The kinetic energy K of a particle is related to its momentum p by K = p2/2μ, where μ is the particle’s mass. In a gas at absolute temperature T, the molecules have a typical kinetic energy of 3k B...

The kinetic energy K of a particle is related to its momentum p by K = p2/2μ, where μ is the particle’s mass. In a gas at absolute temperature T, the molecules have a typical kinetic energy of 3k_BT/2. Derive an expression for the thermal de Broglie wavelength, a typical value for the de Broglie wavelength λ of a molecule in a gas. For helium atoms (μ = 6.7 × 10⁻²⁷
kg), calculate the thermal de Broglie wavelength at room temperature (T = 300 K) and at the boiling point of helium (T = 4 K). Quantum effects become most significant in matter when the thermal de Broglie wavelength of the particles is greater than their separation. At atmospheric pressure, gas molecules are about 1–2 nm apart; in a condensed phase (liquid, solid) they are about ten times closer. How do these compare with the thermal de Broglie wavelengths you calculated for helium?