![]() Question: The half-life of 238 92U undergoing α-decay is 4.5 × 10 9 years. T1/2 = (ln2)/λ = τ ln 2 … (8) Solved Examples for You Therefore, we can summarise the observations as follows: Hence, to obtain the mean life, we integrate this expression over all the times from 0 to ∞ and divide by the total number of nuclei at t = 0 (which is N 0). Hence, the total life of all these nuclei is tλN 0e –λt Δt.The number of nuclei which decay in the time interval: ‘t’ to ‘t + Δt’ is: R(t)Δt = (λN 0e –λt Δt).Next, let’s find the relation between the mean life τ and the disintegration constant λ. For this, let’s input the following values in equation (5), Let’s find the relation between T1/2 and the disintegration constant λ. ![]() Mean life τ – the time at which both R and N have been reduced to, e -1 of their initial values.Half-life T 1/2 – the time at which both R and N are reduced to half of their initial values.There are two ways to measure the time for which a radionuclide can last. The SI unit for measurement of activity is ‘ becquerel’ and is defined as,Īn older unit, the curie, is still in common use:ġ curie = 1 Ci = 3.7 × 10 10 Bq (decays per second) The total decay rate of a sample is also known as the activity of the sample. Where R and the number of radioactive nuclei that have not yet undergone decay must be evaluated at the same instant. Now we can rewrite equation (1) as follows, Equation (5) is the alternative form of the Law of Radioactive Decay. Where, R 0 is the radioactive decay rate at the time t = 0, and R is the rate at any subsequent time t. The decay rate is now defined as,ĭifferentiating equation (4) on both sides, we get, Let’s say that we consider a time interval dt and get a decay count ΔN (= –dN). ![]() Even if we don’t know the number of nuclei in the sample, by simply measuring the number of emissions of α, β or γ particles in 10 or 20 seconds, we can calculate the decay rate. This rate gives us the number of nuclei decaying per unit time. In radioactivity calculations, we are more interested in the decay rate R ( = – dN/dt) than in N itself. Next, we set t 0 = 0 and rearrange the above equation (3) to get,Įquation (4) is the Law of Radioactive Decay. Where, N 0 is the number of radioactive nuclei in the sample at some arbitrary time t 0 and N is the number of radioactive nuclei at any subsequent time t. Now, integrating both the sides of the above equation, we get, Hence, the rate of change of N (in the limit Δt → 0) is, Now, the change in the number of nuclei in the sample is, dN = – ΔN in time Δt. Where λ = radioactive decay constant or disintegration constant. If N = total number of nuclei in the sample and ΔN = number of nuclei that undergo decay in time Δt then, When a radioactive material undergoes α, β or γ-decay, the number of nuclei undergoing the decay, per unit time, is proportional to the total number of nuclei in the sample material.
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