CHAPTER 14 -- STATISTICAL & THERMAL PHYSICS
Statistical physics bridges the gap between the microscopic behavior of particles and the macroscopic properties of materials (thermodynamics).
14.1 THERMAL MOTION
At temperatures above absolute zero, atoms and molecules are in constant random motion.
- Translational Kinetic Energy: For an ideal gas, the average kinetic energy per molecule is:
K_avg = (3/2) * k_B * T
where k_B is the Boltzmann constant (1.38 x 10^-23 J/K) and T is temperature in Kelvin.
- Temperature is essentially a measure of this random kinetic energy.
14.2 BOLTZMANN DISTRIBUTIONS
For a system in thermal equilibrium at temperature T, the probability P_i of finding the system in a state with energy E_i is given by the Boltzmann distribution:
P_i is proportional to e^(-E_i / k_B * T)
or
N_i / N_total = (1/Z) * e^(-E_i / k_B * T)
where Z is the partition function (sum over all states).
This implies that states with lower energy are more populated than states with higher energy.
14.3 RELAXATION TIMES
When a system is perturbed from equilibrium, it takes time to return to the thermal distribution.
- The "relaxation time" (tau) characterizes this rate.
- Rate of change is often proportional to the deviation from equilibrium.
dX/dt = -(X - X_eq) / tau
- This concept is central to NMR (T1 relaxation) and dielectric relaxation.
14.4 POPULATION DIFFERENCES
In many two-level systems (like spin states in NMR), the signal strength depends on the population difference between the lower (N+) and upper (N-) levels.
Ratio: N- / N+ = e^(-Delta_E / k_B * T)
Since Delta_E is often small (for NMR), the ratio is very close to 1. The "excess" population in the lower state is what generates the net magnetization.
- At lower temperatures, the population difference increases, leading to stronger signals.
- At high temperatures, the populations equalize (saturation).
