Answer :
Final Answer:
The wavelength of the electron in the ground state of the hydrogen atom with a velocity of 2.00×10^6 m/s is approximately 3.31×10^−10 meters.
Explanation:
When an electron is in the ground state of a hydrogen atom, it occupies the lowest energy level. According to quantum mechanics, the relationship between the velocity of an electron and its wavelength is given by the de Broglie wavelength formula: λ = h / p, where λ is the wavelength, h is the Planck constant (6.626×10^-34 J*s), and p is the momentum of the electron.
In this case, to find the wavelength, we need to determine the momentum of the electron. Momentum is given by the product of mass (m) and velocity (v): p = m * v. The mass of an electron is approximately 9.109×10^-31 kg.
- 1: Calculate the momentum of the electron:
p = m * v = (9.109×10^-31 kg) * (2.00×10^6 m/s) = 1.8218×10^-24 kg m/s.
- 2: Use the de Broglie wavelength formula to find the wavelength:
λ = h / p = (6.626×10^-34 J*s) / (1.8218×10^-24 kg m/s) ≈ 3.31×10^-10 meters.
The de Broglie wavelength and its significance in understanding the wave-like behavior of particles, such as electrons, can provide insights into the fundamental principles of quantum mechanics. It highlights the duality of particles as both particles and waves, and this concept is crucial in understanding the behavior of subatomic particles in various physical and chemical phenomena. The de Broglie wavelength has been experimentally verified and is a cornerstone of modern physics, playing a pivotal role in fields like quantum mechanics, nanotechnology, and electron microscopy.
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Final answer:
The wavelength of the electron in the ground state of the hydrogen atom is 3.63 × 10^-10 m.
Explanation:
The wavelength of the electron in the ground state of the hydrogen atom can be calculated using the de Broglie equation:
λ = h / p
where λ is the wavelength, h is the Planck's constant (6.626 × 10-34 J·s), and p is the momentum of the electron. The momentum can be calculated using the equation:
p = m × v
where m is the mass of the electron (9.109 × 10-31 kg) and v is the velocity of the electron in the ground state (2.00 × 106 m/s).
Substituting the values into the equations:
p = (9.109 × 10-31 kg) × (2.00 × 106 m/s) = 1.82 × 10-24 kg·m/s
λ = (6.626 × 10-34 J·s) / (1.82 × 10-24 kg·m/s) = 3.63 × 10-10 m
Therefore, the wavelength of the electron in the ground state of the hydrogen atom is 3.63 × 10-10 m.
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