In the IB Physics syllabus, the topic of Discrete Energy and Radioactivity covers several fundamental concepts in atomic and nuclear physics. This study note will break down the complex ideas into smaller, digestible sections for better understanding. We will cover discrete energy levels, photon energy calculations, emission and absorption spectra, isotopes and radioactive decay, background radiation, alpha, beta, and gamma particles, decay equations, half-life, and methods to investigate half-life.
Atoms consist of electrons orbiting a nucleus. These electrons can only occupy specific energy levels. These energy levels are quantized, meaning electrons can only have certain discrete energies.
Note:
Electrons can gain energy through collisions with other particles, absorbing photons, or from thermal energy.
Photons are packets of electromagnetic energy. They are massless and carry energy in discrete amounts. The energy of a photon can be calculated using the formula:
$$ E = hf $$
Where:
Using the wave equation, the energy can also be expressed in terms of wavelength ($\lambda$):
$$ E = \frac{hc}{\lambda} $$
Where:
Tip:
The higher the frequency of the electromagnetic radiation, the higher the energy of the photon.
When an electron transitions between energy levels, it emits or absorbs a photon with energy equal to the difference between the two levels:
$$ \Delta E = E_2 - E_1 = hf $$
Where:
Example:
If an electron in a hydrogen atom transitions from $n=3$ to $n=2$, the energy difference can be calculated using the Rydberg formula: $$ \Delta E = 13.6 \left( \frac{1}{2^2} - \frac{1}{3^2} \right) \text{eV} = 1.89 \text{eV} $$
To calculate the energy of photons emitted or absorbed during electron transitions, use the relationship:
$$ \Delta E = hf = \frac{hc}{\lambda} $$
Example:
Calculate the wavelength of a photon emitted when an electron falls from $E_2 = -3.4$ eV to $E_1 = -13.6$ eV.
$$ \Delta E = -3.4 - (-13.6) = 10.2 \text{ eV} $$
Convert eV to Joules: $$ 10.2 \text{ eV} \times 1.602 \times 10^{-19} \text{ J/eV} = 1.63 \times 10^{-18} \text{ J} $$
Using $E = \frac{hc}{\lambda}$: $$ \lambda = \frac{hc}{E} = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{1.63 \times 10^{-18}} = 1.22 \times 10^{-7} \text{ m} = 122 \text{ nm} $$
When electrons drop to lower energy levels, they emit photons of specific wavelengths, creating an emission spectrum. Each element has a unique emission spectrum that can be used to identify it.
Emission spectrum of hydrogen showing discrete wavelengths.
Conversely, when electrons absorb photons and move to higher energy levels, certain wavelengths are absorbed from a continuous spectrum, creating an absorption spectrum.
Absorption spectrum showing dark lines where wavelengths are absorbed.
Note:
Both emission and absorption spectra provide evidence for quantized energy levels in atoms.
Isotopes are atoms of the same element with different numbers of neutrons. For example, Carbon-12 and Carbon-14 are isotopes of Carbon.
Unstable isotopes undergo radioactive decay to become more stable. This process involves the emission of alpha, beta, or gamma particles.
Common Mistake:
Confusing alpha particles with helium atoms. Alpha particles are helium nuclei without electrons.
Background radiation is the low-level radiation present in the environment. It comes from various sources:
Note:
Background radiation is always present and varies by location.
Alpha particles are helium nuclei consisting of 2 protons and 2 neutrons. They have a high mass and charge, resulting in low penetration and high ionization.
Beta particles are high-energy electrons (beta-minus) or positrons (beta-plus). They have moderate penetration and ionization.
Gamma particles are high-energy photons with no mass or charge. They have high penetration and low ionization.
Example:
A nucleus of iron-59 decays into cobalt-59 by beta emission followed by gamma emission. The total energy released is $2.52 \times 10^{-13}$ J.
Steps:
Decay equations represent the transformation of one element into another during radioactive decay. For example, beta-minus decay:
$$ {}_6^{14}\text{C} \rightarrow {}_7^{14}\text{N} + e^- + \bar{\nu}_e $$
Where:
Half-life is the time required for half of the radioactive nuclei in a sample to decay. It is a measure of the stability of a radioactive isotope.
Example:
If a sample of Carbon-14 has a half-life of 5730 years, in 5730 years, half of the Carbon-14 atoms will have decayed to Nitrogen-14.
To investigate half-life, measure the activity of a radioactive sample over time and plot the decay curve. The time taken for the activity to reduce by half is the half-life.
Note:
Ensure proper safety measures when handling radioactive materials.
Understanding discrete energy levels and radioactivity is crucial in the study of atomic and nuclear physics. These concepts explain the behavior of electrons in atoms, the nature of photons, and the processes of radioactive decay. By mastering these topics, students will gain a deeper insight into the fundamental principles governing the physical world.