Thermal Physics and Ideal Gases
Thermal Physics and Ideal Gases
Thermal physics connects the macroscopic properties of gases (pressure, volume, temperature) to the microscopic behaviour of their molecules through the kinetic theory.
Temperature Scales
Absolute zero (0 K = −273.15°C) is the lowest possible temperature, where particles have minimum kinetic energy (zero-point energy in quantum mechanics).
Conversion: T(K) = T(°C) + 273.15
The kelvin is the SI unit of temperature. Gas law equations always require temperature in kelvin.
The Gas Laws
These are empirical laws valid for fixed amounts of gas:
Boyle's Law (constant temperature): pV = constant or p₁V₁ = p₂V₂
Charles's Law (constant pressure): V/T = constant or V₁/T₁ = V₂/T₂
Pressure Law (constant volume): p/T = constant or p₁/T₁ = p₂/T₂
The Ideal Gas Equation
Combining all three gas laws:
pV = nRT (using moles)
or equivalently:
pV = NkT (using number of molecules)
where:
- p = pressure (Pa)
- V = volume (m³)
- n = number of moles (mol)
- R = molar gas constant = 8.314 J mol⁻¹ K⁻¹
- T = absolute temperature (K)
- N = number of molecules
- k = Boltzmann constant = R/N_A = 1.381 × 10⁻²³ J K⁻¹
- N_A = Avogadro constant = 6.022 × 10²³ mol⁻¹
Worked Example
0.050 mol of an ideal gas occupies 1.2 × 10⁻³ m³ at 20°C. Calculate the pressure.
T = 20 + 273 = 293 K
pV = nRT → p = nRT/V = (0.050 × 8.314 × 293) / (1.2 × 10⁻³)
p = 121.8 / 1.2 × 10⁻³ = 1.015 × 10⁵ Pa ≈ 1.0 atm
Assumptions of Kinetic Theory
An ideal gas is a theoretical model based on these assumptions:
1. The gas consists of a very large number of identical molecules in random motion
2. The molecules are point particles — their volume is negligible compared to the container volume
3. Molecules exert no forces on each other except during collisions
4. Collisions between molecules (and with the walls) are perfectly elastic (kinetic energy is conserved)
5. The duration of collisions is negligible compared to the time between collisions
6. Molecules obey Newton's laws of motion
Deriving the Pressure of an Ideal Gas
Consider N molecules, each of mass m, in a cubic box of side length L.
Step 1: A molecule moving with x-component of velocity u hits a wall. Change in momentum = 2mu (elastic collision, velocity reverses).
Step 2: Time between successive hits on the same wall = 2L/u (molecule travels to opposite wall and back).
Step 3: Force from one molecule on the wall: F = rate of change of momentum = 2mu / (2L/u) = mu²/L
Step 4: Total force from N molecules: F_total = Nm⟨u²⟩/L where ⟨u²⟩ is the mean of u².
Step 5: Pressure = F/A = Nm⟨u²⟩/(L × L²) = Nm⟨u²⟩/V
Step 6: Since motion is random and 3D: ⟨c²⟩ = ⟨u²⟩ + ⟨v²⟩ + ⟨w²⟩ = 3⟨u²⟩
Therefore: pV = ⅓Nm⟨c²⟩
where ⟨c²⟩ is the mean square speed.
Kinetic Energy and Temperature
Combining pV = NkT with pV = ⅓Nm⟨c²⟩:
NkT = ⅓Nm⟨c²⟩
½m⟨c²⟩ = 3/2 kT
This fundamental result states that the average translational kinetic energy of a molecule is directly proportional to the absolute temperature:
E_k = 3/2 kT (per molecule)
or E_k = 3/2 RT (per mole)
Root Mean Square Speed
The root mean square (rms) speed is:
c_rms = √⟨c²⟩ = √(3kT/m) = √(3RT/M)
where M is the molar mass in kg mol⁻¹.
Worked Example — rms Speed
Find the rms speed of nitrogen molecules (M = 0.028 kg mol⁻¹) at 300 K.
c_rms = √(3RT/M) = √(3 × 8.314 × 300 / 0.028)
c_rms = √(266,880) = 517 m s⁻¹
Internal Energy of an Ideal Gas
The internal energy of an ideal gas is the sum of the kinetic energies of all its molecules (no potential energy because ideal gas molecules have no intermolecular forces).
U = N × 3/2 kT = 3/2 nRT
For an ideal gas, internal energy depends only on temperature. Changing volume at constant temperature does not change internal energy.
The Boltzmann Distribution
At a given temperature, not all molecules have the same speed. The Maxwell-Boltzmann distribution shows:
- A few molecules have very low or very high speeds
- The distribution peaks at the most probable speed
- The mean speed is slightly higher than the most probable speed
- The rms speed is higher still: c_rms > c_mean > c_most probable
Increasing temperature:
- Shifts the peak to the right (higher speeds)
- Broadens the distribution
- Lowers the peak height (area under curve = total number of molecules, which is constant)
Real Gases
Real gases deviate from ideal behaviour at:
- High pressures (molecules are close together, volume is not negligible, intermolecular forces matter)
- Low temperatures (molecules move slowly, intermolecular forces become significant relative to kinetic energy)
Gases behave most ideally at high temperatures and low pressures.