Ideal Gas Law: Mathematical Example

The ideal gas law connects the pressure, volume, and temperature:

(1) for a fixed volume, a higher temperature leads to an increase in pressure, P ≈ T,

(2) at constant pressure, a higher temperature leads to an increase volume, V ≈ T, and

(3) at constant temperature, a higher pressure is associated with a smaller volume, P ≈ 1/V. This can be summarized as

PV / T = const,

where T is the absolute temperature [measured in Kelvin]. This law assume that the amount of gas [i.e., the mass] is kept constant.

Example: A glass column is filled with air. At room temperature, the pressue is 2.5-times the atmospheric pressure. What is the pressure inside the glass column when it immersed in boiling water?

Solution:

We have the absolute temperatures T₀ = 298 K and T_f = 373 K.

Since the volume is constant V₀ = V₁ = V, P₀ V / T₀ = P₁ V / T, so that,

P₁ = (T₁/T₀) P₀ = (373K/298K) 2.5atm = 3.1 atm

In an ideal gas, molecules are moving in random directions aat random speeds inside a container. The molecules bounce off the walls. During a collision, the wall of the container exerts a force on a molecule; thus, the molecule exerts a (reaction) force on the container wall: the summation of all forces from all collisions produces the macroscopic pressure.

A simplified derivation starts from a cubic container with volume V = L³ ; we we choose a coordinate system aligned with the cube. We assume that the molecule travels with speed v along the x -coordinate. The collision with the wall gives the change in momentum Δp = (-mv_x) – m(v_x) = – 2mv_x, . Since the time between consecutive collisons is t = 2L/v , the (average) force by one molecule follows F_aye = ( -2mv_x)/ (2 L/v_x) = – mv_x²/L . The particle moves in all directions so that v_x² = (v²) /3, where we introduced the mean square value (v²) . Thus for the force due to N molecules F = Nm (v²) /3L so that for the pressure P = F/L² = Nm (v²) /3V.

We find,

where we introduced the average kinetic energy of the molecule (KE) = m (v²) /2. Comparison with ideal gas law PV/T = const yields

(KE) = (m/2) V²_rms = 3/2 kT,

where k = 1.381 x 10^-23 J/K is the Boltzmann constant.

The number of molecules in a gas is enormous N > > 1. Avogardo’s number is used to quantify large numbers,

N_A = 6.022 x 10²³

The number of moles is then given by n = N/ N_A

The product of the molecular mass and Avogadro’s number is the molar mass M = N_A m. For helium M_he ≈ 4 g; and air [80% N² and 20% O₂]. M_air = 28.8 g. Avogadro’s number and the Boltzmann constant are the connection between microscopic and macroscopic quantities. The unit for masses of molecules is the atomic mass unit:

1 u = (1 g / N_A) = 1.66 x 10^-27 kg. N_A