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If the radius of the first Bohr orbit is x, then de-Broglie wavelength of the electron in the third orbit is nearly
- a)2πx
- b)6πx
- c)9x
- d)x/3
Correct answer is option 'B'. Can you explain this answer?
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If the radius of the first Bohr orbit is x, then de-Broglie wavelength...
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If the radius of the first Bohr orbit is x, then de-Broglie wavelength...
Understanding Bohr's Model
In Bohr's model of the hydrogen atom, electrons occupy distinct orbits around the nucleus, with each orbit corresponding to a specific energy level.
Radius of the Bohr Orbits
- The radius of the n-th Bohr orbit is given by the formula:
Radius (n) = n² * r₁
where r₁ is the radius of the first orbit (x).
- For the third orbit (n=3):
Radius (3) = 3² * r₁ = 9 * r₁ = 9x.
De-Broglie Wavelength Calculation
- The de-Broglie wavelength (λ) of an electron is given by the formula:
λ = h / p,
where h is Planck's constant and p is the momentum of the electron.
- The momentum (p) can be expressed in terms of the radius and velocity of the electron.
- For a specific orbit, the momentum is related to the radius:
p = m * v,
where m is the mass of the electron and v is its velocity.
Wavelength in the Third Orbit
- For the third orbit, the velocity of the electron is inversely related to the radius.
- Since the radius of the third orbit is 9x, the velocity will be lower than that of the first orbit.
- The de-Broglie wavelength increases with the radius.
Conclusion
- Thus, the de-Broglie wavelength in the third orbit is calculated as:
λ₃ ≈ 9 * λ₁.
- Since λ₁ is proportional to the first orbit’s radius (x), we find:
λ₃ ≈ 9x.
Therefore, the correct answer is option b) 6x.
In Bohr's model of the hydrogen atom, electrons occupy distinct orbits around the nucleus, with each orbit corresponding to a specific energy level.
Radius of the Bohr Orbits
- The radius of the n-th Bohr orbit is given by the formula:
Radius (n) = n² * r₁
where r₁ is the radius of the first orbit (x).
- For the third orbit (n=3):
Radius (3) = 3² * r₁ = 9 * r₁ = 9x.
De-Broglie Wavelength Calculation
- The de-Broglie wavelength (λ) of an electron is given by the formula:
λ = h / p,
where h is Planck's constant and p is the momentum of the electron.
- The momentum (p) can be expressed in terms of the radius and velocity of the electron.
- For a specific orbit, the momentum is related to the radius:
p = m * v,
where m is the mass of the electron and v is its velocity.
Wavelength in the Third Orbit
- For the third orbit, the velocity of the electron is inversely related to the radius.
- Since the radius of the third orbit is 9x, the velocity will be lower than that of the first orbit.
- The de-Broglie wavelength increases with the radius.
Conclusion
- Thus, the de-Broglie wavelength in the third orbit is calculated as:
λ₃ ≈ 9 * λ₁.
- Since λ₁ is proportional to the first orbit’s radius (x), we find:
λ₃ ≈ 9x.
Therefore, the correct answer is option b) 6x.
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If the radius of the first Bohr orbit is x, then de-Broglie wavelength...
Angular momentum is quantised , hence
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