Nuclei

The decay of atoms over time can be described using the formula ,
where
represents the number of atoms at time
is the initial number of atoms,
is the decay constant.
For a specific case where equals :

Simplifying this expression, we find:

Taking the natural logarithm (ln) of both sides, we get:

Now, solving for

This equation establishes the relationship between the decay constant and the half-life .

The liquid droplet theory of nuclear fission, developed by George Gamow and Niels Bohr in the 1930s, describes the nucleus as a liquid drop that can undergo division (fission) into two smaller drops, releasing a significant amount of energy in the process. This theory provides a good explanation for many aspects of nuclear fission, including the energy release and the behavior of fissile materials such as uranium-235 when bombarded with neutrons.

So, the correct option is (A): Liquid droplet theory

Binding Energy of :
BE = ∆m × 931
BE = [2(1.0087 + 1.0073) – 4.0015] × 931
BE = [4.032 - 4.0015] × 931
BE = 0.0305 × 931
BE = 28.4 MeV

So, the correct option is (C): 28.4 MeV

Radius of the orbit of an atom
Where Z is the atomic number.
For the first orbit n = 1
Then,
⟹
Since Z is at its maximum value in the case of doubly ionized lithium, this results in the smallest radius for the first orbit in doubly ionized lithium.

So, the correct option is (A): Doubly ionized lithium.

The isotones have same number of neutrons (N) but difference proton number (Z)
i.e , (A-Z)are equal
From the options ₃₄Se⁷⁴, ₃₁Ca⁷¹ each one has same number of neutron i.e 40
Hence isotonic pair is of ₃₄Se⁷⁴, ₃₁Ga⁷¹

Therefore, the correct option is (A): ₃₄Se⁷⁴, ₃₁Ga⁷¹

The graph shows that for high atomic mass nuclei, the binding energy per nucleon falls as the number of mass atoms increases.
Mass and number are reduced as a result of fission. Nuclei's stability will rise with a drop in mass number, allowing for the fission of heavier nuclei.

Therefore, the correct option is (B): decreases with mass number at high mass numbers

we have
Energy released, E = (△m)×931MeV
atomic mass unit = 931MeV
m=mass of product− mass of reactant
m=c−a−b
E=( m)×931
i. e ,E =(c−a−b)
The binding energies of and are respectively a,b and c

Therefore, the correct answer is (B): c-a-b

Huge amounts of energy are generated during the fission process, and according to Einstein's mass-energy equivalent relationship, energy equates to mass of matter.

Therefore, the correct option is (A): less than 1

Radioactive Decay Constant: The radioactive decay of a material follows the formula: N(t) = N0 , where N(t) is the activity at time t, N0 is the initial activity, is the decay constant, and t is time.
If R₁ is the activity at time t₁ and R₂ is the activity at time t₂, we have:
R₁ = N0
R₂ = N0
Dividing these equations: = =

So, the correct option is R₁ = R₂

constant







Recoil . of nucleus

If and are the momenta of thorium and helium nuclei respectively, then according to law of conservation of linear momentum

0= or

-ve sign shows that both are moving in opposite directions.

But in magnitude


If m and m are the masses of thorium and helium
nuclei respectively, then

Kinetic energy of thorium nucleus is and that of helium nucleus is




But and

Thus the helium nucleus has more kinetic energy than the thorium nucleus.

Number of nuclei remaining









minute

So, is generated