Law of Radioactive Decay

A radioactive sample has half-life of $5$ years. The percentage of fraction decayed in $10$ years will be

Activity of a radioactive sample decreases to $\left(\frac{1}{3}\right)^{rd}$ of its original value in $3$ days. Then in $9$ days its activity will become

Activity of a radioactive sample decreases to $\left(\frac{1}{4}\right)^{th}$ of its original value in 4 days. Then in 16 days its activity will become $x$ times the original value. The value of $x$ is

A radioactive element has rate of disintegration $16{,}000$ disintegrations per minute at a particular instant. After four minutes, it becomes $2000$ disintegrations per minute. The decay constant per minute is

Two different radioactive elements with half lives $T_1$ and $T_2$ have undecayed atoms $N_1$ and $N_2$ respectively, present at a given instant. The ratio of their activities at this instant is

Initially $8 \times 10^{10}$ atoms are present in a radioactive sample. The half-life of the sample is 1 hour. The number of atoms that will decay in the time interval $t = 2$ hour to $t = 5$ hour, is

th The half-life of the isotope ${}_{11}\text{Na}^{24}$ is 15 hr. How much time does it take for $\left(\frac{7}{8}\right)$ of a sample of this isotope to decay?

The activity of a radioactive sample is measured as $N_0$ counts per minute at time $t = 0$ and $\frac{N_0}{e}$ counts per minute at time $t = 3$ minute. The time (in minute) in which the activity reduces to half the value, is

A radioactive sample $S_1$ having activity $A_1$ has twice the number of nuclei as another sample of $S_2$ of activity $A_2$. If $A_2 = 2A_1$ then the ratio of half life of $S_1$ to the half life of $S_2$ is

Two radioactive substances $A$ and $B$ have decay constant $5\lambda$ and $\lambda$ respectively. At $t = 0$, they have the same number of nuclei. The ratio of number of nuclei of $A$ to those of $B$ will be $\left(\frac{1}{e}\right)^2$ after a time interval

Two samples A and B contain equal amount of radioactive substances. If $\left(\frac{1}{8}\right)^{th}$ of sample A and $\left(\frac{1}{128}\right)^{th}$ of sample B, remain after 9 hours, then the ratio of half life period of B to that of A is

In an alpha decay, the daughter nucleus produced is itself unstable and undergoes further decay. If the number of parent and daughter nuclei at time $t$ are $N_p$ and $N_d$ respectively and their decay constants are $\lambda_p$ and $\lambda_d$ respectively. What condition needs to be satisfied in order for $N_d$ to remain constant?

A radioactive nucleus X with a half-life $T$, decays into a nucleus Y. At $t = 0$, there is no nucleus Y. After some time '$t$', the ratio of the number of Y to that of X is $0.4$. Then, '$t$' is

A radioactive element having half-life 30 min. is undergoing beta decay. The fraction of radioactive element remains undecayed after 90 min. will be

A radio active element has rate of disintegration 8000 disintegrations per minute at a particular instant. After four minutes it becomes 2000 disintegrations per minute. The decay constant per minute is

A radioactive element has rate of disintegration 9000 disintegration per minute at a particular instant. After two minutes it becomes 3000 disintegration per minute. The decay constant per minute is

The activity of radioactive sample is measured as $\text{N}_0$ counts per minute at time $\text{t} = 0$, and $\frac{\text{N}_0}{\text{e}}$ counts per minute at time $\text{t} = 3$ minute, The activity reduces to half its value in time (in minute)

Two radioactive materials $A$ and $B$ having decay constant ' $7\lambda$ ' and ' $\lambda$ ' respectively, initially have same number of nuclei. The time taken to have the ratio of number of nuclei of material B to that of A as ' e ' is

A radioactive nucleus $n_2$ has 3 times the decay constant as compared to the decay constant of another radioactive nucleus $n_1$. If initial number of both nuclei are the same, what is the ratio of number of nuclei of $n_2$ to the number of nuclei of $n_1$, after one half-life of $n_1$?

A radioactive material $P$ first decays into $Q$ and then $Q$ decays to non-radioactive material $R$. Which of the following figure represents time dependent mass of $P, Q$ and $R$?

A radioactive substance has half-life of 60 minutes. During 3 hours, the amount of substance decayed would be

Two radioactive substances A and B have decay constants ' $5\lambda$ ' and ' $\lambda$ ' respectively. At $t = 0$, they have the same number of nuclei. The ratio of number of nuclei of $A$ to those of $B$ will be $\left(\frac{1}{e}\right)^2$ after a time interval

Half-lives of two radioactive elements A and B are 30 minute and 60 minute respectively. Initially the samples have equal number of nuclei. After 120 minute the ratio of decayed numbers of nuclei of $B$ to that of $A$ will be

If 'T' is the half life of a radioactive substance then its instantaneous rate of change of activity is proportional to

Half life of radio-active element is 1600 years. The fraction of sample remains undecayed after 6400 years will be

A radioactive sample has half-life of 5 years. The percentage of fraction decayed in 10 years will be

The half life of a radioactive substance is 20 minutes. In how much time, the activity of substance drops to $\left(\frac{1}{16}\right)$th of its initial value?

The half-life of a radioactive nucleus is 5 years. The fraction of the original sample that would decay in 15 years is:

The half life of a radioactive substance is T. The time taken, for disintegrating $\frac{7}{8}$ th part of its original mass will be:

A radio active material is reduced to $1/8$ of its original amount in 3 days. If $8 \times 10^{-3}\text{ kg}$ of the material is left after 5 days the initial amount of the material is

If a radioactive element having half-life of $30\text{ min}$ is undergoing beta decay, the fraction of radioactive element remains undecayed after $90\text{ min}$. will be

In a radioactive disintegration, the ratio of initial number of atoms to the number of atoms present at time $t = \frac{1}{2\lambda}$ is [$\lambda =$ decay constant]

Two different radioactive elements with half lives '$\text{T}_1$ ' and '$\text{T}_2$' have undecayed atoms '$\text{N}_1$ ' and '$\text{N}_2$ ' respectively present at a given instant. The ratio of their activities at that instant is

The decay constant for a radioactive nuclide is 1.5 $\times$ 10 $-5$ s$-1$. Atomic weight of the substance is 60 g mole$-1$, ($N_A = 6 \times 10^{23}$). The activity of 1.0 $\mu$g of the substance is ___________ $\times$ 10$10$ Bq.

Two radioactive elements A and B initially have same number of atoms. The half life of A is same as the average life of B. If $\lambda_A$ and $\lambda_B$ are decay constants of A and B respectively, then choose the correct relation from the given options.

A radioactive nucleus decays by two different processes. The half life of the first process is 5 minutes and that of the second process is $30\text{ s}$. The effective half-life of the nucleus is calculated to be $\frac{\alpha}{11}\text{ s}$. The value of $\alpha$ is __________.

Substance A has atomic mass number 16 and half life of 1 day. Another substance B has atomic mass number 32 and half life of $\frac{1}{2}$ day. If both A and B simultaneously start undergo radio activity at the same time with initial mass 320 g each, how many total atoms of A and B combined would be left after 2 days.

Given below are two statements

Statement I: The law of radioactive decay states that the number of nuclei undergoing the decay per unit time is inversely proportional to the total number of nuclei in the sample. Statement II: The half-life of a radionuclide is the sum of the life time of all nuclei, divided by the initial concentration of the nuclei at time t = 0.

In the light of the above statements, choose the most appropriate answer from the options given below:

A radioactive sample decays $\frac{7}{8}$ times its original quantity in 15 minutes. The half-life of the sample is

The half life period of a radioactive substance is 60 days. The time taken for $\frac{7}{8}$ th of its original mass to disintegrate will be:

Following statements related to radioactivity are given below:

(A) Radioactivity is a random and spontaneous process and is dependent on physical and chemical conditions.

(B) The number of un-decayed nuclei in the radioactive sample decays exponentially with time. (C) Slope of the graph of loge (no. of undecayed nuclei) Vs. time represents the reciprocal of mean life time ($\tau$).

(D) Product of decay constant ($\lambda$) and half-life time (T1/2) is not constant.

Choose the most appropriate answer from the options given below:

The half life of a radioactive substance is 5 years. After x years a given sample of the radioactive substance gets reduced to 6.25% of its initial value. The value of x is ____________.

The activity of a radioactive material is $6.4 \times 10^{-4}$ curie. Its half life is 5 days. The activity will become $5 \times 10^{-6}$ curie after:

What is the half-life period of a radioactive material if its activity drops to $1/16$th of its initial value in 30 years?

The activity of a radioactive material is 2.56 $\times$ 10$-$3 Ci. If the half life of the material is 5 days, after how many days the activity will become 2 $\times$ 10$-$5 Ci?

A radioactive nucleus can decay by two different processes. Half-life for the first process is 3.0 hours while it is 4.5 hours for the second process. The effective half-life of the nucleus will be:

A freshly prepared radioactive source of half life 2 hours 30 minutes emits radiation which is 64 times the permissible safe level. The minimum time, after which it would be possible to work safely with source, will be _________ hours.

The disintegration rate of a certain radioactive sample at any instant is 4250 disintegrations per minute. 10 minutes later, the rate becomes 2250 disintegrations per minute. The approximate decay constant is:

$($Take $\log_{10} 1.88 = 0.274)$

At any instant, two elements X1 and X2 have same number of radioactive atoms. If the decay constant of X1 and X2 are 10 $\lambda$ and $\lambda$ respectively, then the time when the ratio of their atoms becomes $\frac{1}{e}$ respectively will be:

Two radioactive materials A and B have decay constants $25\lambda$ and $16\lambda$ respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of B to that of A will be "e" after a time $\frac{1}{a\lambda}$. The value of a is _________.

A sample contains 10$-$2 kg each of two substances A and B with half lives 4 s and 8 s respectively. The ratio of their atomic weights is 1: 2. The ratio of the amounts of A and B after 16 s is $\frac{x}{100}$. The value of x is ___________.

A sample of radioactive element contains $8 \times 10^{16}$ active nuclei. The halt-life of the element is 15 days. The number of nuclei decayed after 60 days is

The nuclear activity of a radioactive element becomes $\left(\frac{1}{8}\right)^{th}$ of its initial value in 30 years. The half- life of radioactive element is _____________ years.

A radioactive substance decays to $\left(\frac{1}{16}\right)^{th}$ of its initial activity in 80 days. The half life of the radioactive substance expressed in days is ____________.

A radioactive sample disintegrates via two independent decay processes having half lives $T_{1/2}^{(1)}$ and $T_{1/2}^{(2)}$ respectively. The effective half-life T1/2 of the nuclei is:

The half life of a radioactive substance is 30 minute. The time taken between 40% decay and 85% decay of the same radioactive substance is

The half life of a radioactive nuclide is 100 hours. The fraction of original activity that will remain after 150 hours would be:

There are 1010 radioactive nuclei in a given radioactive element, its half-life time is 1 minute. How many nuclei will remain after 30 seconds? $\left(\sqrt{2} = 1.414\right)$

If 'f' denotes the ratio of the number of nuclei decayed (Nd) to the number of nuclei at t = 0 (N0) then for a collection of radioactive nuclei, the rate of change of 'f' with respect to time is given as: [$\lambda$ is the radioactive decay constant]

Some nuclei of a radioactive material are undergoing radioactive decay. The time gap between the instances when a quarter of the nuclei have decayed and when half of the nuclei have decayed is given as:

(where $\lambda$ is the decay constant)

Calculate the time interval between 33% decay and 67% decay if half-life of a substance is 20 minutes.

The half life period of radioactive element x is same as the mean life time of another radioactive element y. Initially they have the same number of atoms. Then:

A radioactive sample is undergoing $\alpha$ decay. At any time t1, its activity is A and another time t2, the activity is $\frac{A}{5}$. What is the average life time for the sample?

The half-life of ${}^{198}Au$ is 3 days. If atomic weight of ${}^{198}Au$ is 198 g/mol then the activity of 2 mg of ${}^{198}Au$ is [in disintegration/second]:

For a certain radioactive process the graph between In R and t(sec) is obtained as shown in the figure. Then the value of half life for the unknown radioactive material is approximately:

The half-life of Au198 is 2.7 days. The activity of 1.50 mg of Au198 if its atomic weight is 198 g mol${}^{-1}$ is, (Na = 6 $\times$ 1023/mol).

Two radioactive materials $X_1$ and $X_2$ have decay constants '$5\lambda$' and '$\lambda$' respectively. Initially, they have the same number of nuclei. After time '$t$', the ratio of number of nuclei of $X_1$ to that of $X_2$ is $\frac{1}{e}$. Then $t$ is equal to

Two radioactive substances X and Y originally have N1 and N2 nuclei respectively. Half life of X is half of the half life of Y. After three half lives of Y, number of nuclei of both are equal. The ratio $\frac{N_1}{N_2}$ will be equal to:

A sample of a radioactive nucleus A disintegrates to another radioactive nucleus B, which in turn disintegrates to some other stable nucleus C. Plot of a graph showing the variation of number of atoms of nucleus B versus time is:

(Assume that at t = 0, there are no B atoms in the sample)

A radioactive sample has an average life of 30 ms and is decaying. A capacitor of capacitance 200 $\mu$F is first charged and later connected with resistor 'R'. If the ratio of charge on capacitor to the activity of radioactive sample is fixed with respect to time then the value of 'R' should be _____________ $\Omega$.

At time t = 0, a material is composed of two radioactive atoms A and B, where NA(0) = 2NB(0). The decay constant of both kind of radioactive atoms is $\lambda$. However, A disintegrates to B and B disintegrates to C. Which of the following figures represents the evolution of NB(t) / NB(0) with respect to time t?

[NA (0) = No. of A atoms at t = 0

NB (0) = No. of B atoms at t = 0]

A radioactive material decays by simultaneous emissions of two particles with half lives of 1400 years and 700 years respectively. What will be the time after which one third of the material remains? (Take ln 3 = 1.1)

A radioactive nucleus decays by two different processes. The half life for the first process is 10 s and that for the second is 100 s. The effective half life of the nucleus is close to:

In a radioactive material, fraction of active material remaining after time t is 9/16. The fraction that was remaining after t/2 is

The half life of a radioactive sample undergoing $\alpha$-decay is $1.4 \times 10^{17}\text{ s}$. If the number of nuclei in the sample is $2.0 \times 10^{21}$, the activity of the sample is nearly

The activity of a radioactive sample falls from 700 s–1 to 500 s–1 in 30 minutes. Its half life is close to:

Activities of three radioactive substances A, B and C are represented by the curves A, B and C, in the figure. Then their half-lives

$T_{\frac{1}{2}}(A)$: $T_{\frac{1}{2}}(B)$: $T_{\frac{1}{2}}(C)$ are in the ratio:

The rate of radioactive disintegration at an instant for a radioactive sample of half life $2.2 \times 10^{9}\text{ s}$ is $10^{10}\text{ s}^{-1}$. The number of radioactive atoms in that sample at that instant is,

Using a nuclear counter the count rate of emitted particles from a radioactive source is measured. At t = 0 it was 1600 counts per second and t = 8 seconds it was 100 counts per second. The count rate observed, as counts per second, at t = 6 seconds is close to -

Half lives of two radioactive nuclei A and B are 10 minutes and 20 minutes, respectively, If initially a sample has equal number of nuclei, then after 60 minutes, the ratio of decayed numbers of nuclei A and B will be:

Two radioactive substances A and B have decay constants 5$\lambda$ and $\lambda$ respectively. At t = 0, a sample has the same number of the two nuclei. The time taken for the ratio of the number of nuclei to become $\left(\frac{1}{e}\right)^2$ will be:

Two radioactive materials A and B have decay constants 10$\lambda$ and $\lambda$, respectively. It initially they have the same number of nuclei, then the ratio of the number of nuclei of A to that of B will be 1/e after a time:

A sample of radioactive material A, that has an activity of 10 mCi(1 Ci = 3.7 $\times$ 1010 decays/s), has twice the number of nuclei as another sample of a different radioactive materail B which has an activity of 20 mCi. The correct choices for half-lives of A and B would then be respectively:

At a given instant, say t = 0, two radioactive substance A and B have equal activities. the ratio $\frac{R_B}{R_A}$ of their activities after time t itself decays with time t as e${}^{-3}$t. If the half-life of A is ln2, the half-life of B is:

For a radioactive material, half-life is 10 minutes. If initially there are 600 number of nuclei, the time taken (in minutes) for the disintegration of 450 nuclei is

At some instant, a radioactive sample S1 having an activity 5 $\mu$Ci has twice the number of nuclei as another sample S2 which has an activity of 10 $\mu$Ci. The half lives of S1 and S2 are:

A containing active cobalt ${}^{60}_{27}\text{Co}$ having activity of $0.8\ \mu\text{Ci}$ and decay constant $\lambda$ is injected in an animal's body. If $1\text{ cm}^3$ of blood is drawn from the animal's body after $10$ hrs of injection, the activity found was $300$ decays per minute What is the volume of blood that is flowing in the body? $(\text{Ci} = 3.7 \times 10^{10}$ decays per second and at $t = 10$ hrs $e^{-\lambda t} = 0.84)$

Radioactive material 'A' has decay constant '8 $\lambda$' and material 'B' has decay constant '$\lambda$'. Initially they have same number of nuclei. After what time, the ratio of number of nuclei of material 'B' to that 'A' will be e?

A radioactive nucleus A with a half life T, decays into a nucleus B. At t = 0, there is no nucleus B. At sometime t, the ratio of the number of B to that of A is 0.3. Then, t is given by:

The half-life of a radioactive substance is 30 minutes. The time (in minutes) taken between 40% decay and 85% decay of the same radioactive substance is

Half-lives of two radioactive elements $A$ and $B$ are $20$ minutes and $40$ minutes, respectively. Initially, the samples have equal number of nuclei. After $80$ minutes, the ratio of decayed number of $A$ and $B$ nuclei will be:

Let N$\beta$ be the number of $\beta$ particles emitted by 1 gram of Na24 radioactive nuclei (half life = 15 hrs) in 7.5 hours, N$\beta$ is close to (Avogadro number = 6.023 $\times$ 1023 / g. mole):

A radioactive X with a half life 1.4 $\times$ 109 years decays to Y which is stable. A sample of the rock from a cave was found to contain X and Y in the ratio 1: 7. The age of the rock is

A piece of wood from a recently cut tree shows 20 decays per minute. A wooden piece of same size placed in a museum (obtained from a tree cut many years back) shows 2 decays per minute. If half life of C14 is 5730 years, then age of the wooden piece placed in the museum is approximately:

A piece of bone of an animal from a ruin is found to have 14C activity of 12 disintegrations per minute per gm of its carbon content. The 14C activity of a living animal is 16 disintegrations per minute per gm. How long ago nearly did the animal die? (Given half life of 14C is t1/2 = 5760 years)

A radioactive nuclei with decay constant 0.5/s is being produced at a constant rate of 100 nuclei/s. If at t = 0 there were no nuclei, the time when there are 50 nuclei is:

The half life of a radioactive isotope 'X' is 20 years. It decays to another element 'Y' which is stable. The two elements 'X' and 'Y' were found to be in the ratio 1: 7 in a sample of a given rock. The age of the rock is estimated to be

The half-life of a radioactive element A is the same as the mean-life of another radioactive element B. Initially both substances have the same number of atoms, then:

The half life of a radioactive nucleus is 50 days. The time invertal (t2 $-$ t1) between the time t2 when $\frac{2}{3}$ of it has decayed and the time t1 when $\frac{1}{3}$ of it had decayed is

A mixture consists of two radioactive materials A1 and A2 with half lives of 20 s and 10 s respectively. Initially the mixture has 40 g of A1 and 160 g of A2. The amount of the two in the mixture will become equal after

The half life of a radioactive isotope X is 50 years. It decays to another element Y which is stable. The two elements X and Y were found to be in the ratio of 1: 15 in a sample of a given rock. The age of the rock was estimated to be

The half life of a radioactive substance is $20$ minutes. The approximate time interval $(t_2 - t_1)$ between the time $t_2$ when $\frac{2}{3}$ of it had decayed and time $t_1$ when $\frac{1}{3}$ of it had decayed is: