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Physics Nuclei Nuclei

At time t = 0, a material is composed of two radioactive atoms A and, B, where N_A(0) = 2N_B(0). The decay constant of both kind of radioactive atoms is $λ .$ However, A disintegrates to B and B disintegrates to C. Which of the following figures represents the evolution of N_B(t)/N_B(0) with respect to time t?

$[\begin{matrix} N_{A} (0) = N o . o f A a t o m s a t t = 0 & N_{B} (0) = N o . o f B a t o m s a t t = 0 \end{matrix}]$

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alok kumar singh

Contributor-Level 10

$\frac{d N_{B}}{d t} = λ N_{A} - λ N_{B} = 2 λ N_{B} (0) . e^{- λ t} - λ N_{B}$

$\Rightarrow \frac{d N_{B}}{d t} + λ N_{B} = 2 λ N_{B} (0) e^{- λ t}$

Solving it, $N_{B} (t) = N_{B} (0) (1 + 2 λ t) e^{- λ t}$

N_B (t) is maximum when,

$\frac{d N_{B}}{d t} = 0 \Rightarrow t = \frac{1}{2 λ}$

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Physics Nuclei Class 12th

Which of the following figure represents the variation of $I_{n} (\frac{R}{R_{0}})$ with I_nA (if R = radius of a nucleus and A = its mass number)

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alok kumar singh

Contributor-Level 10

We know, R = R_oA₃₌

_{$\Rightarrow \frac{R}{R_{O}} = A^{3}$}

Taking log both side

$l o g_{e} (\frac{R}{R_{O}}) = 3 l o g A$

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Physics Nuclei Class 12th

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 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.

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Payal Gupta

Contributor-Level 10

Radius, R = R₀A^1/3

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Physics Nuclei Nuclei

The activity of a radioactive sample falls from $700 s^{- 1}$ to $500 s^{- 1}$ in 30 minutes. Its half life is close to

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alok kumar singh

Contributor-Level 10

$A = A_{0} e^{- λ t}$

$\begin{array}{r} 500 = 700 e^{- λ t} \\ λ t = l n ? \frac{7}{5} \\ \frac{l n ? 2}{t_{1 / 2}} * 30 = l n ? \frac{7}{5} \\ ? t_{1 / 2} = \frac{l n ? 2 * 30}{l n ? \frac{7}{5}} \end{array}$

$t_{1 / 2} = 61.8 m i n \approx 62 m i n$

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3 months ago

Physics Nuclei Class 12th

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

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alok kumar singh

Contributor-Level 10

$A = \frac{A_{0}}{16} = \frac{A_{0}}{2^{30 / t_{1 / 2}}} \Rightarrow 2^{4} = 2^{30 / t_{1 / 2}}$

$\Rightarrow \frac{A_{0}}{16} = \frac{A_{0}}{2^{30 / t_{1 / 2}}} \Rightarrow 2^{4} = 2^{30 / t_{1 / 2}}$

$\Rightarrow t_{1 / 2} = \frac{30}{4} = 7.5$ years

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New answer posted

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Physics Nuclei Class 12th

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 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

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alok kumar singh

Contributor-Level 10

Radius, R = R₀A^1/3

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Physics Nuclei Class 12th

The half life period of a radioactive substance is 60 days. The time taken for $\frac{7}{8} t h$ of its original mass to disintegrate will be:

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Vishal Baghel

Contributor-Level 10

$t_{\frac{1}{2}} = 60 d a y s$

at t = 0 $\frac{m}{m_{0}} \to$ 0

$a t t \to \frac{m}{m_{0}} (1 - \frac{7}{8})$

= $\frac{m}{8 m_{0}} \to \frac{7}{8} (\frac{m}{m_{0}})$

$\Rightarrow \frac{1}{2^{3}} = \frac{1}{2^{(\frac{t}{60})}} \Rightarrow \frac{t}{60} = 3$

t = 180 days

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Physics Nuclei Nuclei

Two radioactive materials A and B have decay constants 25 $λ$ and 16 $λ$ 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 $\frac{1}{a λ} .$ The value of a is__________.

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alok kumar singh

Contributor-Level 10

$λ_{A} = 25 λ,$ $λ_{B} = 16 λ$

At t = 0 Þ N_A = N_B = N₀

after t = $\frac{1}{a λ} : - \frac{N_{B}}{N_{A}} = \frac{N_{0} e^{- 16 λ t}}{N_{0} e^{- 25 λ t}} = e^{(25 λ - 16 λ) t}$

->e = $e^{(9 λ \frac{1}{a λ})}$

$\Rightarrow \frac{9}{a} = 1 \Rightarrow a = 9$

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4 months ago

Physics Nuclei Class 12th

Two radioactive materials A and B have decay constants 25 $λ$ and 16 $λ$ 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 $\frac{1}{a λ} .$ The value of a is__________.

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Payal Gupta

Contributor-Level 10

$λ_{A} = 25 λ,$ $λ_{B} = 16 λ$

At t = 0 N_A = N_B = N₀

after t = $\frac{1}{a λ} : - \frac{N_{B}}{N_{A}} = \frac{N_{0} e^{- 16 λ t}}{N_{0} e^{- 25 λ t}} = e^{(25 λ - 16 λ) t}$

e = $e^{(9 λ \frac{1}{a λ})}$

$\Rightarrow \frac{9}{a} = 1 \Rightarrow a = 9$

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4 months ago

Physics Nuclei Class 12th

A sample contains 10^-2kg each of two substances A and B with half lives 4s and 8s respectively. The ration of their atomic weights is 1 : 2. The ratio of the amounts of A and B after16s is $\frac{x}{100}$ . The value of x is ______.

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Payal Gupta

Contributor-Level 10

$f o r A, t_{\frac{1}{2}} = 4 s e c$

$= m A_{0} e^{-} \frac{0.693}{4} * 16$

$m_{A} = m_{A 0} e^{- 4 * 0.693}$ - (1)

for B,

for B, $t_{\frac{1}{2}} = 8 s e c$

$m_{B} = m_{B_{0}} e^{- 2 * 0.693}$ - (2)

$\frac{m_{A}}{m_{B}} = \frac{m_{A O}}{m_{B O}} \frac{e^{- 4} * 0.693}{e^{- 2} * 0.693}$

$\frac{m_{A}}{m_{B}} = \frac{25}{100} = \frac{x}{100} x = 25$

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