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Mean-Life

Mean Life is an important topic of radioactivity. Just as the half-life of a radioactive substance is important, formula to calculate mean life is also equally important. The IIT JEE often picks up questions on the calculation of average life of radioactive substances. It is important to have expertise in this area as well in order to remain competitive in the JEE.

Mean life of radioactive elements is expected to be somewhat longer than the half-life. If in a given radioactive element, half of its elements have decayed after one half life, some well-defined average life expectancy can be assumed which is the mean life of the atoms.

Formulas of Calculating Radioactivity Mean Life

The mean life of an element equals the half-life of the substance divided by the natural logarithm of 2 which is about 0.693. In fact, the mean life turns out to equal the number τ which appears in the exponential term e^−t/τ involved in the description of decay or growth. It is termed as the time constant.

Mean lifetime is a very significant quantity that can be measured directly for small number of atoms. If there are ‘n’ active nuclei, (atoms) (of the same type, of course), the mean life is

τ = τ

₁ + τ₃ + ... + τ₂ / n

Where τ₁, τ₂,....... τ_n represent the observed lifetime of the individual nuclei and n is a very large number. It can also be calculated as a weighted average:

τ = τ₁N₁ + τ₃N₂ + ... + τ₂N_n / N₁ + ... + N_n

Where N₁ nuclei live for time τ₁,

N₂ nuclei live for time τ₂....... and so on.

This quantity may be related with γ.

Using calculus we may rewrite it as:

Formulas of calculating radioactivity mean life

Where |dN| is the number of nuclei decaying between t, t + dt; the modulus sign is required to ensure that it is positive.

dN = –λN₀e^–λtdt

and |dN| = λN₀e^–λtdt

or τ = 1/λ.

The half-life and the mean life of substances are related to each other by the formula

T = τ ln 2 ≈ 0.693 τ

The two parameters vary drastically for different substances. For example the half-life of Polonium-212 is less than 1 microseconds, while for Thorium-232, the half-life crosses even 1 billion years.

Watch this Video for more reference

The JEE often asks questions on mean life equation or the average mean life of radioactivity. Let us consider one such proble

Illustration:

Cascade Decays: Frequently, a nucleus A decays into another nucleus B, which again decays into C, and so on until the series ends in a stable nuclide. These are known as cascade decays. We will consider a simple example of cascade decay where a nuclide A decays into B, and then, B decays into C:

λ_A λ_B

A ———> B ———> C

Suppose that there are N₀ atoms of A to begin with. The problem is to find the numbers 0f atoms of A, B, C respectively as a function of time t.

Solution:

Suppose that at time t, there are N_A atoms of A and N_batoms of B. We write the equations:

dN_A / dt –activity of A = –λ_AN_A … (i)

dN_B / dt = + (activity of A) – (activity of B)

= λ_AN_A – λ_BN_B … (ii)

Since each decay of A increases the number of atoms of B

dN_C / dt = λ_BN_B … (iii)

Solving equation (i), we get

N_A (t) = N₀ e^–λ_A¹ … (iv)

Substituting this in equation (ii), we get. the solution,

N_B (t) =

… (v)

Putting the initial value (t = 0) of N_B, we get C₁^, and substituting this value of C₁we get the expression for N_B:

N_B(t) =
… (vi)

Substitution the value of N_B in equation (iii) and solving, we get

N_C =
… (vii)

Using these values of N_A, N_B and N_C, the activity can be calculated, if required

askIITians offers comprehensive study material containing numerous examples on calculation of mean life. The JEE aspirants can refer the notes for the use of mean life formula.

Related Resources

Click here for the Complete Syllabus of IIT JEE Physics.
Look into the Model Papers with Solutions to get a hint of the kinds of questions asked in the exam.
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