Quick Summary
This question tests the concept of atoms and nuclei, specifically the properties of muonic hydrogen atoms, with a difficulty rating of 3 stars, requiring 6 minutes to solve. The key formula is $$r_n = \frac{n^2 h^2 \epsilon_0}{\pi \mu e^2}$$. The answer is D, as the radius of the muonic hydrogen atom is inversely proportional to the mass of the orbiting particle.
The Question
Given that the mass of a muon is $m_\mu = 200 m_e$, where $m_e$ is the mass of an electron, and the charge of the muon is $|q_\mu| = |e|$, which of the following statements about the muonic hydrogen atom are correct?
(A) Radius of hydrogen (A) Radius of muon = 200
(B) Velocity relation given is wrong
(C) Ionization energy
(D) Radius of hydrogen (A) Radius of muon = 200, Velocity relation given is wrong, Ionization energy
Quick Answer
The correct answer is D. This is because the radius of the muonic hydrogen atom is indeed inversely proportional to the mass of the muon, the velocity relation given is incorrect as it doesn’t account for the mass independence in the Bohr model, and the ionization energy is directly proportional to the mass of the orbiting particle.
Why Other Options Are Incorrect
Option A
Option A is partially correct in stating the relationship between the radius of hydrogen and the radius of the muon but does not address the velocity relation or ionization energy, making it an incomplete choice.
Option B
Option B is incorrect because the velocity relation given is indeed wrong. The velocity of the electron or muon in the Bohr model of a hydrogen-like atom does not depend on the mass of the orbiting particle, contradicting the statement.
Option C
Option C is partially correct in addressing the ionization energy but does not consider the radius or the incorrect velocity relation, thus it is not a comprehensive choice.
Video Solution
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Understanding the Concept
The concept tested here involves understanding the properties of muonic hydrogen atoms, which are hydrogen atoms where the electron is replaced by a muon. The key principle is the Bohr model of the atom, where the energy levels and radius of the orbiting particle can be calculated using the formulas $$r_n = \frac{n^2 h^2 \epsilon_0}{\pi \mu e^2}$$ and $$E_n = -\frac{\mu e^4}{8 \epsilon_0^2 n^2 h^2}$$, with $\mu$ being the reduced mass of the system.
Detailed Step-by-Step Solution
Step 1: Calculate the Radius of the Muonic Hydrogen Atom
The radius of a hydrogen-like atom is given by $$r_n = \frac{n^2 h^2 \epsilon_0}{\pi \mu e^2}$$. For a muonic hydrogen atom, $\mu = m_\mu$, so $$r_\mu = \frac{n^2 h^2 \epsilon_0}{\pi m_\mu e^2}$$. Given $m_\mu = 200 m_e$, $$r_\mu = \frac{r_H}{200}$$, where $r_H$ is the radius of the hydrogen atom.
Step 2: Examine the Velocity Relation
The velocity of the electron or muon in a hydrogen-like atom is given by $$v = \frac{e^2}{2 \epsilon_0 n h}$$. This formula shows that the velocity is independent of the mass of the orbiting particle, thus the statement about the velocity relation being wrong is correct.
Step 3: Calculate the Ionization Energy
The ionization energy of a hydrogen-like atom is $$E = \frac{\mu e^4}{8 \epsilon_0^2 n^2 h^2}$$. For a muonic hydrogen atom, this becomes $$E_\mu = \frac{m_\mu e^4}{8 \epsilon_0^2 n^2 h^2} = 200 \frac{m_e e^4}{8 \epsilon_0^2 n^2 h^2} = 200 E_H$$, where $E_H$ is the ionization energy of hydrogen.
Final Answer
The final answer is: $\boxed{D}$
Essential Formulas for This Topic
$$r_n = \frac{n^2 h^2 \epsilon_0}{\pi \mu e^2}$$
$$E_n = -\frac{\mu e^4}{8 \epsilon_0^2 n^2 h^2}$$
$$v = \frac{e^2}{2 \epsilon_0 n h}$$
Common Mistakes to Avoid
Mistake 1: Assuming Velocity Scales with Mass
Students often mistakenly think that the velocity of the orbiting particle in a hydrogen-like atom scales with the mass of the particle, which is incorrect according to the Bohr model.
Mistake 2: Incorrectly Scaling Ionization Energy
Some students incorrectly believe that the ionization energy scales with the square of the mass of the orbiting particle, when in fact it scales linearly with mass.
Mistake 3: Forgetting Angular Momentum Quantization
Students may overlook that the angular momentum of the orbiting particle must satisfy $$m v r = \frac{n h}{2 \pi}$$, which is crucial for understanding the properties of hydrogen-like atoms.
Key Concept Summary
- The Bohr model applies to all hydrogen-like atoms.
- The radius of the orbit is inversely proportional to the reduced mass of the system.
- The velocity of the orbiting particle is independent of its mass.
- The ionization energy is directly proportional to the mass of the orbiting particle.
- The angular momentum of the orbiting particle is quantized.
The Golden Rule for hydrogen-like atoms is that their properties can be predicted using the Bohr model with the reduced mass of the system.
Frequently Asked Questions
Q: What is the difference between a hydrogen atom and a muonic hydrogen atom?
A: The main difference is that in a muonic hydrogen atom, the electron is replaced by a muon, which has a mass about 200 times that of an electron.
Q: How does the mass of the muon affect the radius of the muonic hydrogen atom?
A: The radius of the muonic hydrogen atom is inversely proportional to the mass of the muon, so it is 200 times smaller than the radius of a regular hydrogen atom.
Q: Is the velocity of the muon in a muonic hydrogen atom different from the velocity of the electron in a hydrogen atom?
A: No, according to the Bohr model, the velocity of the orbiting particle is independent of its mass, so the velocity of the muon and the electron are the same in their respective atoms.
Q: How does the ionization energy of a muonic hydrogen atom compare to that of a hydrogen atom?
A: The ionization energy of a muonic hydrogen atom is 200 times that of a hydrogen atom because it is directly proportional to the mass of the orbiting particle.
Prerequisites to Solve This Question
- Understanding of the Bohr model of the atom.
- Knowledge of the formulas for the radius and ionization energy of hydrogen-like atoms.
- Familiarity with the concept of reduced mass and its application to muonic hydrogen atoms.
After Solving This, You Can:
- ✔ Calculate the properties of other hydrogen-like atoms.
- ✔ Understand the implications of replacing an electron with a different particle in atomic physics.
- ✔ Apply the Bohr model to solve problems involving muonic atoms.
Study Tips for This Topic
To excel in this topic, focus on understanding the Bohr model and its applications to different types of hydrogen-like atoms. Practice calculating the radius, velocity, and ionization energy for various atoms, and make sure to grasp the concept of reduced mass and its significance.
Difficulty Rating & Exam Frequency
This question has a difficulty rating of 3 stars. It is moderately common in JEE Main and Advanced exams, with an importance level of 7 out of 10, as understanding the properties of hydrogen-like atoms is crucial for atomic physics and chemistry.
Written by Minal Kumari
Senior Physics Educator at Padho Likho JEE. Specializes in breaking down complex mechanics and electromagnetism concepts with structured problem-solving techniques for JEE and NEET aspirants.
Last Updated: July 2026
Question Source: JEE Main 2026 PYQ
Topic: Atoms And Nuclei