Advertisements
Advertisements
प्रश्न
Show that for radioactive decay N(t) = `"N"_"O" "e"^{-λ"t"}`, where symbols have their usual meaning.
उत्तर
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.
संबंधित प्रश्न
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.
Decay constant of 197Hg is 0.017 h-1. What is its half-life?
The half-life of 18F is 110 minutes. What fraction of 18F sample decays in 20 minutes?
The half-life of 67Ga is 78 h. How long will it take to decay 12% of the sample of Ga?
65% of 111In sample decays in 4.2 d. What is its half-life?
A 3/4 of the original amount of radioisotope decays in 60 minutes. 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)
Show that half life period of radioactive material varies inversely to decay constant λ.
Obtain an expression for the half-lifetime of radioactive material. Hence state the relation between an average life and half lifetime of radioactive material.
A radioactive substance decays to (1/10)th of its original value in 56 days. Calculate its decay constant.
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]
A radioactive element has rate of disintegration 10,000 disintegrations per minute at a particular instant. After four minutes it become 2500 disintegrations per minute. The decay constant per minute is ______.
For radioactive substances, the fraction of its initial quantity (N0) which will disintegrate in its average lifetime is about ______.
(e = 2.71)
A radioactive sample S1 having the activity A1 has twice the number of nuclei as another sample S2 of activity A2 If A2 = 2A1 then the ratio of half-life of S1 to the half-life of S2 is ______.
The half-life of a radioactive substance is 10 days. The time taken for the `(7/8)^"th"` of the sample of disintegrates is ______.
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.
The half-life of \[\ce{^238_92U}\] undergoing ∝- -decay is 4.5 × 109 years. What is the activity of 1g sample of \[\ce{^238_92U}\]?
Define half life period.
In one mean lifetime of a radioactive element the fraction of the nuclei that has disintegrated is ______. [e is the base of natural logarithm]
