Two radioactive nuclei $P$ and $Q,$ in a given sample decay into a stable nucleus $R .$ At time $t = 0,$ number of $P$ species are $4 N_{0}$ and that of $Q$ are $N_{0} .$ Half-life of $P$ (for conversion to $R$ ) is $1 m i n u t e$ where as that of $Q$ is $2 m i n u t e s .$ Initially there are no nuclei of $R$ present in the sample. When number of nuclei of $P$ and $Q$ are equal, the number of nuclei of $R$ present in the sample would be :

Question

Verified by Toppr

Answer 1

No. of nuclei of $P$ after time, $t$ is given by, $N_{P} = N_{o} (\frac{1}{2})^{t / T_{1 / 2}}$
$⟹ N_{P} = 4 N_{o} (\frac{1}{2})^{t}$ (eqaution $01$ )
No. of nuclei of $Q$ after time, $t$ is given by, $N_{Q} = N_{o} (\frac{1}{2})^{t / T_{1 / 2}}$
$⟹ N_{Q} = N_{o} (\frac{1}{2})^{t / 2}$ (eqaution $02$ )
When nuclei of $P = Q$ or $N_{P} = N_{Q}$
$⟹ N_{o} (\frac{1}{2})^{t / 2} = 4 N_{o} (\frac{1}{2})^{t}$
$⟹ (\frac{1}{2})^{t / 2} = 4 (\frac{1}{2})^{t}$
$⟹ t = 4$ $m i n$
After $4$ $m i n$ , $N_{P} = N_{Q}$
Therefore, after $4$ $m i n$ , no. of nuclei of $R$ present is given by,
$N_{R} = (N_{o} - N_{P}) + (N_{o} - N_{Q})$
$⟹ N_{R} = ⎝ ⎛ 4 N_{o} - 4 N_{o} (\frac{1}{2})^{4} ⎠ ⎞ + ⎝ ⎛ N_{o} - N o (\frac{1}{2})^{2} ⎠ ⎞$
$⟹ N_{R} = 4 N_{o} (1 - \frac{1}{1 6}) + N_{o} (1 - \frac{1}{4})$
$⟹ N_{R} = N_{o} {\frac{1 5}{4} + \frac{3}{4}}$
$⟹ N_{R} = \frac{9 N _{o}}{2}$

Answer 2

No. of nuclei of

P

after time,

t

is given by,

N_{P} = N_{o} (\frac{1}{2})^{t / T_{1 / 2}}

⟹ N_{P} = 4 N_{o} (\frac{1}{2})^{t}

(eqaution

01

)

No. of nuclei of

Q

after time,

t

is given by,

N_{Q} = N_{o} (\frac{1}{2})^{t / T_{1 / 2}}

⟹ N_{Q} = N_{o} (\frac{1}{2})^{t / 2}

(eqaution

02

)

When nuclei of

P = Q

or

N_{P} = N_{Q}

⟹ N_{o} (\frac{1}{2})^{t / 2} = 4 N_{o} (\frac{1}{2})^{t}

⟹ (\frac{1}{2})^{t / 2} = 4 (\frac{1}{2})^{t}

⟹ t = 4

m i n

After

4

m i n

,

N_{P} = N_{Q}

Therefore, after

4

m i n

, no. of nuclei of

R

present is given by,

N_{R} = (N_{o} - N_{P}) + (N_{o} - N_{Q})

⟹ N_{R} = ⎝ ⎛ 4 N_{o} - 4 N_{o} (\frac{1}{2})^{4} ⎠ ⎞ + ⎝ ⎛ N_{o} - N o (\frac{1}{2})^{2} ⎠ ⎞

⟹ N_{R} = 4 N_{o} (1 - \frac{1}{1 6}) + N_{o} (1 - \frac{1}{4})

⟹ N_{R} = N_{o} {\frac{1 5}{4} + \frac{3}{4}}

⟹ N_{R} = \frac{9 N _{o}}{2}

$2 N_{0}$

$3 N_{0}$

$\frac{9 N _{0}}{2}$

$\frac{5 N _{0}}{2}$

Two radioactive materials $X_{1}$ $X_{1}$ contain same number of nuclei. If $6 λ s^{- 1}$ and $4 λ s^{- 1}$ are the decay constants of $X_{1}$ and $X_{2}$ respectively, the ratio of number of 48 nuclei, undecayed of $X_{1}$ to that of $X_{2}$ will be (1/e) after a time.

Two radioactive materials $X_{1}$ and $X_{2}$ contain same number of nuclei. If $6 λ s^{- 1}$ and $4 λ s^{- 1}$ are the decay constants of $X_{1}$ and $X_{2}$ respectively, the ratio of number of nuclei, undecayed of $X_{1}$ to that of $X_{2}$ will be (1/e) after a time:

There are two radioactive substances $A$ and $B$ . Decay constant of $B$ is two times that of $A$ . Initially, both have equal number of nuclei. After $n$ half lives of $A$ , rate of disintegration of both are equal. The value of $n$ is

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:

Revise with Concepts

Law of Radioactive Decay

Plot Number of Neutrons vs Protons for Radioactive Decay

Half Life

2N0​

3N0​

29N0​​

25N0​​

Two radioactive materials X1​ X1​ contain same number of nuclei. If 6λs−1 and 4λs−1 are the decay constants of X1​ and X2​ respectively, the ratio of number of 48 nuclei, undecayed of X1​ to that of X2​ will be (1/e) after a time.

Two radioactive materials X1​ and X2​ contain same number of nuclei. If 6λs−1 and 4λs−1 are the decay constants of X1​ and X2​ respectively, the ratio of number of nuclei, undecayed of X1​ to that of X2​ will be (1/e) after a time:

There are two radioactive substances A and B. Decay constant of B is two times that of A. Initially, both have equal number of nuclei. After n half lives of A, rate of disintegration of both are equal. The value of n is

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:

Revise with Concepts

Law of Radioactive Decay

Plot Number of Neutrons vs Protons for Radioactive Decay

Half Life

$2 N_{0}$

$3 N_{0}$

$\frac{9 N _{0}}{2}$

$\frac{5 N _{0}}{2}$

Two radioactive materials $X_{1}$ $X_{1}$ contain same number of nuclei. If $6 λ s^{- 1}$ and $4 λ s^{- 1}$ are the decay constants of $X_{1}$ and $X_{2}$ respectively, the ratio of number of 48 nuclei, undecayed of $X_{1}$ to that of $X_{2}$ will be (1/e) after a time.

Two radioactive materials $X_{1}$ and $X_{2}$ contain same number of nuclei. If $6 λ s^{- 1}$ and $4 λ s^{- 1}$ are the decay constants of $X_{1}$ and $X_{2}$ respectively, the ratio of number of nuclei, undecayed of $X_{1}$ to that of $X_{2}$ will be (1/e) after a time:

There are two radioactive substances $A$ and $B$ . Decay constant of $B$ is two times that of $A$ . Initially, both have equal number of nuclei. After $n$ half lives of $A$ , rate of disintegration of both are equal. The value of $n$ is

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: