# In a sample of radioactive material, what fraction of an initial number of active nuclei will remain undisintegrated after half-life of a half-life of the sample?

A.\[\dfrac{1}{4}\]

B. \[\dfrac{1}{{2\sqrt 2 }}\]

C.\[\dfrac{1}{{\sqrt 2 }}\]

D. \[2\sqrt 2 \]

Answer

Verified

57.9k+ views

**Hint:**Use the expression of decay constant\[\left( \lambda \right)\]deduced from the relation between half-life and decay constant in the radioactive decay formula.

**Formula used:**

\[N = {N_0}{e^{ - \lambda t}}\] and \[t_{\frac {1}{2}}=\dfrac {\ln 2}{\lambda}\]

Where,

N = Number of atoms after time t, \[{N_0} = \] Number of atoms at time t= 0

\[\lambda = \]Decay constant and \[t_{\frac {1}{2} }\]= half life.

**Complete step by step solution:**

Given here is a sample of radioactive material and we have to calculate the number of nuclei that will remain after half of the half-life \[t_{\frac{1}{2}}\] of the sample.

From the information given in question we know that time \[t = \dfrac {t_{\frac {1}{2}}}{2}\].

As we know that decay constant \[\left( \lambda \right)\] is related with half life by equation

\[t_{\frac {1}{2}}=\dfrac {\ln 2}{\lambda}\].

The decay constant can be expressed as

\[\lambda=\dfrac {\ln 2}{t_{\frac {1}{2}}}\].

From the law of radioactive disintegration we have,

\[N = {N_0}{e^{ - \lambda t}}\,..........(1)\]

Where,

N = Number of atoms after time t, \[{N_0} = \] Number of atoms at time t= 0

Substituting \[t = \dfrac {t_{1/2}}{2}\] and \[\lambda=\dfrac {\ln 2}{t_{\frac {1}{2}}}\] in \[{e^{ - \lambda t}}\] we get,

\[e^{ - \lambda t}=e^{-\left (\dfrac {\ln 2}{t_{1/2}}\right) \left(\dfrac {t_{1/2}}{2}\right)}= e^{-\dfrac {\ln 2}{2}}\]

\[e^{ - \lambda t}= e^{-\dfrac {\ln 2}{2}}\]

Substituting \[{e^{ - \lambda t}} = {e^{ - \left( {\dfrac{{\ln 2}}{2}} \right)}}\] in equation (1) we get,

\[N = {N_0}{e^{ - \left( {\dfrac{{\ln 2}}{2}} \right)}}\,.........(2)\]

Equation (2) can be expressed as,

\[N = {N_0}{e^{ - \dfrac{1}{2}\left( {\ln 2} \right)}}\, \Rightarrow {N_0}{e^{\ln {2^{ - \frac{1}{2}}}}}\]

\[N = {N_0}{e^{\ln {2^{ - \dfrac{1}{2}}}}}\]

Using in the above equation it can be expressed as,

\[N = {N_0}\,{2^{ - \left( {\dfrac{1}{2}} \right)}}\,\]

Further solving the above equation we get,

\[N = {N_0}\,{2^{ - \dfrac{1}{2}}} \Rightarrow \,N\, = \dfrac{{{N_0}}}{{{2^{\dfrac{1}{2}}}}}\,\,\,\,\,\,\,\,\,\,\,\]

As \[{2^{\frac{1}{2}}} = \sqrt 2 \]

Above equation can be written as,

\[\,N\, = \dfrac{{{N_0}}}{{\sqrt 2 }} \Rightarrow N = \dfrac{1}{{\sqrt 2 }}{N_0}\,\,\,\,\,\,\,\,\,\,\,\]

So, after half of the half-life of the given sample amount of undisintegrated nuclei will be \[\dfrac{1}{{\sqrt 2 }}\]of the initial amount of the given sample.

**Therefore, option C is the correct option.**

**Note:**Remaining amount of radioactive sample after half of half life will not be equal to the exact half of \[\dfrac{{{N_0}}}{2}\] where, \[{N_0}\] is the initial amount of radioactive sample.

Last updated date: 02nd Jun 2023

•

Total views: 57.9k

•

Views today: 0.13k

Recently Updated Pages

Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts

Ray optics is valid when characteristic dimensions class 12 physics CBSE

A ball impinges directly on a similar ball at rest class 11 physics CBSE

What is the Full Form of PVC, PET, HDPE, LDPE, PP and PS ?

Alfred Wallace worked in A Galapagos Island B Australian class 12 biology CBSE

Imagine an atom made up of a proton and a hypothetical class 12 chemistry CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

How do you define least count for Vernier Calipers class 12 physics CBSE

Why is the cell called the structural and functional class 12 biology CBSE

Two balls are dropped from different heights at different class 11 physics CBSE