The mass number of nucleus having radius equal to half of the radius of nucleus with mass number 192 is:
- 24
- 32
- 40
- 20
The Correct Option is A
Approach Solution - 1
To solve this problem, we need to apply the formula relating the radius of a nucleus to its mass number. The radius \( R \) of a nucleus is given by the formula:
\(R = R_0 A^{1/3}\)
where \( R_0 \) is a constant (~1.2-1.3 femtometers) and \( A \) is the mass number.
We are given that the radius of the first nucleus is half of the radius of a nucleus with mass number 192, denoted as \( R_1 = \frac{1}{2} R_2 \). Let \( A_1 \) be the mass number of the first nucleus, and \( A_2 = 192 \).
- For the nucleus with mass number 192, its radius \( R_2 \) is: \(R_2 = R_0 (192)^{1/3}\)
- For the nucleus with mass number \( A_1 \), its radius \( R_1 \) is: \(R_1 = R_0 (A_1)^{1/3}\)
- According to the problem, the relation given is: \(R_0 (A_1)^{1/3} = \frac{1}{2} R_0 (192)^{1/3}\)
- Canceling \( R_0 \) from both sides: \((A_1)^{1/3} = \frac{1}{2} (192)^{1/3}\)
- Cubing both sides to eliminate the cube root gives: \(A_1 = \left( \frac{1}{2} (192)^{1/3} \right)^3\)
- Calculate \( (192)^{1/3} \): \((192)^{1/3} = 5.82\) (approximately)
- Thus: \(A_1 = \left( \frac{5.82}{2} \right)^3 = \left( 2.91 \right)^3 \approx 24.68\)
Rounding 24.68 to the nearest whole number gives 24. Therefore, the mass number of the nucleus having a radius equal to half of the radius of the nucleus with mass number 192 is 24.
Approach Solution -2
The radius \( R \) of a nucleus is proportional to the cube root of its mass number \( A \):
\[ R \propto A^{1/3}. \]Let \( R_1 \) and \( R_2 \) be the radii of two nuclei with mass numbers \( A_1 \) and \( A_2 \), respectively. Given:
\[ R_1 = \frac{1}{2} R_2 \quad \text{and} \quad A_2 = 192. \]Using the proportionality,
\[ \frac{R_1}{R_2} = \left( \frac{A_1}{A_2} \right)^{1/3}. \]Substitute \( R_1 = \frac{1}{2} R_2 \):
\[ \frac{1}{2} = \left( \frac{A_1}{192} \right)^{1/3}. \]Cubing both sides:
\[ \frac{1}{8} = \frac{A_1}{192}. \]Solving for \( A_1 \):
\[ A_1 = 192 \times \frac{1}{8} = 24. \]Thus, the answer is:
\[ 24. \]Top JEE Main Physics Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
A black body is at a temperature of 2880 K. The energy of radiation emitted by this body with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien's constant, b = 2.88×106 nm-K. Then,
- $I _{ CM }$ is the moment of inertia of a circular disc about an axis (CM) passing through its center and perpendicular to the plane of disc $I_{A B}$ is it's moment of inertia about an axis $AB$ perpendicular to plane and parallel to axis $CM$ at a distance $\frac{2}{3} R$ from center Where $R$ is the radius of the dise The ratio of $I _{ AB }$ and $I _{ CM }$ is $x: 9$ The value of $x$ is______
- A nucleus disintegrates into two smaller parts, which have their velocities in the ratio $3: 2$ The ratio of their nuclear sizes will be $\left(\frac{x}{3}\right)^{\frac{1}{3}}$ The value of ' $x$ ' is:-
Top JEE Main Nuclei Questions
- A nucleus disintegrates into two smaller parts, which have their velocities in the ratio $3: 2$ The ratio of their nuclear sizes will be $\left(\frac{x}{3}\right)^{\frac{1}{3}}$ The value of ' $x$ ' is:-
The electric potential at the surface of an atomic nucleus \( (z = 50) \) of radius \( 9 \times 10^{-13} \) cm is \(\_\_\_\_\_\_\_ \)\(\times 10^{6} V\).
In a nuclear fission reaction of an isotope of mass \( M \), three similar daughter nuclei of the same mass are formed. The speed of a daughter nuclei in terms of mass defect \( \Delta M \) will be:
- The mass defect in a particular reaction is \( 0.4 \, \text{g} \). The amount of energy liberated is \( n \times 10^7 \, \text{kWh} \), where \( n = \) _____. (speed of light \( = 3 \times 10^8 \, \text{m/s} \))
- A nucleus at rest disintegrates into two smaller nuclei with their masses in the ratio of 2:1. After disintegration they will move
Top JEE Main Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.