First slide

Wave nature of matter

Question

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

Moderate

Solution

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

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