Advertisements
Advertisements
Question
Show that for radioactive decay N(t) = `"N"_"O" "e"^{-λ"t"}`, where symbols have their usual meaning.
Advertisements
Solution
If ‘N(t)’ is the number of parent nuclei present at any instant ‘t’, ‘dN’ is the number of nuclei disintegrated in the short interval of time ‘dt’, then,
dN ∝ −N(t) dt
dN = −λ N(t) dt
where λ is known as decay constant or disintegration constant.
The negative sign indicates the disintegration of atoms.
Integrating both sides of the equation,
`int_{"N"_0}^{"N"("t")} "dN"/("N"("t")) = int_0^"t" -lambda "dt"`
where N0 is a number of parent atoms at time t = 0.
∴ loge `("N"("t"))/"N"_0 = -lambda"t"`
∴ N(t) = N0 `"e"^{-lambda"t"}`
This is the required relation.
RELATED QUESTIONS
What is gamma decay?
Sample of carbon obtained from any living organism has a decay rate of 15.3 decays per gram per minute. A sample of carbon obtained from very old charcoal shows a disintegration rate of 12.3 disintegrations per gram per minute. Determine the age of the old sample given the decay constant of carbon to be 3.839 × 10−12per second.
Describe alpha decay and write down the formula for the energy generated in this decay.
Complete the following equation describing nuclear decay.
\[\ce{_8^19O->e^- { +}}\] _____
Complete the following equation describing nuclear decay.
\[\ce{_7^12N -> _6^12C {+}}\] ______
Write relation between decay constant of a radioelement and its half-life.
Derive the relationship between half-life and decay constant of a radioelement.
Decay constant of 197Hg is 0.017 h-1. What is its half-life?
The half-life of 35S is 87.8 d. What percentage of 35S sample remains after 180 d?
0.5 g sample of 201Tl decays to 0.0788 g in 8 days. What is its half-life?
65% of 111In sample decays in 4.2 d. What is its half-life?
The half-life of radon is 3.82 d. By what time would 99.9% of radon will be decayed?
A sample of old wood shows 7.0 dps/g. If the fresh sample of tree shows 16.0 dps/g, how old is the given sample of wood? (Half-life of 14C is 5730 y)
The decay constant λ of a certain radioactive material is 0.2166 per day. The average life τ of the radioactive material is ______
Show that half life period of radioactive material varies inversely to decay constant λ.
A radioactive nucleus emits 4 α-particles and 7 β-particles in succession. The ratio of number of neutrons of that of protons, is
[A = mass number, Z =atomic number]
Most excited states of an atom have life times of about ____________.
The activity of a radioactive sample ____________.
What is alpha decay?
The radioactivity of a sample is R1 at a time T1 and R2 at a time T2. If the half-life of the specimen is T, the number of atoms that have disintegrated in the time (T1 - T2) is proportional to ______.
For radioactive substances, the fraction of its initial quantity (N0) which will disintegrate in its average lifetime is about ______.
(e = 2.71)
The graph obtained by plotting loge (A) [A is the activity of a radioactive sample] against t (time) out of the following is:
The disintegration rate of a radio-active sample is 1010 per hour at 20 hours from the start. It reduces to 5 × 109 per hour after 30 hours. Calculate the decay constant.
If the number of nuclei of a radioactive substance becomes `1/e` times the initial number in 10 days, what is the decay constant of the substance?
Show that the number of nuclei of a radioactive material decreases exponentially with time.
In one mean lifetime of a radioactive element the fraction of the nuclei that has disintegrated is ______. [e is the base of natural logarithm]
