A stationary thorium nucleus (A=220, Z=90) emits an alpha particle with kinetic energy E_{\alpha} . What is the kinetic energy of the recoiling nucleus?

^{220} Z _{90} \rightarrow \alpha + X m_X = 220 - m_{\alpha} = 216 E =\cfrac{p^2}{2m} p = \sqrt{2mE} Momentum Conservation: \sqrt{2m_{\alpha}E_{\alpha}}= \sqrt{2m_{X}E_{X}} m_{\alpha}E_{\alpha}= m_{X}E_{X} 4E_{\alpha}= 216E_{X} E(X) = \cfrac{E_{\alpha}}{54}

The above is a plot of binding energy per nucleon \mathrm{E}_{\mathrm{b}} , against the nuclear mass \mathrm{M};\mathrm{A},\ \mathrm{B},\ \mathrm{C},\ \mathrm{D},\ \mathrm{E},\ \mathrm{F} correspond to different nuclei. Consider four reactions :(i) \mathrm{A}+\mathrm{B}\rightarrow \mathrm{C}+\epsilon (ii) \mathrm{C}\rightarrow \mathrm{A}+\mathrm{B}+\epsilon (iii) \mathrm{D}+\mathrm{E}\rightarrow \mathrm{F}+\epsilon (iv) \mathrm{F}\rightarrow \mathrm{D}+\mathrm{E}+\epsilon where \epsilon is the energy released ? In which reaction is \epsilon positive ?

The reactions for which product is/are more stable than reactant, energy & will be positive. (Energy will be released)As B.E/nuclear decides stability \therefore Stability :C>B C>A D>F and E>F \left( i \right) A+B\longrightarrow C+\varepsilon \left( iv \right) F\longrightarrow D+E+\varepsilon are the reactions for which \varepsilon will be positive

Nuclei - Hots Questions

Q: Consider two nuclei of same radioactive nuclide. One of the nuclei was created in a supernova explosion 5 billion years ago. The other was created in a nuclear reaction 5 minutes ago. The probability of decay during next time is

The half life t_{1/2} and decay constant \lambda remain constant, independent of time of creation of radioactive nuclei. So the probability of decay remains same.

Q: STATEMENT-1: The plot of atomic number (y-axis) versus number of neutrons (x-axis) for stable nuclei shows a curvature towards x-axis from the line of 45^{o} slope as the atomic number is increased.STATEMENT-2: Proton-proton electrostatic repulsions begin to overcome attractive forces involving protons and neutrons and neutrons in heavier nuclides.

If the curve does not bend down towards the x axis then the proton-proton repulsion would overcome the attractive force of proton and neutron. Therefore, the curve bends down.

Q: For the \beta^+ (positron) emission from a nucleus, there is another competing process known as electron capture (electron from an inner orbit, say, the K shell, is captured by the nucleus and a neutrino is emitted). e^+\, +\, ^A_ZX\, \rightarrow\, ^A_{Z - 1}Y\, +\, v Show that if \beta^+ emission is energetically allowed, electron capture is necessarily allowed but not vice versa.

Consider the competing processes: ^A_ZW\, \rightarrow\, ^A_{Z - 1}\, Y\, +\, e^+\, +\, v_e\, +\, Q_1 (positron cature) e^-\, +\, ^A_ZW\, \rightarrow\, ^A_{Z - 1}Y\, +\, v_e\, +\, Q_2 (electron capture) Q_1\, =\, [m_N(^A_ZX)\, -\, m_N\, (^A_{Z - 1}Y)\, -\, m_e]\, c^2 =\, [m_N\,(^A_ZX)\, -\, Zm_e\, -\, m(^A_{Z - 1}Y)\,-\, (Z\,-\,1)\,m_e\,-\, m_e ]\, c^2 =\, [m_N(^A_ZX)\, -\, m_N\, (A_{Z - 1}Y)\, -\, 2m_e]\,c^2 Q_2\, =\, [m_N(^A_ZX)\, +\, m_e\, -\, m_N\, (^A_{Z - 1}Y)]\,c^2\, =\, [m(^A_ZX)\, -\, m\, (^A_{Z - 1}Y)]\, c^2 This means Q_1\, >\, 0 implies Q_2\, >\, 0 , but Q_2\, >\, 0 does not necessarily mean Q_1\, >\, 0 .

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Hots Questions

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Nuclei

- Think before you solve

Radioactive decay law states that radioactive samples decay exponentially. But what can you say about the disintegration probability of a single nuclide. Is the disintegration probability the same for every nuclide?

Consider two nuclei of same radioactive nuclide. One of the nuclei was created in a supernova explosion

5

billion years ago. The other was created in a nuclear reaction

5

minutes ago. The probability of decay during next time is

different of each nuclei

nuclei created in explosion decays first

nuclei created in the reactor decays first

independent of time of creation

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What determines the decay rate of a of a radioactive sample? Is it related to the energy released?

Assertion

In alpha decay of different radioactive nuclides, the energy of alpha particles has been compared. It is found that as the energy of alpha particle increases the half-life of the decay goes on decreasing.

Reason

More is the energy in any decay process, more is the probability of decaying the nuclide which leads to faster rate of decay.

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Assertion is correct but Reason is incorrect

Assertion is incorrect but Reason is correct

View solution

Does the process of spontaneous alpha decay obey the law of conservation of momentum?

A stationary thorium nucleus

(A = 220, Z = 90)

emits an alpha particle with kinetic energy

E_{α}

. What is the kinetic energy of the recoiling nucleus?

\frac{E _{α}}{1 0 8}

\frac{E _{α}}{1 1 0}

\frac{E _{α}}{5 5}

\frac{E _{α}}{5 4}

View solution

What is the reason behind stable nuclei not having more protons than neutrons?

STATEMENT-1:
The plot of atomic number (y-axis) versus number of neutrons (x-axis) for stable nuclei shows a curvature towards x-axis from the line of

4 5^{o}

slope as the atomic number is increased.
STATEMENT-2:
Proton-proton electrostatic repulsions begin to overcome attractive forces involving protons and neutrons and neutrons in heavier nuclides.

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is correct explanation for STATEMENT-1

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1

STATEMENT-1 is True, STATEMENT-2 is False

STATEMENT-1 is False, STATEMENT-2 is True

View solution

How can we check the feasibility of a nuclear reaction using the binding energy per nucleon curve?

The above is a plot of binding energy per nucleon

E_{b}

, against the nuclear mass

M; A, B, C, D, E, F

correspond to different nuclei. Consider four reactions :
(i)

A + B \to C + ϵ

(ii)

C \to A + B + ϵ

(iii)

D + E \to F + ϵ

(iv)

F \to D + E + ϵ

where

ϵ

is the energy released ? In which reaction is

ϵ

positive ?

(i) and (iv)

(i) and (iii)

(ii) and (iv)

(ii) and (iii).

View solution

$β^{+}$ decay and electron capture are two competing processes. Can you show that one reaction is energetically possible and the other is not?

For the

β^{+}

(positron) emission from a nucleus, there is another competing process known as electron capture (electron from an inner orbit, say, the K shell, is captured by the nucleus and a neutrino is emitted).

e^{+} +_{Z}^{A} X \to_{Z - 1}^{A} Y + v

Show that if

β^{+}

emission is energetically allowed, electron capture is necessarily allowed but not vice versa.

View solution