$

After 2 half-lives:

$N_2=\frac{500}{2}=250$

After 3 half-lives:

$N_3=\frac{250}{2}=125$

Therefore, 125 nuclei will remain after 3 half-lives.

5. Quick Recap

• The half-life is the time required for half of the radioactive nuclei in a sample to decay.
• It is calculated using the formula $t_{1/2}=\frac{0.693}{\lambda }$.
• Radioactive decay follows first-order kinetics, meaning the rate of decay depends on the number of undecayed nuclei.
• Half-life applications include carbon dating and medical uses such as cancer treatment.

6. Practice Questions

Question 1:

A radioactive substance has a half-life of 10 minutes. If the initial amount is 800 nuclei, how many will remain after 30 minutes?

Question 2:

The half-life of a radioactive isotope is 20 seconds, and its decay constant is $0.034{\text{s}}^{-1}$. Calculate its half-life.

Question 3:

A radioactive substance decays to 25% of its original quantity in 60 minutes. What is its half-life?

Question 4:

After 4 half-lives, what fraction of a radioactive substance remains undecayed?

Question 5:

The half-life of a sample of iodine-131 is 8 days. How long will it take for a 100 mg sample to decay to 12.5 mg?

Answers:

1. After 30 minutes (which is 3 half-lives), 100 nuclei will remain.
2. $t_{1/2}=\frac{0.693}{0.034}=20.4\text{seconds}$.
3. Half-life is 30 minutes.
4. $\frac{1}{16}$ of the sample remains after 4 half-lives.
5. It will take 24 days for the sample to decay to 12.5 mg.

7. Glossary

• Half-Life ($t_{1/2}$): The time required for half of the radioactive nuclei in a sample to decay.
• Decay Constant ($\lambda$): The probability per unit time that a nucleus will decay.
• Radioactive Decay: The spontaneous process by which unstable nuclei emit radiation and transform into different elements or isotopes.

8. Final Recommendations

1. Add more visual aids: Incorporating a graph of exponential decay and decay curves would enhance understanding.
2. Expand practice questions: Add more NEET-style multiple-choice questions to simulate exam conditions.
3. Incorporate more interactive learning techniques: Use mnemonics and analogies to help students retain key formulas and concepts.