Question

Neutrons in thermal equilibrium with matter at $27^{\circ} C$ can be thought to behave like ideal gas. Assuming them to have a speed of $v_{\text {rms }}$, If the de-Broglie wavelength $\lambda$ is $\left(\frac{11 N}{156}\right) nm$. The value of $N$ is______. [Take $m_{n}=1.69 \times 10^{-27} kg , k=1.44 \times 10^{-23} J / K , h=6.60 \times 10^{-34} J\, \sec$]

Answer 1

$v_{r m s}=\sqrt{\frac{3 k T}{m}} ; \lambda=\frac{h}{p}$
$\Rightarrow \lambda=\frac{h}{m_{n} \times v_{r m s}}=\frac{h}{\sqrt{3 k T m_{n}}}$
$\Rightarrow \lambda=\frac{6.6 \times 10^{-34}}{\sqrt{3 \times 1.44 \times 10^{-23} \times 1.69 \times 10^{-27} \times 300}}$
$=\frac{2.2 \times 10^{-10}}{1.2 \times 1.3}=\frac{11 \times 2 \times 10^{-9}}{156}$
$=\frac{11 \times 2}{156} nm$