Atoms & Nuclei

ATOMIC STRUCTURE

1. Rutherford’s Alpha Scattering Experiment

In this experiment, fast moving alpha particles were fired at a thin gold foil. The way these particles scattered helped scientists understand the internal structure of the atom for the first time.

Most alpha particles passed straight through the foil without any deviation. This indicate that most of the atom is empty space. Some particles were slightly deflected, proving that a positive charge exists inside the atom. A very few particles bounced back, indicating a very small, dense, and positively charged center called the nucleus.

This experiment completely rejected Thomson’s model of the atom. It established that the nucleus contains almost all the mass of the atom. The electrons are present outside the nucleus and occupy a large empty region. This model became the foundation of modern atomic physics.

Distance of Closest Approach

The distance of closest approach is the minimum distance an alpha particle can reach before being completely stopped by the electrostatic repulsion of the nucleus. At this point, the kinetic energy of the alpha particle becomes zero temporarily.

After stopping, the particle reverses its direction due to strong repulsive force from the nucleus. This concept helps in estimating the size of the nucleus, which is extremely small compared to the size of the atom.

\[ \frac{1}{2}mv^2 = \frac{1}{4\pi\epsilon_0} \frac{(Ze)(2e)}{r_0} \]

This equation represents energy conservation. The kinetic energy of the alpha particle is completely converted into electrostatic potential energy at the closest distance. It clearly shows how strong nuclear repulsion is at very small distances.

Impact Parameter

The impact parameter is the perpendicular distance between the initial path of the alpha particle and the center of the nucleus. It plays a key role in determining how much deflection occurs during scattering.

If the impact parameter is large, the particle passes far from the nucleus and experiences very weak force. If it is small, the particle comes closer and undergoes strong deflection. For zero impact parameter, the particle directly hits the nucleus and rebounds backward.

2. Bohr’s Model of Hydrogen Atom

Bohr introduced his model to solve the failure of Rutherford’s model. According to classical physics, electrons should lose energy continuously and fall into the nucleus. However, atoms are stable, so Bohr introduced quantization rules.

Bohr proposed that electrons can only move in specific fixed orbits around the nucleus. In these orbits, electrons do not emit energy. This explained why atoms do not collapse and remain stable for a long time.

Bohr’s Postulates

Stable Orbits : Electrons don’t lose energy while staying in fixed paths around the nucleus.

Electrons move in fixed orbits without radiating energy. Each orbit corresponds to a definite energy level. This is why energy inside atom is quantized instead of continuous.

Quantization of Angular Momentum

\[ L = mvr = \frac{nh}{2\pi} \]

This means angular momentum of an electron is restricted to fixed values. Only specific orbits are allowed where this condition is satisfied. This idea introduced quantum theory into atomic structure.

Energy Transition

\[ h\nu = E_i - E_f \]

When an electron jumps from a higher energy level to a lower one, energy is emitted in the form of light. When it absorbs energy, it moves to a higher level. This explains line spectra of hydrogen.

Important Results

Radius of Orbit

\[ r_n \propto \frac{n^2}{Z} \]

The radius increases with the square of principal quantum number. Higher orbits are larger and less tightly bound. As nuclear charge increases, electrons are pulled closer due to stronger attraction.

Energy of Electron

\[ E_n = -\frac{13.6 Z^2}{n^2} \, eV \]

Energy is negative because the electron is bound to the nucleus. More negative value means more stable system. Ground state (n=1) is the most stable configuration of hydrogen atom.

3. Atomic Spectra

Atomic spectra are produced when electrons jump between energy levels and emit photons of specific energies. Each element produces a unique spectrum, acting like its fingerprint.

\[ \frac{1}{\lambda} = RZ^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]

This equation gives wavelength of emitted radiation. It shows that spectral lines depend only on energy level difference. This confirms quantized nature of energy inside atoms.

Spectral Series

Lyman series corresponds to transitions ending at n=1 and lies in ultraviolet region. Balmer series corresponds to transitions ending at n=2 and lies in visible region, which is important for spectroscopy. Paschen series corresponds to infrared region and is not visible to human eye.

NUCLEI

1. Nuclear Size and Density

The nucleus is a very small but extremely dense central region of an atom containing protons and neutrons. Despite its small size, it contains almost all the mass of the atom.

\[ R = R_0 A^{1/3} \]

This shows that nuclear radius increases slowly with mass number. Even if the nucleus becomes heavier, its size increases only slightly because of the cube root relation.

\[ \rho \approx 2.3 \times 10^{17} \, kg/m^3 \]

This extremely high density means nuclear matter is unimaginably compact. A small amount of nuclear material contains enormous mass compared to normal matter.

2. Mass Defect and Binding Energy

When protons and neutrons combine to form a nucleus, the total mass decreases slightly. This missing mass is called mass defect and is converted into binding energy.

\[ \Delta m = [Z m_p + (A-Z)m_n] - M \]

Binding energy represents the energy required to break a nucleus into individual nucleons. Higher binding energy means more stable nucleus. This is why iron nucleus is extremely stable.

\[ E_b = \Delta m c^2 \]

3. Radioactivity

Radioactivity is a natural process in which unstable nuclei decay spontaneously to achieve stability. During decay, particles or electromagnetic radiation is emitted.

\[ N = N_0 e^{-\lambda t} \]

This equation shows exponential decay law. It means number of undecayed nuclei decreases continuously with time, never becoming exactly zero.

\[ T_{1/2} = \frac{0.693}{\lambda} \]

Half-life is the time required for half of radioactive nuclei to decay. It is independent of temperature, pressure, or chemical state.

\[ A = \lambda N \]

Activity represents the rate of decay. Higher activity means faster disintegration of radioactive material.

4. Nuclear Forces

Nuclear forces are the strongest forces in nature that bind protons and neutrons inside the nucleus. Without them, nucleus would not exist due to repulsion between protons.

These forces act only over very short distances, typically inside the nucleus. They are independent of charge and show saturation effect, meaning each nucleon interacts only with nearby nucleons.

5. Fission and Fusion

Nuclear fission is the process in which a heavy nucleus splits into two lighter nuclei releasing large amount of energy. This process is used in nuclear reactors.

Nuclear fusion is the process in which two light nuclei combine to form a heavier nucleus. This process releases even more energy and powers stars like the Sun.

REVISION POINTS

\(E = mc^2\) → mass-energy equivalence
\(L = \frac{nh}{2\pi}\) → quantized angular momentum
\(T_{1/2} = \frac{0.693}{\lambda}\) → half-life
\(A = \lambda N\) → activity
\(2\pi r = n\lambda\) → wave nature of electron

SOME IMPORTANT NUMERICALS

Question No. 1:
An alpha particle of kinetic energy 5 MeV approaches a gold nucleus (Z = 79). Find distance of closest approach.

Answer & Explanation No. 1:
At closest approach, kinetic energy converts completely into electrostatic potential energy.

\[ \frac{1}{2}mv^2 = \frac{1}{4\pi\epsilon_0} \frac{(Ze)(2e)}{r} \]
This is based on **energy conservation principle** in Rutherford scattering.

\[ r = \frac{2Ze^2}{4\pi\epsilon_0 \cdot KE} \]
Substituting values: \[ r \approx \frac{2 \cdot 79 \cdot 1.44}{5} = 45.5 \, fm \]

Question No. 2:
Find the ratio of radii of 2nd orbit and 1st orbit of hydrogen atom.

Answer & Explanation No. 2:
Bohr’s model states: \[ r_n \propto n^2 \]
Radius increases with square of principal quantum number.

\[ \frac{r_2}{r_1} = \frac{2^2}{1^2} = 4 \]

Question No. 3:
Find energy of electron in n = 2 orbit of hydrogen atom.

Answer & Explanation No. 3:
Bohr energy formula: \[ E_n = -\frac{13.6}{n^2} \, eV \]
Negative sign shows electron is **bound to nucleus**.

\[ E_2 = -\frac{13.6}{4} = -3.4 \, eV \]

Question No. 4:
Electron falls from n = 3 to n = 2. Find energy of emitted photon.

Answer & Explanation No. 4:
Energy difference gives photon energy: \[ E = 13.6 \left(\frac{1}{2^2} - \frac{1}{3^2}\right) \]
Emission occurs when electron drops to lower level.

\[ E = 13.6 \cdot \frac{5}{36} = 1.89 \, eV \]

Question No. 5:
Find wavelength for transition n = 3 to n = 2 (R = \(1.1 \times 10^7\)).

Answer & Explanation No. 5:
Rydberg formula: \[ \frac{1}{\lambda} = R\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \]
This explains **hydrogen spectral lines**.

\[ \lambda \approx 6.55 \times 10^{-7} m \]

Question No. 6:
Find radius of nucleus with A = 64 (R₀ = 1.2 fm).

Answer & Explanation No. 6:
Nuclear radius: \[ R = R_0 A^{1/3} \]
Shows nucleus grows slowly with mass number.

\[ R = 1.2 \times 4 = 4.8 \, fm \]

Question No. 7:
Find mass defect if binding energy is 28 MeV.

Answer & Explanation No. 7:
Relation: \[ E = \Delta m c^2 \]
Mass defect converts into binding energy.

\[ \Delta m = \frac{28}{931} = 0.030 \, u \]

Question No. 8:
If decay constant is 0.0231 s⁻¹, find half-life.

Answer & Explanation No. 8:
Radioactive decay is exponential: \[ T_{1/2} = \frac{0.693}{\lambda} \]
Independent of temperature and pressure.

\[ T_{1/2} = 30 \, s \]

Question No. 9:
Initial nuclei = 1000, λ = 0.1, time = 10 s. Find remaining nuclei.

Answer & Explanation No. 9:
Decay law: \[ N = N_0 e^{-\lambda t} \]
Represents continuous decrease of nuclei.

\[ N = 1000 e^{-1} \approx 368 \]

Question No. 10:
Find activity if N = \(5 \times 10^6\), λ = \(2 \times 10^{-3}\).

Answer & Explanation No. 10:
Activity: \[ A = \lambda N \]
Shows rate of nuclear decay.

\[ A = 10^4 \, Bq \]