Question #88429
Free neutrons of mass, 1.67 × 10–27 kg and de Broglie wavelength 1 nm have a mean life
time of about 900 s. The distance from the source where the number of neutrons have
decayed to one-half its initial value is (h = 6.62 × 10–34 Js)
(1) 2.4 × 105 m (2) 2.4 × 10–4 m
(3) 4.8 × 105 m (4) 4.8 × 10–4 m
1
Expert's answer
2019-04-25T09:18:49-0400

In order to define the distance from the source that is


l=vT,l = v T,

one has to obtain both velocity and half-life period of neutrons.

1) according to the de Broglie theory, there exists the following relation for the momentum of a particle:


p=hλ,p=mvp = \frac{h}{\lambda}, \quad p = mv

v=hmλ\Rightarrow \, v = \frac{h}{m \lambda}

2) there exists the following relation between the half-life and average lifetime of a decaying object:


T1/2T=τln2T_{1/2}\equiv T = \tau \ln{2}

Finally, we obtain:


l=hτln2mλ=6.6210349102ln21.6710271092.4105ml=\frac{h \tau \ln{2}}{m \lambda} = \frac{6.62 \cdot 10^{-34} \cdot 9 \cdot 10^{2} \ln{2}}{1.67 \cdot 10^{-27} \cdot 10^{-9}} \approx 2.4 \cdot 10^{5} \,m

Answer: (1) 2.4 * 105 m.


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