Dual Nature of Matter and Radiation Part 1

Given, Planck's constant, $h=6.6 \times 10^{-34}\,\mathrm{Js}$ Boltzmann constant, $k_{B}=1.38 \times 10^{-23}\,\mathrm{J/K}$ Mass of an electron, $m_{e}=9 \times 10^{-31}\,\mathrm{kg}$ Temperature of an ideal gas, $T=300\,\mathrm{K}$ As we know that, de-Broglie wavelength, $\lambda=\frac{h}{mv}=\frac{h}{\sqrt{2mE}}$ ... (i) Here, E is the kinetic energy. $E=\frac{3 K_{B} T}{2}$ Substituting value of $E$ in Eq. (i), we get, $\lambda=\frac{h}{\sqrt{3 m K_{B} T}}$ Substituting the given values in the above equation, we get $\lambda=\frac{6.6 \times 10^{-34}}{\sqrt{3 \times 9 \times 10^{-31} \times 138 \times 10^{-23} \times 300}}$ = 6.26 nm ∴ The corresponding de-Broglie wavelength of an electronapproximately at 300 K is 6.26 nm.