$\left[ \frac{1}{4\pi\epsilon_0} = 9 \times 10^9 \text{ Nm}^2/\text{C}^2 \text{ and electron charge} = 1.6 \times 10^{-19} \text{ C} \right]$
Nuclear Physics: Alpha Scattering Diameter Estimation
This question involves estimating the size of an atomic nucleus using the concept of the distance of closest approach derived from Rutherford's alpha scattering experiment.
Physics Principles
When an alpha particle ($\alpha$-particle) approaches a nucleus, it experiences electrostatic repulsion (Coulomb force). The distance of closest approach ($r_0$) is the minimum distance the $\alpha$-particle gets to the center of the nucleus before its kinetic energy is momentarily zero and it reverses direction. This distance provides an estimate for the nuclear radius.
The formula for the distance of closest approach ($r_0$) for a head-on collision is:
$r_0 = \frac{Zze^2}{4\pi\epsilon_0 E}$Where:
- $Z$ is the atomic number of the target nucleus.
- $z$ is the atomic number (charge number) of the incident particle (for $\alpha$-particle, $z=2$).
- $e$ is the elementary charge.
- $E$ is the initial kinetic energy of the incident particle.
- $\frac{1}{4\pi\epsilon_0}$ is Coulomb's constant, denoted as $k$.
The question asks for the estimated *diameter*, which is twice the radius (or distance of closest approach).
Calculation Steps
Given Data:
- Energy of $\alpha$-particle, $E = 7.9 \text{ MeV}$
- Atomic number of target, $Z = 79$
- Atomic number of $\alpha$-particle, $z = 2$
- Coulomb's constant, $k = \frac{1}{4\pi\epsilon_0} = 9 \times 10^9 \text{ Nm}^2/\text{C}^2$
- Electron charge, $e = 1.6 \times 10^{-19} \text{ C}$
Step 1: Convert Energy to Joules
First, convert the kinetic energy from Mega-electron Volts (MeV) to Joules (J):
$E = 7.9 \text{ MeV} = 7.9 \times 10^6 \text{ eV}$ $E = 7.9 \times 10^6 \times (1.6 \times 10^{-19} \text{ J}) = 12.64 \times 10^{-13} \text{ J}$Step 2: Calculate Distance of Closest Approach ($r_0$)
Substitute the values into the formula:
$r_0 = \frac{k Z z e^2}{E}$ $r_0 = \frac{(9 \times 10^9 \text{ Nm}^2/\text{C}^2) \times 79 \times 2 \times (1.6 \times 10^{-19} \text{ C})^2}{12.64 \times 10^{-13} \text{ J}}$ $r_0 = \frac{9 \times 10^9 \times 79 \times 2 \times (2.56 \times 10^{-38})}{12.64 \times 10^{-13}}$ $r_0 = \frac{3639.12 \times 10^{-29}}{12.64 \times 10^{-13}}$ $r_0 \approx 287.9 \times 10^{-16} \text{ m}$ $r_0 \approx 2.88 \times 10^{-14} \text{ m}$This value ($r_0$) represents the estimated radius of the nucleus.
Step 3: Calculate Nuclear Diameter
The diameter ($d$) is twice the radius:
$d = 2 \times r_0$ $d = 2 \times (2.88 \times 10^{-14} \text{ m})$ $d = 5.76 \times 10^{-14} \text{ m}$Conclusion
The estimated diameter of the nuclei of the target material is approximately $5.76 \times 10^{-14}$ m.
Similar Questions
Assuming in forward bias condition there is a voltage drop of $0.7 \text{ V}$ across a silicon diode, the current through diode $D_1$ in the circuit is ________$\text{mA}$.
(Assume all diodes in the given circuit are identical)
- The ratio of de Broglie wavelength of a deutron with kinetic energy $E$ to that of an alpha particle with kinetic energy $2E$, is $n : 1$. The value of $n$ is ________.
(Assume mass of proton = mass of neutron) : - The wave numbers of three spectral lines of H atom are considered. Identify the set of spectral lines belonging to Balmer series.
(R = Rydberg constant) Find the correct combination of A, B, C and D inputs which can cause the LED to glow.
- The minimum frequency of photon required to break a particle of mass 15.348 amu into 4 $\alpha$ particles is _________ kHz.
[mass of He nucleus = 4.002 amu, $1 \text{ amu} = 1.66 \times 10^{-27} \text{ kg}, h = 6.6 \times 10^{-34} \text{ J.s} \text{ and } c = 3 \times 10^8 \text{ m/s}$] The correct truth table for the given input data of the following logic gate is :
- Light is incident on a metallic plate having work function $110 \times 10^{-20} \text{ J}$. If the produced photoelectrons have zero kinetic energy then the angular frequency of the incident light is _________ rad/s.
(h = $6.63 \times 10^{-34} \text{ J.s}$). - The smallest wavelength of Lyman series is 91 nm. The difference between the largest wavelengths of Paschen and Balmer series is nearly _________ nm.
- Two electrons are moving in orbits of two hydrogen like atoms with speeds $3 \times 10^5 \text{ m/s}$ and $2.5 \times 10^5 \text{ m/s}$ respectively. If the radii of these orbits are nearly same then the possible order of energy states are ______ respectively.
- Given below are two statements:
Statement I: For all elements, greater the mass of the nucleus, greater is the binding energy per nucleon.
Statement II: For all elements, nuclei with less binding energy per nucleon transforms to nuclei with greater binding energy per nucleon.
In the light of the above statements, choose the correct answer from the options given below
Important Questions from Modern Physics
Assuming in forward bias condition there is a voltage drop of $0.7 \text{ V}$ across a silicon diode, the current through diode $D_1$ in the circuit is ________$\text{mA}$.
(Assume all diodes in the given circuit are identical)- The ratio of de Broglie wavelength of a deutron with kinetic energy $E$ to that of an alpha particle with kinetic energy $2E$, is $n : 1$. The value of $n$ is ________.
(Assume mass of proton = mass of neutron) : - The wave numbers of three spectral lines of H atom are considered. Identify the set of spectral lines belonging to Balmer series.
(R = Rydberg constant) Find the correct combination of A, B, C and D inputs which can cause the LED to glow.
- The minimum frequency of photon required to break a particle of mass 15.348 amu into 4 $\alpha$ particles is _________ kHz.
[mass of He nucleus = 4.002 amu, $1 \text{ amu} = 1.66 \times 10^{-27} \text{ kg}, h = 6.6 \times 10^{-34} \text{ J.s} \text{ and } c = 3 \times 10^8 \text{ m/s}$]
