The masses of two radioactive substances are same and their half lives are $1$ year and $2$ years respectively. The ratio of their activities after $6$ years will be

Question

Verified by Toppr

Answer 1

Given : $T_{A} = 1$ years and $T_{B} = 2$ years
Time of observation $t = 6$ years
Let the initial mass (or nuclei) of two radioactive substance be $N_{o}$ .
Initial activity of substance A $R_{o_{A}} = λ_{A} N_{o} = \frac{( ln 2 ) N _{o}}{T _{A}}$
Activity of substance A after time $t$ , $R_{A} = R_{o_{A}} e^{\frac{- ( l n 2 ) t}{T _{A}}}$
Initial activity of substance B $R_{o_{B}} = λ_{B} N_{o} = \frac{( ln 2 ) N _{o}}{T _{B}}$
Activity of substance B after time $t$ , $R_{B} = R_{o_{B}} e^{\frac{- ( l n 2 ) t}{T _{B}}}$
Thus ratio of activities after 6 years, we get $\frac{R _{A}}{R _{B}} = \frac{R _{o_{A}}}{R _{o_{B}}} e^{- (l n 2) [\frac{t}{T _{A}} - \frac{t}{T _{B}}]}$
Or $\frac{R _{A}}{R _{B}} = \frac{T _{B}}{T _{A}} e^{- (l n 2) [\frac{6}{1} - \frac{6}{2}]}$
Or $\frac{R _{A}}{R _{B}} = \frac{2}{1} e^{- 3 l n 2} = 2 \times e^{l n \frac{1}{8}}$
Or $\frac{R _{A}}{R _{B}} = 2 \times \frac{1}{8}$
$⟹$ $R_{A} : R_{B} = 1 : 4$

Answer 2

Given :

T_{A} = 1

years and

T_{B} = 2

years

Time of observation

t = 6

years

Let the initial mass (or nuclei) of two radioactive substance be

N_{o}

.

Initial activity of substance A

R_{o_{A}} = λ_{A} N_{o} = \frac{( ln 2 ) N _{o}}{T _{A}}

Activity of substance A after time

t

,

R_{A} = R_{o_{A}} e^{\frac{- ( l n 2 ) t}{T _{A}}}

Initial activity of substance B

R_{o_{B}} = λ_{B} N_{o} = \frac{( ln 2 ) N _{o}}{T _{B}}

Activity of substance B after time

t

,

R_{B} = R_{o_{B}} e^{\frac{- ( l n 2 ) t}{T _{B}}}

Thus ratio of activities after 6 years, we get

\frac{R _{A}}{R _{B}} = \frac{R _{o_{A}}}{R _{o_{B}}} e^{- (l n 2) [\frac{t}{T _{A}} - \frac{t}{T _{B}}]}

Or

\frac{R _{A}}{R _{B}} = \frac{T _{B}}{T _{A}} e^{- (l n 2) [\frac{6}{1} - \frac{6}{2}]}

Or

\frac{R _{A}}{R _{B}} = \frac{2}{1} e^{- 3 l n 2} = 2 \times e^{l n \frac{1}{8}}

Or

\frac{R _{A}}{R _{B}} = 2 \times \frac{1}{8}

⟹

R_{A} : R_{B} = 1 : 4

The masses of two radioactive substances are same and their half lives are $1$ year and $2$ years respectively. The ratio of their activities after $6$ years will be

$1 : 4$

$1 : 2$

$1 : 3$

$1 : 6$

At a given instant, say t = 0, two radioactive substances 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 $l n 2$ , the half-life of B is:

The ratio of molecular mass of two radioactive substances is 3/2 and the ratio of their decay constants is 4/3. Then, the ratio of their initial activity per mole will be.

Two radioactive samples consist of equal number of atoms.The half lives are 2 hours and 4 hours respectively.The ratio of their activities then is

Two radioactive samples $A_{1}$ and $A_{2}$ having half life 3 years and 2 years respectively have been decaying for many years. today the number of atoms in the sample $A_{1}$ is twice the number of atoms in the sample $A_{2}$ . both the samples had same number of atoms X years ago, calculate X.

half life of a radioactive substance A is 4 days. the probability that a nucleus will decay in two half lives is:

Revise with Concepts

Law of Radioactive Decay

Plot Number of Neutrons vs Protons for Radioactive Decay

Half Life

The masses of two radioactive substances are same and their half lives are 1 year and 2 years respectively. The ratio of their activities after 6 years will be

1:4

1:2

1:3

1:6

At a given instant, say t = 0, two radioactive substances A and B have equal activities. The ratio RA​RB​​ of their activities after time t itself decays with time t as e−3t. If the half-life of A is ln2, the half-life of B is:

The ratio of molecular mass of two radioactive substances is 3/2 and the ratio of their decay constants is 4/3. Then, the ratio of their initial activity per mole will be.

Two radioactive samples consist of equal number of atoms.The half lives are 2 hours and 4 hours respectively.The ratio of their activities then is

Two radioactive samples A1​ and A2​ having half life 3 years and 2 years respectively have been decaying for many years. today the number of atoms in the sample A1​ is twice the number of atoms in the sample A2​. both the samples had same number of atoms X years ago, calculate X.

half life of a radioactive substance A is 4 days. the probability that a nucleus will decay in two half lives is:

Revise with Concepts

Law of Radioactive Decay

Plot Number of Neutrons vs Protons for Radioactive Decay

Half Life

The masses of two radioactive substances are same and their half lives are $1$ year and $2$ years respectively. The ratio of their activities after $6$ years will be

$1 : 4$

$1 : 2$

$1 : 3$

$1 : 6$

At a given instant, say t = 0, two radioactive substances 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 $l n 2$ , the half-life of B is:

Two radioactive samples $A_{1}$ and $A_{2}$ having half life 3 years and 2 years respectively have been decaying for many years. today the number of atoms in the sample $A_{1}$ is twice the number of atoms in the sample $A_{2}$ . both the samples had same number of atoms X years ago, calculate X.