###### Advertisements

###### Advertisements

A source contains two species of phosphorous nuclei, \[\ce{_15^32P}\] (T_{1/2} = 14.3 d) and \[\ce{_15^33P}\] (T_{1/2} = 25.3 d). At time t = 0, 90% of the decays are from \[\ce{_15^32P}\]. How much time has to elapse for only 15% of the decays to be from \[\ce{_15^32P}\]?

###### Advertisements

#### Solution

**Data: **\[\ce{_15^32P}\] : T_{1/2} = 14.3 d

∴ `lambda_1 = 0.693/(14.3 "d") = 0.04846 "d"^-1`

\[\ce{_15^32P}\] : T_{1/2} = 25.3 d

∴ `lambda_2 = 0.693/(25.3 "d") = 0.02739 "d"^-1`

At time t = 0, `("N"_"O1" lambda_1)/("N"_"O2"lambda_2) = (90%)/(10%) = 9` ...(1) and

at time t, `("N"_"O1" lambda_1"e"^(-lambda_1"t"))/("N"_"O2" lambda_2"e"^(-lambda_2"t")) = (15%)/(85%) = 3/17` ...(2)

Dividing Eq. (1) by Eq. (2), we get,

`("N"_"O1" lambda_1)/("N"_"O2"lambda_2) * ("N"_"O1" lambda_1"e"^(-lambda_1"t"))/("N"_"O2" lambda_2"e"^(-lambda_2"t")) = 9/(3//17) = 153/3`

∴ `"e"^((lambda_1 - lambda_2)"t") = 153/3`

∴ `(lambda_1 - lambda_2)"t" = 2.303 log_10(153/3) = 2.303(log_10 153 - log_10 3)`

∴ (0.04846 - 0.02739) t = 2.303 (2.1847 - 0.4771)

∴ t = `((2.303)(1.7076))/0.02107` = 186.6 days

#### APPEARS IN

#### RELATED QUESTIONS

The decay constant of radioactive substance is 4.33 x 10^{-4} per year. Calculate its half life period.

State the law of radioactive decay.

Derive the mathematical expression for law of radioactive decay for a sample of a radioactive nucleus

Under certain circumstances, a nucleus can decay by emitting a particle more massive than an α-particle. Consider the following decay processes:

\[\ce{^223_88Ra -> ^209_82Pb + ^14_6C}\]

\[\ce{^223_88 Ra -> ^219_86 Rn + ^4_2He}\]

Calculate the Q-values for these decays and determine that both are energetically allowed.

Represent Radioactive Decay curve using relation `N = N_o e^(-lambdat)` graphically

A radioactive nucleus 'A' undergoes a series of decays as given below:

The mass number and atomic number of A_{2} are 176 and 71 respectively. Determine the mass and atomic numbers of A_{4} and A.

Using the equation `N = N_0e^(-lambdat)` obtain the relation between half-life (T) and decay constant (`lambda`) of a radioactive substance.

(a) Derive the relation between the decay constant and half life of a radioactive substance.

(b) A radioactive element reduces to 25% of its initial mass in 1000 years. Find its half life.

Two different radioactive elements with half lives T_{1} and T_{2} have N_{1} and N_{2} undecayed atoms respectively present at a given instant. Derive an expression for the ratio of their activities at this instant in terms of N_{1} and N_{2 ?}

A radioactive nucleus ‘A’ undergoes a series of decays according to the following scheme:

The mass number and atomic number of A are 180 and 72 respectively. What are these numbers for A_{4}?

In a radioactive decay, neither the atomic number nor the mass number changes. Which of the following particles is emitted in the decay?

Calculate the maximum kinetic energy of the beta particle emitted in the following decay scheme:^{12}N → ^{12}C* + *e*^{+} + *v*^{12}C* → ^{12}C + γ (4.43MeV).

The atomic mass of ^{12}N is 12.018613 u.

(Use Mass of proton m_{p} = 1.007276 u, Mass of `""_1^1"H"` atom = 1.007825 u, Mass of neutron m_{n} = 1.008665 u, Mass of electron = 0.0005486 u ≈ 511 keV/c^{2},1 u = 931 MeV/c^{2}.)

The decay constant of `""_80^197`Hg (electron capture to `""_79^197`Au) is 1.8 × 10^{−4} S^{−1}. (a) What is the half-life? (b) What is the average-life? (c) How much time will it take to convert 25% of this isotope of mercury into gold?

The decay constant of ^{238}U is 4.9 × 10^{−18} S^{−1}. (a) What is the average-life of ^{238}U? (b) What is the half-life of ^{238}U? (c) By what factor does the activity of a ^{238}U sample decrease in 9 × 10^{9} years?

^{57}Co decays to ^{57}Fe by β^{+}- emission. The resulting ^{57}Fe is in its excited state and comes to the ground state by emitting γ-rays. The half-life of β^{+}- decay is 270 days and that of the γ-emissions is 10^{−8} s. A sample of ^{57}Co gives 5.0 × 10^{9} gamma rays per second. How much time will elapse before the emission rate of gamma rays drops to 2.5 × 10^{9}per second?

When charcoal is prepared from a living tree, it shows a disintegration rate of 15.3 disintegrations of ^{14}C per gram per minute. A sample from an ancient piece of charcoal shows ^{14}C activity to be 12.3 disintegrations per gram per minute. How old is this sample? Half-life of ^{14}C is 5730 y.

A radioactive isotope is being produced at a constant rate dN/dt = R in an experiment. The isotope has a half-life t_{1}_{/2}. Show that after a time t >> t_{1}_{/2} the number of active nuclei will become constant. Find the value of this constant.

Consider the situation of the previous problem. Suppose the production of the radioactive isotope starts at t = 0. Find the number of active nuclei at time t.

The half-life of ^{40}K is 1.30 × 10^{9} y. A sample of 1.00 g of pure KCI gives 160 counts s^{−1}. Calculate the relative abundance of ^{40}K (fraction of ^{40}K present) in natural potassium.

Obtain a relation between the half-life of a radioactive substance and decay constant (λ).

Define the term 'decay constant' of a radioactive sample. The rate of disintegration of a given radioactive nucleus is 10000 disintegrations/s and 5,000 disintegrations/s after 20 hr. and 30 hr. respectively from start. Calculate the half-life and the initial number of nuclei at t= 0.

What is the amount of \[\ce{_27^60Co}\] necessary to provide a radioactive source of strength 10.0 mCi, its half-life being 5.3 years?

Disintegration rate of a sample is 10^{10} per hour at 20 hours from the start. It reduces to 6.3 x 10^{9} per hour after 30 hours. Calculate its half-life and the initial number of radioactive atoms in the sample.

The isotope \[\ce{^57Co}\] decays by electron capture to \[\ce{^57Fe}\] with a half-life of 272 d. The \[\ce{^57Fe}\] nucleus is produced in an excited state, and it almost instantaneously emits gamma rays.

(a) Find the mean lifetime and decay constant for ^{57}Co.

(b) If the activity of a radiation source ^{57}Co is 2.0 µCi now, how many ^{57}Co nuclei does the source contain?

c) What will be the activity after one year?

Before the year 1900 the activity per unit mass of atmospheric carbon due to the presence of ^{14}C averaged about 0.255 Bq per gram of carbon.

(a) What fraction of carbon atoms were ^{14}C?

(b) An archaeological specimen containing 500 mg of carbon, shows 174 decays in one hour. What is the age of the specimen, assuming that its activity per unit mass of carbon when the specimen died was equal to the average value of the air? The half-life of ^{14}C is 5730 years.

Obtain an expression for the decay law of radioactivity. Hence show that the activity A(t) =λN_{O} e^{-λt}.

Two radioactive materials X_{1} and X_{2} have decay constants 10λ and λ respectively. If initially, they have the same number of nuclei, then the ratio of the number of nuclei of X_{1} to that of X_{2} will belie after a time.

A radioactive element disintegrates for an interval of time equal to its mean lifetime. The fraction that has disintegrated is ______

Which one of the following nuclei has shorter meant life?

'Half-life' of a radioactive substance accounts for ______.

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

After 1 hour, `(1/8)^"th"` of the initial mass of a certain radioactive isotope remains undecayed. The half-life of the isotopes is ______.

Two radioactive materials Y_{1} and Y_{2} have decay constants '5`lambda`' and `lambda` respectively. Initially they have same number of nuclei. After time 't', the ratio of number of nuclei of Y_{1} to that of Y_{2 }is `1/"e"`, then 't' is equal to ______.

What percentage of radioactive substance is left after five half-lives?

Two electrons are ejected in opposite directions from radioactive atoms in a sample of radioactive material. Let c denote the speed of light. Each electron has a speed of 0.67 c as measured by an observer in the laboratory. Their relative velocity is given by ______.

The half-life of a radioactive nuclide is 20 hrs. The fraction of the original activity that will remain after 40 hrs is ______.

If 10% of a radioactive material decay in 5 days, then the amount of original material left after 20 days is approximately :

The half-life of the radioactive substance is 40 days. The substance will disintegrate completely in

Suppose we consider a large number of containers each containing initially 10000 atoms of a radioactive material with a half life of 1 year. After 1 year ______.

A piece of wood from the ruins of an ancient building was found to have a ^{14}C activity of 12 disintegrations per minute per gram of its carbon content. The ^{14}C activity of the living wood is 16 disintegrations per minute per gram. How long ago did the tree, from which the wooden sample came, die? Given half-life of ^{14}C is 5760 years.

The radioactivity of an old sample of whisky due to tritium (half-life 12.5 years) was found to be only about 4% of that measured in a recently purchased bottle marked 10 years old. The age of a sample is ______ years.

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

The half-life of `""_82^210Pb` is 22.3 y. How long will it take for its activity 0 30% of the initial activity?