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Wave Optics

Question
CBSEENPH12039051

a) Deduce the expression, N = N0 e−λt, for the law of radioactive decay.

(b) (i) Write symbolically the process expressing the β+ decay of Na presubscript 11 presuperscript 22 . Also write the basic nuclear process underlying this decay.

(ii) Is the nucleus formed in the decay of the nucleus Na presubscript 11 presuperscript 22 , an isotope or isobar?

Solution

As per the law of radioactive decay, we have
fraction numerator increment straight N over denominator increment straight t end fraction proportional to straight N

where,

N = Number of nuclei in the sample

ΔN = Amount of nuclei undergoing decay

Δt = Time taken for decay 

So, fraction numerator increment straight N over denominator increment straight t end fraction equals λN space semicolon spacewhere λ is Decay constant or disintegration constant.
Δt = 0
therefore space dN over dt equals space minus straight lambda space straight N
dN over straight N equals space minus straight lambda space dt

Integrating space both space sides comma space we space get

integral subscript straight N subscript straight o end subscript superscript straight N dN over straight N space equals space minus space straight lambda space integral subscript straight t subscript straight o end subscript superscript straight t dt space

ln space straight N space minus space ln space straight N subscript straight o space equals space minus straight lambda space left parenthesis straight t minus straight t subscript straight o right parenthesis

At space straight t subscript 0 space equals space 0 colon

ln space straight N over straight N subscript straight o space equals space minus straight lambda space straight t space

therefore space straight N space left parenthesis straight t right parenthesis space equals space straight N subscript straight o space straight e to the power of negative λt end exponent space 

(b)
(i) The β+ decay for  is given below:
Na presubscript 11 presuperscript 22 space rightwards arrow space Ne presubscript 10 presuperscript 22 straight space plus straight space straight beta to the power of plus straight space plus straight space straight nu

A proton is converted into neutron if, the unstable nucleus has excess protons than required for stability.

In the process, a positron e+ (or a β+) and a neutrino ν are created and emitted from the nucleus.
p → n + β+ + ν
This process is called beta plus decay.
(ii) The nucleus so formed is an isobar of Na presubscript 11 presuperscript 22 because the mass number is same, whereas the atomic numbers are different.

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