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The de-Broglie wavelength of material particles of mass ‘m’ moving with velocity ‘u’ which are in thermal equilibrium at temperature T is

Question and options

The de-Broglie wavelength of material particles of mass ‘m’ moving with velocity ‘u’ which are in thermal equilibrium at temperature T is
  1. h/2mKTh/\sqrt{2mKT}
  2. h/3mKTh/\sqrt{3mKT}
  3. h/muTh/\sqrt{muT}
  4. h/2KTh/\sqrt{2KT}

Answer and explanation

Correct answer: B - \(h/\sqrt{3mKT}\)
Thermal kinetic energy 12mu2=32kT\frac{1}{2}mu^2 = \frac{3}{2}kT gives momentum p=3mkTp = \sqrt{3mkT}, so λ=h/p=h/3mkT\lambda = h/p = h/\sqrt{3mkT}. The expression h/2mKTh/\sqrt{2mKT} corresponds to KE=kT\mathrm{KE} = kT, not the mean thermal energy; h/muTh/\sqrt{muT} and h/2KTh/\sqrt{2KT} are dimensionally wrong.

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