Outside the nucleus, the neutron itself is radioactive and decays into a proton, an electron, and an antineutrino. The half-life of a neutron (mass = 1.675 × 10-27 kg) outside the nucleus is 10.4 min. On average, over what distance x would a beam of 8.32-eV neutrons travel before the number of neutrons decreased to 75.0% of its initial value? Ignore relativistic effects.
let,
half life, t1/2=10.4 min
energy,K.E=8.32ev
decay cosstant, lambda=ln(2)/t/12
lambda=ln(2)/(10.4*60) = 0.00111
and
N(t)=No*e^-lambda*t
75% of No=No*e^-lambda*t
0.75*No=No*e^-(0.00111)*t
0.75=e^-(0.00111*t)
===> t=259.17 sec
time required to decay 75* of neutron is t=259.17 sec
and
K.E=8.32ev
1/2*m*v^2=8.32/(6.24*10^18)
1/2*1.675*10^-27*v^2=8.32/(6.24*10^18)
===> speed, v=3.99*10^4 m/sec
distance traveled by the beam, x=v*t
x=3.99*10^4*(259.17)
x=1.034*10^7 m
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