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Q2.28:
The unit of length convenient on the nuclear scale is a fermi : (1 f = 10^{-15} m). Nuclear sizes obey roughly the following empirical relation :
(r = r_{0} A^{1/3})
where (r) is the radius of the nucleus, (A) its mass number, and (r_{o}) is a constant equal to about, 1.2 f. Show that the rule implies that nuclear mass density is nearly constant for different nuclei. Estimate the mass density of sodium nucleus. Compare it with the average mass density of a sodium atom obtained in Exercise. 2.27.
Solution :
Step 1: Theoretical Derivation of Nuclear Density
To demonstrate that nuclear density is constant, we define the following variables:
- Nuclear radius: $r = r_0 A^{1/3}$
- Mass number: $A$
- Nuclear mass: $M_{nuc} \approx A \cdot m_p$, where $m_p \approx 1.67 \times 10^{-27} \, \text{kg}$ [Assuming mass of nucleon $\approx$ mass of proton].
- Nuclear volume: $V = \frac{4}{3} \pi r^3$
Substituting the expression for $r$ into the volume formula:
$V = \frac{4}{3} \pi (r_0 A^{1/3})^3 = \frac{4}{3} \pi r_0^3 A$ [By substitution of radius empirical relation]Now, define the nuclear density ($\rho_{nuc}$):
$\rho_{nuc} = \frac{\text{Mass}}{\text{Volume}} = \frac{A \cdot m_p}{\frac{4}{3} \pi r_0^3 A} = \frac{3 m_p}{4 \pi r_0^3}$Since $m_p$, $\pi$, and $r_0$ are constants, $\rho_{nuc}$ is independent of the mass number $A$. Thus, nuclear density is constant for all nuclei.
Step 2: Estimation of Sodium Nucleus Density
Given values:
- $r_0 = 1.2 \, \text{f} = 1.2 \times 10^{-15} \, \text{m}$
- $m_p \approx 1.67 \times 10^{-27} \, \text{kg}$
Calculating the density:
$\rho_{nuc} = \frac{3 \times 1.67 \times 10^{-27}}{4 \times 3.14159 \times (1.2 \times 10^{-15})^3}$ $\rho_{nuc} = \frac{5.01 \times 10^{-27}}{4 \times 3.14159 \times 1.728 \times 10^{-45}}$ $\rho_{nuc} = \frac{5.01 \times 10^{-27}}{2.171 \times 10^{-44}} \approx 2.3 \times 10^{17} \, \text{kg/m}^3$Step 3: Comparison with Sodium Atom Density
From Exercise 2.27, the radius of a sodium atom is approximately $R_a \approx 1.9 \times 10^{-10} \, \text{m}$ and the mass is $M_a \approx 3.8 \times 10^{-26} \, \text{kg}$.
Density of sodium atom ($\rho_{atom}$):
$\rho_{atom} = \frac{M_a}{\frac{4}{3} \pi R_a^3} = \frac{3.8 \times 10^{-26}}{\frac{4}{3} \times 3.14159 \times (1.9 \times 10^{-10})^3}$ $\rho_{atom} = \frac{3.8 \times 10^{-26}}{2.87 \times 10^{-27}} \approx 1.3 \times 10^3 \, \text{kg/m}^3$| System | Density ($\text{kg/m}^3$) |
|---|---|
| Sodium Nucleus | $\approx 2.3 \times 10^{17}$ |
| Sodium Atom | $\approx 1.3 \times 10^{3}$ |
Geometric Representation of Relative Scales
Final Solution: The derivation shows that nuclear density is constant because $A$ cancels out in the ratio of mass to volume. The estimated density of the sodium nucleus is $2.3 \times 10^{17} \, \text{kg/m}^3$, which is approximately $10^{14}$ times greater than the density of the sodium atom, illustrating that the mass of an atom is concentrated almost entirely in its nucleus while the volume is dominated by electron orbits.
More Questions from Class 11 Physics Units and Measurements Additional Exercises
- Q2.25: A man walking briskly in rain with speed $v$ must slant his umbrella forward making an angle $\theta$ with the vertical. A student derives the following relation between $\theta$ and $v$ : $\tan \theta = v$ and checks that the relation has a correct limit: as $v \to 0$, $\theta \to 0$, as expected. (We are assuming there is no strong wind and that the rain falls vertically for a stationary man). Do you think this relation can be correct ? If not, guess the correct relation.
- Q2.25: A man walking briskly in rain with speed (v) must slant his umbrella forward making an angle (\theta) with the vertical. A student derives the following relation between (\theta) and (v) : (\tan \theta = v) and checks that the relation has a correct limit: as (v \rightarrow 0), (\theta \rightarrow 0), as expected. (We are assuming there is no strong wind and that the rain falls vertically for a stationary man). Do you think this relation can be correct ? If not, guess the correct relation.
- Q2.26: It is claimed that two cesium clocks, if allowed to run for 100 years, free from any disturbance, may differ by only about 0.02 s. What does this imply for the accuracy of the standard cesium clock in measuring a time-interval of 1 s ?
- Q2.26: It is claimed that two cesium clocks, if allowed to run for 100 years, free from any disturbance, may differ by only about 0.02 s. What does this imply for the accuracy of the standard cesium clock in measuring a time-interval of 1 s ?
- Q2.27: Estimate the average mass density of a sodium atom assuming its size to be about 2.5 Å. (Use the known values of Avogadro’s number and the atomic mass of sodium). Compare it with the mass density of sodium in its crystalline phase : 970 kg m$^{–3}$. Are the two densities of the same order of magnitude ? If so, why ?
- Q2.27: Estimate the average mass density of a sodium atom assuming its size to be about 2.5 (Å). (Use the known values of Avogadro’s number and the atomic mass of sodium). Compare it with the mass density of sodium in its crystalline phase : 970 (kg m^{-3}). Are the two densities of the same order of magnitude ? If so, why ?
- Q2.28: The unit of length convenient on the nuclear scale is a fermi : 1 f = $10^{–15}$ m. Nuclear sizes obey roughly the following empirical relation : $r = r_0 A^{1/3}$ where $r$ is the radius of the nucleus, $A$ its mass number, and $r_o$ is a constant equal to about, 1.2 f. Show that the rule implies that nuclear mass density is nearly constant for different nuclei. Estimate the mass density of sodium nucleus. Compare it with the average mass density of a sodium atom obtained in Exercise. 2.27.
- Q2.29: A LASER is a source of very intense, monochromatic, and unidirectional beam of light. These properties of a laser light can be exploited to measure long distances. The distance of the Moon from the Earth has been already determined very precisely using a laser as a source of light. A laser light beamed at the Moon takes 2.56 s to return after reflection at the Moon’s surface. How much is the radius of the lunar orbit around the Earth ?
- Q2.29: A LASER is a source of very intense, monochromatic, and unidirectional beam of light. These properties of a laser light can be exploited to measure long distances. The distance of the Moon from the Earth has been already determined very precisely using a laser as a source of light. A laser light beamed at the Moon takes 2.56 s to return after reflection at the Moon’s surface. How much is the radius of the lunar orbit around the Earth ?
- Q2.30: A SONAR (sound navigation and ranging) uses ultrasonic waves to detect and locate objects under water. In a submarine equipped with a SONAR, the time delay between generation of a probe wave and the reception of its echo after reflection from an enemy submarine is found to be 77.0 s. What is the distance of the enemy submarine? (Speed of sound in water = 1450 m s$^{–1}$).
- Q2.30: A SONAR (sound navigation and ranging) uses ultrasonic waves to detect and locate objects under water. In a submarine equipped with a SONAR, the time delay between generation of a probe wave and the reception of its echo after reflection from an enemy submarine is found to be 77.0 s. What is the distance of the enemy submarine? (Speed of sound in water = 1450 (m s^{-1})).
- Q2.31: The farthest objects in our Universe discovered by modern astronomers are so distant that light emitted by them takes billions of years to reach the Earth. These objects (known as quasars) have many puzzling features, which have not yet been satisfactorily explained. What is the distance in km of a quasar from which light takes 3.0 billion years to reach us ?
- Q2.31: The farthest objects in our Universe discovered by modern astronomers are so distant that light emitted by them takes billions of years to reach the Earth. These objects (known as quasars) have many puzzling features, which have not yet been satisfactorily explained. What is the distance in km of a quasar from which light takes 3.0 billion years to reach us ?
- Q2.32: It is a well known fact that during a total solar eclipse the disk of the moon almost completely covers the disk of the Sun. From this fact and from the information you can gather from examples 2.3 and 2.4, determine the approximate diameter of the moon.
- Q2.32: It is a well known fact that during a total solar eclipse the disk of the moon almost completely covers the disk of the Sun. From this fact and from the information you can gather from examples 2.3 and 2.4, determine the approximate diameter of the moon.
- Q2.33: A great physicist of this century (P.A.M. Dirac) loved playing with numerical values of Fundamental constants of nature. This led him to an interesting observation. Dirac found that from the basic constants of atomic physics ($c$, $e$, mass of electron, mass of proton) and the gravitational constant $G$, he could arrive at a number with the dimension of time. Further, it was a very large number, its magnitude being close to the present estimate on the age of the universe ($\sim$15 billion years). From the table of fundamental constants in this book, try to see if you too can construct this number (or any other interesting number you can think of ). If its coincidence with the age of the universe were significant, what would this imply for the constancy of fundamental constants?
- Q2.33: A great physicist of this century (P.A.M. Dirac) loved playing with numerical values of Fundamental constants of nature. This led him to an interesting observation. Dirac found that from the basic constants of atomic physics ((c), (e), mass of electron, mass of proton) and the gravitational constant (G), he could arrive at a number with the dimension of time. Further, it was a very large number, its magnitude being close to the present estimate on the age of the universe (~15 billion years). From the table of fundamental constants in this book, try to see if you too can construct this number (or any other interesting number you can think of ). If its coincidence with the age of the universe were significant, what would this imply for the constancy of fundamental constants?
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