Radioactive decay

A nuclear reactor starts producing a radioactive nuclide $X$ from $t=0$, at a constant rate of $\alpha$ per second. Each decay of $X$ produces energy $E_0$, which is utilized to heat a liquid of mass $m$ and specific heat $s$. Assuming no heat loss from the liquid and taking $\lambda$ as the decay constant of $X$, the rate of increase in the temperature of the liquid is:

Two radioactive substances A and B of mass numbers 200 and 212 respectively, shows spontaneous α-decay with same $Q$ value of 1 MeV. The ratio of energies of $\alpha$-rays produced by A and B is ________.

A sample initially contains only U-238 isotope of uranium. With time, some of the U-238 radioactively decays into Pb-206 while the rest of it remains undisintegrated.

When the age of the sample is $\mathbf{P}\times10^8$ years, the ratio of mass of Pb-206 to that of U-238 in the sample is found to be 7. The value of P is _____.

[Given: Half-life of U-238 is $4.5\times10^9$ years; $\log_e 2 = 0.693$]

List-I shows different radioactive decay processes and List-II provides possible emitted particles. Match each entry in List-I with an appropriate entry from List-II, and choose the correct option.

In a radioactive decay process, the activity is defined as $A = -\dfrac{dN}{dt}$, where $N(t)$ is the number of radioactive nuclei at time $t$. Two radioactive sources, $S_1$ and $S_2$ have same activity at time $t = 0$. At a later time, the activities of $S_1$ and $S_2$ are $A_1$ and $A_2$, respectively. When $S_1$ and $S_2$ have just completed their 3rd and 7th half-lives, respectively, the ratio $A_1/A_2$ is _____.

The binding energy of nucleons in a nucleus can be affected by the pairwise Coulomb repulsion. Assume that all nucleons are uniformly distributed inside the nucleus. Let the binding energy of a proton be $E_b^p$ and the binding energy of a neutron be $E_b^n$ in the nucleus.

Which of the following statement(s) is(are) correct?

In a radioactive decay chain reaction, ${}^{230}_{90}\mathrm{Th}$ nucleus decays into ${}^{214}_{84}\mathrm{Po}$ nucleus. The ratio of the number of $\alpha$ to number of $\beta^-$ particles emitted in this process 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 $\dfrac x{100}$. The value of $x$ is ______.

How many alpha and beta particles are emitted when Uranium $\mathrm{{}_{92}U^{238}}$ decays to lead $\mathrm{{}_{82}Pb^{206}}$ ?

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 :

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 $\log_e$ (no. of undecayed nuclei) Vs. time represents the reciprocal of mean life time ($\tau$).

(D) Product of decay constant ($\lambda$) and half-life time ($T_{1/2}$) is not constant.

Choose the most appropriate answer from the options given below :

The activity of a radioactive material is $2.56\times10^{-3}$ Ci. If the half life of the material is 5 days, after how many days the activity will become $2\times10^{-5}$ Ci ?

In the following nuclear reaction,
$$D\xrightarrow{\alpha}D_1\xrightarrow{\beta^-}D_2\xrightarrow{\alpha}D_3\xrightarrow{\gamma}D_4$$
Mass number of D is 182 and atomic number is 74. Mass number and atomic number of $D_4$ respectively will be ______

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 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 $\log_{10}1.88=0.274$)

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

The activity of a radioactive material is $6.4\times10^{-4}$ curie. Its half life is 5 days. The activity will become $5\times10^{-6}$ curie after :

The half life period of a radioactive substance is 60 days. The time taken for $\dfrac78$th of its original mass to disintegrate 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.

A radioactive sample decays $\dfrac78$ times its original quantity in 15 minutes. The half-life of the sample is