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Suppose the potential energy between electron and proton at a distance r is vary as U ∝ r2. Assuming Bohr's model of quantization of angular momentum and circular orbits, radius of the nth allowed orbit is proportional to
- a)n
- b)√n
- c)n1/3
- d)n2
Correct answer is option 'B'. Can you explain this answer?
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Suppose the potential energy between electron and proton at a distance...
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Suppose the potential energy between electron and proton at a distance...
Explanation:
Bohr's Model of Quantization of Angular Momentum:
- According to Bohr's model, the angular momentum of an electron in a circular orbit is quantized and given by:
\[ mvr = \dfrac{nh}{2\pi} \]
where m is the mass of the electron, v is the velocity of the electron, r is the radius of the orbit, n is a positive integer called the principal quantum number, and h is the Planck's constant.
Relationship between Potential Energy and Radius:
- The potential energy between an electron and a proton at a distance r is given to vary as \( U \propto r^2 \).
- The potential energy is given by:
\[ U = -\dfrac{kq_1q_2}{r} \]
where k is the electrostatic constant, q1 and q2 are the charges of the electron and proton respectively.
- Since the potential energy is proportional to \( r^2 \), we can write:
\[ U \propto r^2 \]
Relation between Radius and Principal Quantum Number:
- From Bohr's quantization condition, we have:
\[ mvr = \dfrac{nh}{2\pi} \]
- The angular momentum is quantized in integer multiples of \( \dfrac{h}{2\pi} \), where n is the principal quantum number.
- The radius of the nth allowed orbit is proportional to:
\[ r \propto n^2 \]
Conclusion:
- Therefore, the radius of the nth allowed orbit is proportional to \( \sqrt{n} \), as the radius is proportional to the square of the principal quantum number. Hence, the correct answer is option B.
Bohr's Model of Quantization of Angular Momentum:
- According to Bohr's model, the angular momentum of an electron in a circular orbit is quantized and given by:
\[ mvr = \dfrac{nh}{2\pi} \]
where m is the mass of the electron, v is the velocity of the electron, r is the radius of the orbit, n is a positive integer called the principal quantum number, and h is the Planck's constant.
Relationship between Potential Energy and Radius:
- The potential energy between an electron and a proton at a distance r is given to vary as \( U \propto r^2 \).
- The potential energy is given by:
\[ U = -\dfrac{kq_1q_2}{r} \]
where k is the electrostatic constant, q1 and q2 are the charges of the electron and proton respectively.
- Since the potential energy is proportional to \( r^2 \), we can write:
\[ U \propto r^2 \]
Relation between Radius and Principal Quantum Number:
- From Bohr's quantization condition, we have:
\[ mvr = \dfrac{nh}{2\pi} \]
- The angular momentum is quantized in integer multiples of \( \dfrac{h}{2\pi} \), where n is the principal quantum number.
- The radius of the nth allowed orbit is proportional to:
\[ r \propto n^2 \]
Conclusion:
- Therefore, the radius of the nth allowed orbit is proportional to \( \sqrt{n} \), as the radius is proportional to the square of the principal quantum number. Hence, the correct answer is option B.
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Suppose the potential energy between electron and proton at a distance r is vary as U ∝r2. Assuming Bohrs model of quantization of angular momentum and circular orbits, radius of the nth allowed orbit is proportional toa)nb)√nc)n1/3d)n2Correct answer is option 'B'. Can you explain this answer?
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