Definition of decay constant
Calculation of half life
Calculation of initial number of nuclei at t=0
The decay constant (λ) of a radioactive nucleus equals the ratio of the instantaneous rate of decay (‌

∆N

∆t

) to the corresponding instantaneous number of radioactive nuclei.
Alternatively,
The decay constant (λ) of a radioactive nucleus is the constant of proportionality in the relation between its rate of decay and number of its nuclei at any given instant.
Alternatively,
‌(‌

∆N

∆t

)∝N
‌(‌

∆N

∆t

)=λN
The constant (λ) is known as the decay constant
Alternatively,
The decay constant equals the reciprocal of the mean life of a given radioactive nucleus.
λ=‌

1

τ

where, τ=‌ mean life ‌
Alternatively,
The decay constant equal the ratio of ln‌2 to the half life of the given radioactive element.
λ=‌

ln‌2

T1∕2

where T1∕2= Half life
Alternatively,
The decay constant of a radioactive element, is the reciprocal of the time in which the number of its nuclei reduces to ‌

1

e

of its original number.
We have
‌R=λN
‌R(20‌hrs)=10000=λN20
‌R(30‌hrs)=5000=λN30
‌∴‌‌‌

N20

N30

=2
This means that the number of nuclei, of the given radioactive nucleus, gets halved in a time of (30−20) hours =10 hours
∴ Half life =10 hours
This means that in 20 hours ( =2 half lives), the original number of nuclei must have gone down by a factor of 4 .
Hence Rate of decay at t=0
λN0=4λN20
=4×10000=40,000 disintegrations per second
[Note : Award full marks of the last part of this question even if student does not calculate initial number of nuclei and calculates correctly rate of disintegration at t=0 ]
i.e., R0=40,000 disintegrations per second
‌N0=‌

40000

λ

=‌

40000

ln‌2

×10×60×60
‌N0=‌

144×107

0.693

=2.08×109‌ nuclei ‌