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Two nuclei have mass numbers in the ratio 27:125. What is the ratio of their nuclear radii?

Given,
Rato of mass numbers ,

$⇒$

Therefore,

Thus, Ratio of radii is 3:5.

1181 Views

Define nuclear force. Give its two most important characteristics.What is the energy released if all the deuterium atoms in a lake of cross sectional area 2.56 × 105 (km)2 and depth 80 m is used in fusion? Given abundance of  of hydrogen density of water = 103 kg m-3 energy released due to fusion of one atom of

Nuclear force is a strong attractive force acting between the nucleons of the atomic nucleus that holds the nucleus together.

Characteristics of nuclear force :

i) Nuclear forces are short range forces operating upto a distance which is of the order of a few fermi.

ii) Nuclear forces are strongest force in nature. It's magnitude is 100 times that of the electrostatic force and 1038 times that of the gravitational force.

Numerical:

The volume  V of lake is,

$\therefore$ Mass of water, M in lake  = 2.048 x 1013 x 103
= 2.048 x 1016 kg
= 2.048 x 1019 gm

Now, 18 g of water contains 2 g of hydrogen.

$\therefore$ Mass of hydrogen atom in lake = $\frac{1}{9}×2.048×{10}^{19}\mathrm{g}$

$\therefore$ Number of atoms of hydrogen in lake =

Since abundance of ${}_{1}{}^{2}\mathrm{H}$ is only 0.0156% of hydrogen atoms, the number of ${}_{1}{}^{2}\mathrm{H}$ atoms in lake,

Energy released due to fusion of one atom of

$\therefore$ Energy released when all ${}_{1}{}^{2}\mathrm{H}$ atoms present undergo fusion,

94 Views

A radioactive substance has a half-life period of 30 days. Calculate (i) time taken for $\frac{3}{4}$ of original number of atoms to disintegrate and (ii) time taken for $\frac{1}{8}$ of the original number of atoms to remain unchanged.

In any radioactive sample, undergoing decay, the number of nucleii undergoing decay per unit time is directly proportional to the total number of nuclei in the sample. This is known as the radioactive decay law.

Let, N be the number of nucleii in the sample,
$∆$N is the sample undergoing decay and,
is the time then,

$⇒$

where, $\mathrm{\lambda }$ is the decay constant.

Numerical:

Half -period of radioactive substance = 30 days
Number of atoms disintegrated =
Number of atoms left after time t,

Number of half lives in time t days,

where,

T = Half life time
n = no. of half lives
t = time for disintegrates

Number of nuclei left after n half lives is given by,

Therefore,

$⇒$

$⇒$

(ii) Now, using the formula,

$\therefore$

$⇒$

$⇒$
i.e.,                      , is the time taken for 1/8 of the original number of atoms to remain unchanged.

176 Views

Define the term decay constant of a radioactive nucleus.
Two nuclei P, Q have equal number of atoms at t = 0. Their half lives are 3 hours and 9 hours respectively. Compare their rates of disintegration, after 18 hours from the start.

Decay constant of a radioactive element is the reciprocal of the time during which the number of atoms left in the sample reduces to $\frac{1}{\mathrm{e}}$ times the number of atoms in the original sample.

Given, two nucleii  P and Q.
P and Q have equal number of atoms at t=0.
Half life of P = 3 hours.
Half life of Q = 9 hours.

and, t = 18 hours

Number of half lives of P in 18 hours  =$\frac{\mathrm{t}}{{\mathrm{T}}_{1}{2}}}$$\frac{18}{3}=6$

Number of nuclei P left undecayed after 6 half lives is,

Number of half lives of Q in 18 hours=$\frac{\mathrm{t}}{{\mathrm{T}}_{1}{2}}}$ = $\frac{18}{9}=2$

Number of nuclei of Q left undecayed after 2 half lives is,

Ratio of their decay rate is,

$\therefore$

Hence ratio of disintegration is,
1204 Views

Group the following six nuclides into three pairs of (i) isotones, (ii) isotopes and (iii) isobars:

How does the size of a nucleus depend on its mass number? Hence explain why the density of nuclear matter should be independent of the size of the nucleus.

Classification of nuclides is as follows:

Isotopes ; has the same atomic number but, different mass number.

Isobars  ; has the same mass number but, different atomic number.

Isotones ; has same neutron number but, different proton number.

Now,

where,  is the range of nuclear force (or Nuclear Unit Radius)

Density is given by,                         ...(ii)

But,                                                        ...(iii)

From (i) and (iii), we get

where,

...(iv)

From (ii) and (iv), we have

Thus, we can see that nuclear density is clearly independent of mass number A.

153 Views

An unstable element is produced in nuclear reactor at a constant rate R. If its half-life β--decay is T1/2, how much time, in terms of T1/2, is required to produce 50% of the equilibrium quantity?

We have,

Rate of increase of element =

That is,

The solution to this is the sum of the homogeneous solution,
where c is a constant, and

a particular solution,

Therefore, the required solution is,

The constant c is obtained from the requirement that the initial number of nuclei be zero,

so that,

The equilibrium value is

Setting N equal to 1/2 of this value gives,

$⇒$

The result is independent of R.

120 Views