Nuclei

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 $\log_{10}1.88=0.274$)

Mass numbers of two nuclei are in the ratio of 4:3. Their nuclear densities will be in the ratio of

Two lighter nuclei combine to form a comparatively heavier nucleus by the relation given below :

$${}^2_1X+{}^2_1X={}^4_2Y$$

The binding energies per nucleon for ${}^2_1X$ and ${}^4_2Y$ are 1.1 MeV and 7.6 MeV respectively. The energy released in this process is ______ MeV.

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

Read the following statements:

(A) Volume of the nucleus is directly proportional to the mass number.
(B) Volume of the nucleus is independent of mass number.
(C) Density of the nucleus is directly proportional to the mass number.
(D) Density of the nucleus is directly proportional to the cube root of the mass number.
(E) Density of the nucleus is independent of the mass number.

Choose the correct option from the following options.

Two radioactive materials A and B have decay constants 25λ and 16λ 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 $\dfrac1{a\lambda}$. The value of a is _______.

A heavy nucleus $Q$ of half-life 20 minutes undergoes alpha-decay with probability of 60% and beta-decay with probability of 40%. Initially, the number of $Q$ nuclei is 1000. The number of alpha-decays of $Q$ in the first one hour is

A heavy nucleus $N$, at rest, undergoes fission $N \to P + Q$, where $P$ and $Q$ are two lighter nuclei. Let $\delta = M_N - M_P - M_Q$, where $M_P$, $M_Q$ and $M_N$ are the masses of $P$, $Q$ and $N$, respectively. $E_P$ and $E_Q$ are the kinetic energies of $P$ and $Q$, respectively. The speeds of $P$ and $Q$ are $v_P$ and $v_Q$, respectively. If $c$ is the speed of light, which of the following statement(s) is(are) correct?

In a radioactive sample, $\ce{^{40}_{19}K}$ nuclei either decay into stable $\ce{^{40}_{20}Ca}$ nuclei with decay constant $4.5\times10^{-10}$ per year or into stable $\ce{^{40}_{18}Ar}$ nuclei with decay constant $0.5\times10^{-10}$ per year. Given that in this sample all the stable $\ce{^{40}_{20}Ca}$ and $\ce{^{40}_{18}Ar}$ nuclei are produced by the $\ce{^{40}_{19}K}$ nuclei only. In time $t\times10^9$ years, if the ratio of the sum of stable $\ce{^{40}_{20}Ca}$ and $\ce{^{40}_{18}Ar}$ nuclei to the radioactive $\ce{^{40}_{19}K}$ nuclei is 99, the value of $t$ will be,

[Given: $\ln10 = 2.3$]

Suppose a $\ce{^{226}_{88}Ra}$ nucleus at rest and in ground state undergoes α-decay to a $\ce{^{222}_{86}Rn}$ nucleus in its excited state. The kinetic energy of the emitted α particle is found to be 4.44 MeV. $\ce{^{222}_{86}Rn}$ nucleus then goes to its ground state by γ-decay. The energy of the emitted γ photon is ______ keV.

[Given: atomic mass of $\ce{^{226}_{88}Ra}$ = 226.005 u, atomic mass of $\ce{^{222}_{86}Rn}$ = 222.000 u, atomic mass of α particle = 4.000 u, 1 u = 931 MeV/c², c is speed of the light]

In an experiment the initial number of radioactive nuclei is 3000. It is found that $1000 \pm 40$ nuclei decayed in the first 1.0 s. For $|x| \ll 1$, $\ln(1+x) = x$ up to first power in $x$. The error $\Delta\lambda$, in the determination of the decay constant $\lambda$, in $\mathrm{s^{-1}}$, is

In a radioactive decay chain, ${}^{232}_{90}\mathrm{Th}$ nucleus decays to ${}^{212}_{82}\mathrm{Pb}$ nucleus. Let $N_\alpha$ and $N_\beta$ be the number of $\alpha$ and $\beta^-$ particles, respectively, emitted in this decay process. Which of the following statements is (are) true?