Correct answer option is (C)

$\begin{array}{rl}& L=mvr\propto n\\ & \therefore mvr\propto \sqrt{r}\end{array}$

Correct answer option is (C)

In the context of an electron orbiting around a nucleus with a positive charge of $\mathrm{Ze}$, we are dealing with classical physics approximations and the electrostatic force between the electron and the nucleus. In such a setup, the electron's potential energy (U) is due to electrostatic interaction, and it is given by Coulomb's law:

$U=-\frac{kZ{e}^2}{r}$

Where:

• $U$ is the potential energy of the electron,
• $k$ is Coulomb's constant,
• $Z$ is the atomic number (number of protons in the nucleus),
• $e$ is the charge of an electron, and
• $r$ is the radius of the orbit of the electron around the nucleus.

The negative sign indicates that the potential energy is negative because the electron and nucleus attract each other.

The total energy (E) of the electron in orbit is the sum of its kinetic energy (K) and its potential energy (U). Since the electron is in a stable orbit, its kinetic energy can be shown to be exactly half the magnitude of its potential energy but positive:

$K=-\frac{1}{2}U$

Therefore,

$E=K+U=-\frac{1}{2}U+U=\frac{1}{2}U$

To find a relation between total energy (E) and potential energy (U), we rearrange the equation as follows:

$2E=U$

This is to say, the total energy (E) is half the magnitude of potential energy (U) but negative, and the correct relationship between them, when looking for a positive proportionality, yields to $2E=U$. Hence, the correct option is:

Option C: $2\mathrm{E}=\mathrm{U}$

Correct answer option is (A)

According to Bohr's theory, one of the postulates specifies that the angular momentum of an electron in orbit around a nucleus is quantized. This quantization can be expressed by the formula:

$L=n\frac{h}{2\pi }$

Where:

• $L$ is the angular momentum of the electron,
• $n$ is the principal quantum number (or the orbit number in simpler terms), which can take positive integer values (1, 2, 3, ...),
• $h$ is Planck's constant ($6.62607015\times {10}^{-34}{m}^{2}kg/s$), and
• $\frac{h}{2\pi }$ is often denoted as $\hslash$ (h-bar), known as the reduced Planck's constant.

For an electron in the 4th orbit ($n=4$) of a hydrogen atom, we substitute $n=4$ into the equation:

$L=4\frac{h}{2\pi }$

Therefore, the moment of momentum (or angular momentum) of an electron in the $4^{\text{th}}$ orbit of a hydrogen atom is:

$L=4\frac{h}{2\pi }=2\frac{2h}{2\pi }=2\frac{h}{\pi }$

Hence, the correct option is:

Option A: $2\frac{h}{\pi }$

Correct answer option is (C)

Let's analyze each of the given statements to determine which ones are correct:

(A) The angular momentum of an electron in $n^{\text{th}}$ orbit is an integral multiple of $\hslash$.

This statement reflects Bohr's quantization rule for angular momentum in the Bohr model of the hydrogen atom. According to this rule, the angular momentum of an electron in a stationary orbit is quantized and given by:

$L=n\hslash$

where $n$ is a principal quantum number (which can be any positive integer), and $\hslash$ is the reduced Planck's constant. Therefore, this statement (A) is correct.

(B) Nuclear forces do not obey inverse square law.

Nuclear forces, specifically strong nuclear forces, do act over short ranges within the nucleus but do not obey the inverse square law, which is characteristic of the electromagnetic and gravitational forces. Thus, statement (B) is correct.

(C) Nuclear forces are spin dependent.

The strength of the nuclear force can depend on the spin alignment of the nucleons. This is why some isotopes are more stable than others depending on spin-related factors. Thus, statement (C) is correct.

(D) Nuclear forces are central and charge independent.

The strong nuclear force is indeed charge independent, meaning it is the same regardless of the types of nucleons involved (neutrons or protons). However, nuclear forces are not always central; they can be tensor forces too, which involve more complex interactions that are not purely central. Therefore, the entirety of statement (D) is not correct; the statement should specify that nuclear forces are charge independent but may not always be central.

(E) Stability of nucleus is inversely proportional to the value of packing fraction.

This statement (E) is correct.

Based on the above explanations, the correct statements are (A), (B), (C) and (E). Hence, the correct answer is:

$\text{Option C : (A), (B), (C), (D) only}$

Correct answer option is (D)

$\begin{array}{rl}& \text{ Magnetic moment }=\mathrm{i}\pi {\mathrm{r}}^2\\ & \mu =\frac{\text{ evr }}{2}\\ & \mu \propto \left(\frac{1}{\mathrm{n}}\right){\mathrm{n}}^2\\ & \mu \propto \mathrm{n}\\ & \mathrm{x}=1\end{array}$