Physics : Gas Law II

Avogadro’s law hypothesized that for any gas molecules, under same n, P, T, occupies the same V. Under s.t.p. (standard T = 273.15K and P = 1atm), 1 mole of gas occupies 22.4L (0.0224m³) of the spaces.

Ideal gas -- Macroscopic definition

Boyle’s law exactly true for all temperature and pressure, and PV is proportional to T.

Quantitative definition

-          Molecules collide perfectly elastically with the walls of container.

-          Time during collisions is negligible while comparing with time between collisions.

-          Size of molecules is negligible; and no intermolecular force among gas molecules exist.

Derivation of PV=Nmc²/3(Note: c² is the sum of square root of the speed of molecules, and should be c-bar formally)

In a cubic container with side length L and N gas molecules inside, each of mass m, moving randomly in the container, with velocity c, with components v_x, v_y, v_z in the three directions.

Consider one of the molecules in direction of x. Since it collides elastically, speed before and after collision unchanged, but with opposite direction. Hence Δp = -2mv_x.

Time between successive collisions on the wall is given by Δt = 2L/v_x.

Force exerted on the gas molecule F_g = Δp/Δt = -mv_x²/L, then force exerted on the wall = F_w = -F_g = mv_x²/L. Total force exerted on the wall = ΣF_i = m(Σv_ix²)/L = Nmv_x²/L (v_x² is the sum of square root of the components of velocity of the molecules)

P_x = F/A = (Nmv_x²/L)/L² = Nmv_x²/L³ = Nmv_x²/V

When considering pressure on all direction, we know that pressure are the same for all direction, take average on force exerted on each wall:

P = (P_x+P_y+P_z)/3 = (Nmv_x²/V+ Nmv_y²/V+ Nmv_z²/V)/3 = Nm(v_x²+ v_y²+ v_z²)/3V

When the three components add up it’s the original velocity. Therefore arranging the terms gives PV=Nmc²/3, considering density ρ=Nm/V, P=ρc²/3.

Taking square root on c²gives the root-mean-square speed of the gas molecules, written as c_rms. It’s inversely proportional to square root of molar mass of the gas.

Considering the average transitional kinetic energy, KE_avg=mc²/2, putting into PV=Nmc²/3,

PV=NkT= Nmc²/3 = N(KE_avg)(2/3), therefore KE_avg = mean ε = 2/3(kT).

For the internal energy of the whole system, ε = N(2kT/3) = PV(2/3).

The distribution of molecular speeds refers to the Maxwell-Boltzmann distribution, where the highest probability exist in √(2/3) c_rms, and the higher the temperature, the more disperse of the distribution. (Highest prob. decrease with temperature)

Note that real gas molecules can have other forms of energies such as rotational KE, vibrational energy and PE due to intermolecular attraction.

By KE_avg = 2/3(kT), the molecular KE of the ideal gas become zero at 0K, which called the absolute zero where it’s the lowest possible temperature reached by any matter. However in real gas the molecular KE has a minimum value, which is the zero-point energy.

This is really hardcore orz