Statistical Mechanics

Statistical Mechanics is the fifth lecture within the Properties of Matter section of PH1011. It covers the role of statistical mechanics, the Boltzmann and Boltzmann-Maxwell distributions, and the differences between real and ideal gases.

Previous: The Kinetic Theory of Gases II

Next: Thermodynamics and Molar Specific Heats

Boltzmann Distribution
Statistical mechanics are used to determine information about group behaviour of molecules, rather than averages based on indivdual particles. Here this is considered in the context of gases in atmosphere; the density of air molecules varies according to height in the relation N(h) = N0exp(-Mgh/RT). This is the Barometric Formula; N 0 is the number of molecules at sea level. If ε (potential energy) is substituted in (as Ep = mgh), N = N0exp(-ε/kT). This is the Boltzmann distribution, relating the number of molecules with a specific potential energy rather than at a specific height.

Maxwell Distribution
The Maxwell-Boltzmann distribution considers the distribution of atomic speeds in a gas: P(v) = 4π(M/2πRT)3/2v2exp(-(Mv2/2RT). P(v) is the probability distribution function; taking its integral allows calculation of the fraction of molecules within a sample with a speed within the range of v1 to v2. To find the number of molecules at approximately v1, P(v)dv = the fraction of molecules with speed in interval dv around v. P(v) can also be used to calculate the average or root mean square speeds of gas molecules via [v] = ∫0∞vP(v)dv and vrms2 = ∫0∞v2P(v)dv.

Real Gases
Gases act ideally when highly diluted - however at high concentrations the finite volume available causes the interactions and attractions between molecules to have noticable effects. This is due to their sizes and forces being taken into consideration; molecules larger than infinitely small will reduce the overall available volume, causing the Vm of the ideal gas law (PVm = RT) to be Vm' = Vm - b. b is the co-volume; a gas specific constant.

Van der Waals' forces act between the molecules in a gas, attracting them to one another and therefore reducing the pressure of the gas. This causes P' = P + PB, PB being the cohesion pressure due to attraction between molecules. This additional pressure is proportional to the square of particle density N/V. Equating N/V to NA/Vm gives PB = a/Vm2, a being another gas specific constant. Substituting these into the ideal gas equation creates a real gas equation (the Van der Waals equation):

(P + a/Vm2)(Vm - b) = RT

Ideal and real gases behave similarly at high temperatures (see graph) because the molecules are fairly far apart, negating their volumes and attractive forces. However at low temperatures this is untrue; Van der Waals law predicts a phase transition and deviates greatly from the ideal gas law.

Summary
Statistical mechanics allows consideration of molecules in a group rather than an average. The Boltzmann distribution (N = N0exp(-ε/kT)) describes the spread of molecules with a specific energy in comparison to molecules at sea level. The Maxwell distribution gives the probability of finding specific speeds of molecules, and can be integrated to find averages.

Real gases must be handled differently to ideal gases; their size and forces must be taken into account. This gives rise to a different gas law - (P + a/Vm2)(Vm - b) = RT- which explains the behavioural differences.