The smallest atoms can themselves exhibit quantum-mechanical behavior. Calculate the de Broglie wavelength (in pm) of a hydrogen atom traveling at 475 m/s.

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Identify the formula for the de Broglie wavelength: \( \lambda = \frac{h}{mv} \), where \( \lambda \) is the wavelength, \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \) Js), \( m \) is the mass of the particle, and \( v \) is the velocity.

Determine the mass of a hydrogen atom. The mass of a hydrogen atom is approximately \( 1.67 \times 10^{-27} \) kg.

Substitute the given velocity of the hydrogen atom, \( v = 475 \) m/s, into the de Broglie wavelength formula.

Substitute the values for Planck's constant, the mass of the hydrogen atom, and the velocity into the formula: \( \lambda = \frac{6.626 \times 10^{-34}}{1.67 \times 10^{-27} \times 475} \).

Calculate the de Broglie wavelength in meters and then convert it to picometers (1 pm = \( 10^{-12} \) m) for the final answer.

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de Broglie Wavelength

The de Broglie wavelength is a fundamental concept in quantum mechanics that describes the wave-like behavior of particles. 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. For a moving particle, momentum is calculated as the product of its mass and velocity. This concept illustrates the dual nature of matter, where particles exhibit both wave and particle characteristics.

Planck's Constant

Planck's constant (h) is a fundamental physical constant that relates the energy of a photon to the frequency of its associated electromagnetic wave. Its value is approximately 6.626 x 10^-34 J·s. In the context of the de Broglie wavelength, it serves as a bridge between classical and quantum physics, allowing for the calculation of wavelengths associated with moving particles, such as atoms and electrons.

Momentum

Momentum is a vector quantity defined as the product of an object's mass and its velocity (p = mv). In the context of quantum mechanics, momentum plays a crucial role in determining the behavior of particles, including their de Broglie wavelength. For a hydrogen atom, knowing its mass and velocity allows us to calculate its momentum, which is essential for finding its associated wavelength and understanding its quantum behavior.