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An electron is moving in Bohr's fourth orbit. Its de Broglie wavelength is λ. What is the circumference of the fourth orbit?
- a)2/λ
- b)2λ
- c)4λ
- d)4/λ
Correct answer is option 'C'. Can you explain this answer?
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An electron is moving in Bohr's fourth orbit. Its de Broglie wavelengt...
De Broglie Wavelength and the Circumference of the Fourth Orbit
Explanation:
According to Bohr's model of the atom, electrons revolve around the nucleus in circular orbits. Each orbit is characterized by a fixed energy level. The energy of an electron is quantized, meaning that it can only have certain discrete values.
De Broglie wavelength is the wavelength associated with a moving particle. It is given by the formula λ=h/p, where λ is the wavelength, h is Planck's constant, and p is the momentum of the particle.
The momentum of an electron in the nth orbit is given by the formula p=n(h/2π)r, where r is the radius of the orbit. Therefore, the de Broglie wavelength of an electron in the nth orbit is given by the formula λ=n(h/2πr).
We are given that the electron is moving in Bohr's fourth orbit, which means that n=4. Therefore, the de Broglie wavelength of the electron is given by the formula λ=4(h/2πr).
We need to find the circumference of the fourth orbit. The circumference of a circle is given by the formula C=2πr, where C is the circumference and r is the radius.
Solution:
We know that λ=4(h/2πr)
Multiplying both sides by 2πr, we get:
2πrλ=4h
Dividing both sides by λ, we get:
2πr=4h/λ
Using the formula for the circumference of a circle, we get:
C=2πr=4h/λ
Therefore, the circumference of the fourth orbit is 4 times the ratio of Planck's constant to the de Broglie wavelength of the electron.
Hence, the correct answer is option (c) 4λ.
Explanation:
According to Bohr's model of the atom, electrons revolve around the nucleus in circular orbits. Each orbit is characterized by a fixed energy level. The energy of an electron is quantized, meaning that it can only have certain discrete values.
De Broglie wavelength is the wavelength associated with a moving particle. It is given by the formula λ=h/p, where λ is the wavelength, h is Planck's constant, and p is the momentum of the particle.
The momentum of an electron in the nth orbit is given by the formula p=n(h/2π)r, where r is the radius of the orbit. Therefore, the de Broglie wavelength of an electron in the nth orbit is given by the formula λ=n(h/2πr).
We are given that the electron is moving in Bohr's fourth orbit, which means that n=4. Therefore, the de Broglie wavelength of the electron is given by the formula λ=4(h/2πr).
We need to find the circumference of the fourth orbit. The circumference of a circle is given by the formula C=2πr, where C is the circumference and r is the radius.
Solution:
We know that λ=4(h/2πr)
Multiplying both sides by 2πr, we get:
2πrλ=4h
Dividing both sides by λ, we get:
2πr=4h/λ
Using the formula for the circumference of a circle, we get:
C=2πr=4h/λ
Therefore, the circumference of the fourth orbit is 4 times the ratio of Planck's constant to the de Broglie wavelength of the electron.
Hence, the correct answer is option (c) 4λ.
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An electron is moving in Bohr's fourth orbit. Its de Broglie wavelength is λ. What is the circumference of the fourth orbit?a)2/λb)2λc)4λd)4/λCorrect answer is option 'C'. Can you explain this answer?
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