For a nucleus of mass number $ A $ and radius $ R $, the mass density of the nucleus can be represented as:
Show Hint
- \( \frac{2}{3} A \)
- \( \frac{1}{3} A \)
- \( A^3 \)
- Independent of \( A \)
The Correct Option is D
Approach Solution - 1
To solve this question, we need to understand how the mass density of a nucleus relates to its mass number \( A \) and its radius \( R \).
The mass density \( \rho \) of a nucleus is given by the formula:
\(\rho = \frac{\text{mass of nucleus}}{\text{volume of nucleus}}\)
We know that the mass of a nucleus is approximately proportional to its mass number \( A \), because the mass number represents the sum of protons and neutrons, which have nearly equal mass.
The volume \( V \) of a spherical nucleus is calculated using the formula:
\(V = \frac{4}{3} \pi R^3\)
For a nucleus, it is empirically found that the radius \( R \) is related to the mass number \( A \) by the equation:
\(R = R_0 A^{1/3}\)
where \( R_0 \) is a constant. Thus, the volume \( V \) can be expressed as:
\(V = \frac{4}{3} \pi (R_0 A^{1/3})^3 = \frac{4}{3} \pi R_0^3 A\)
Substituting this into the formula for density, we get:
\(\rho = \frac{A}{\frac{4}{3} \pi R_0^3 A}\)
Upon simplification, it becomes:
\(\rho = \frac{1}{\frac{4}{3} \pi R_0^3}\)
From this expression, we can see that the density \( \rho \) is independent of \( A \) since the \( A \) terms cancel each other out.
Therefore, the correct answer is:
Independent of \( A \)
Approach Solution -2
For a nucleus with mass number \( A \) and radius \( R \), we can express the mass and volume as follows: - The mass of the nucleus is proportional to the mass number \( A \), i.e., the mass \( M \) is proportional to \( A \), - The volume \( V \) of the nucleus is proportional to \( R^3 \), where \( R \) is the radius of the nucleus.
The volume is given by the formula for the volume of a sphere: \[ V = \frac{4}{3} \pi R^3 \] Since the mass number \( A \) is proportional to the volume, the mass density is given by: \[ \rho = \frac{\text{mass}}{\text{volume}} = \frac{A}{\frac{4}{3} \pi R^3} \] We know from the liquid drop model of the nucleus that the radius \( R \) is proportional to \( A^{1/3} \).
Thus, we have: \[ R \propto A^{1/3} \] Substituting this into the equation for density: \[ \rho \propto \frac{A}{R^3} = \frac{A}{A} = 1 \] Therefore, the mass density \( \rho \) is independent of \( A \). Thus, the correct answer is Option (4), which states that the mass density is independent of \( A \).
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 Nuclear physics Questions
- The atomic mass of $_{6}^{12}C$ is 12.000000 u and that of $_{6}^{13}C$ is 13.003354 u. The required energy to remove a neutron from $_{6}^{13}C$, if mass of neutron is 1.008665 u, will be :
- The explosive in a hydrogen bomb is a mixture of \( ^2\text{H} \), \( ^3\text{H} \), and \( ^6\text{Li} \) in some condensed form. The chain reaction is given by\[_3^6\text{Li} + _0^1\text{n} \rightarrow _2^4\text{He} + _1^3\text{H}\]\[_1^2\text{H} + _1^3\text{H} \rightarrow _2^4\text{He} + _0^1\text{n}\]During the explosion, the energy released is approximately ____.[Given: \( M(\text{Li}) = 6.01690 \, \text{amu} \), \( M(^3\text{H}) = 2.01471 \, \text{amu} \), \( M(_2^4\text{He}) = 4.00388 \, \text{amu} \), and \( 1 \, \text{amu} = 931.5 \, \text{MeV} \)]
In a nuclear fission process, a high mass nuclide (A ≈ 236) with binding energy 7.6 MeV/Nucleon dissociated into middle mass nuclides (A ≈ 118), having binding energy of 8.6 MeV/Nucleon. The energy released in the process would be ____ MeV.
- The radius of a nucleus of mass number 64 is 4.8 fermi. Then the mass number of another nucleus having radius of 4 fermi is \(\frac{1000}{x}\) , where x is _____.
- If the mass defect in a nuclear reaction is \(0.4\) \(gm\) then find the \(Q\) value of the reaction.
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 ______.