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Standard X

Physics

Nuclear Fusion

a In a typica...

Question

(a) In a typical nuclear reaction e.g
$_{1}^{2} H +_{1}^{2} H \to_{2}^{3} H +_{0}^{1} n + 3.27 M e V$ ,
although number of nucleons is conserved, yet energy is released, How? Explain.
(b) Show that nuclear density in a given nucleus is independent of mass number A.

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Solution

In a nuclear reaction, the aggregate of the masses of the target nucleus $(_{1}^{2} H)$ and the bombarding particle may be greater or less than the aggregate of the masses of the product nucleus $(_{2}^{3} H e)$ and the outgoing particle $(_{0}^{1} n)$ .
So, from the law of conservation of mass- energy some energy ( $3.27 M e v$ ) is evolved or involved in a nuclear reaction.
This energy is called Q-value of the nuclear reaction.

Density of the nucleus $= \frac{m a s s o f n u c l e u s}{v o l u m e o f n u c l e u s}$

Mass of the nucleus $= A a m u$ $= A \times 1.666 \times 10^{- 27} k g$

Volume of the nucleus, $V = \frac{4}{3} π R^{3} = \frac{4}{3} π {(R_{0} A^{\frac{1}{3}})}_{0}^{3}$

Where, $R = R_{0} A^{1 / 3}$ and $V = \frac{4}{3} π R_{0}^{3} A$

Thus, density $= \frac{A \times 1.66 \times 10^{- 27}}{(\frac{4}{3} π R_{0}^{3}) A} = \frac{1.66 \times 10^{- 27}}{(\frac{4}{3} π R_{0}^{3})}$

Which shows that the density is independent of mass number A.

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Similar questions

Q. (i) What characteristic property of nuclear force explains the constancy of binding energy per nucleon

(B E / A)

in the range of mass number

^{'} A^{'}

lying

30 < A < 170

?
(ii) Show that the density of nucleus over a wide range of nuclei is constant-independent of mass number

A

Q. Nuclear binding energy is the energy released during the hypothetical formation of the nucleus by the condensation of individual nucleons. Thus, binding energy per nucleon =

\frac{Total binding energy}{Number of nucleons}

For example, the mass of hydrogen atom is equal to the sum of the masses of a proton and an electron. For other atoms, the atomic mass is less than the sum of the masses of protons, neutrons and electrons present. This difference in mass, termed as mass defect, is a measure of the binding energy of protons and neutrons in the nucleus. The mass-energy relationship postulated by Einstein is expressed as:

Δ E = Δ m c^{2}

Where

Δ E

is the energy liberated,

Δ m

is the loss of mass, and c is the speed of light.
In the reaction

_{1}^{2} H +_{1}^{3} H \to_{2}^{4} H e +_{0}^{1} n

, if binding energies

_{1}^{2} H,_{1}^{3} H and_{2}^{4} H e

are respectively a, b and c (in MeV), then the energy released in this reaction is:

Q. The binding energy per nucleon of

_{1}^{2} H

and

_{2}^{4} H e

are

1.1 M e V

and

7.0 M e V

respectively. Energy released in the process

_{1}^{2} H +_{1}^{2} H \to_{2}^{4} H e

The binding energy of deuteron $(_{1}^{2} H)$ is 1.15 MeV per nucleon and an alpha particle $(_{2}^{4} H e)$ has a binding energy of 7.1 MeV per nucleon. Then in the reaction
$_{1}^{2} H +_{1}^{2} H \to_{2}^{4} H e + Q$
the energy Q released is

Q. From the relation R =

R_{0} A^{\frac{1}{3}}

, where

R_{0}

is a constant and A is the mass number of a nucleus, show that the nuclear matter density is nearly constant (i.e. independent of A).

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(a) In a typical nuclear reaction e.g21H+21H→32H+10n+3.27MeV,although number of nucleons is conserved, yet energy is released, How? Explain. (b) Show that nuclear density in a given nucleus is independent of mass number A.

(a) In a typical nuclear reaction e.g
$_{1}^{2} H +_{1}^{2} H \to_{2}^{3} H +_{0}^{1} n + 3.27 M e V$ ,
although number of nucleons is conserved, yet energy is released, How? Explain.
(b) Show that nuclear density in a given nucleus is independent of mass number A.