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 (0.0224m

^{3}) 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=Nmc**(Note:^{2}/3**c**, and should be c-bar formally)^{2}is the sum of square root of the speed of moleculesIn 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**, then force exerted on the wall = F_{g}= Δp/Δt = -mv_{x}^{2}/L_{w}= -F_{g}= mv_{x}^{2}/L. Total force exerted on the wall =**ΣF**(v_{i}= m(Σv_{ix}^{2})/L = Nmv_{x}^{2}/L_{x}^{2}is the sum of square root of the components of velocity of the molecules)**P**

_{x}= F/A = (Nmv_{x}^{2}/L)/L^{2}= Nmv_{x}^{2}/L^{3}= Nmv_{x}^{2}/VWhen 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_{x}^{2}/V+ Nmv_{y}^{2}/V+ Nmv_{z}^{2}/V)/3 = Nm(v_{x}^{2}+ v_{y}^{2}+ v_{z}^{2})/3VWhen the three components add up it’s the original velocity. Therefore arranging the terms gives

**PV=Nmc**, considering density ρ=Nm/V, P=ρc^{2}/3^{2}/3.Taking square root on c

^{2}gives 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}=mc^{2}/2, putting into PV=Nmc^{2}/3,PV=NkT= Nmc

^{2}/3 = N(KE_{avg})(2/3), therefore**KE**= mean ε_{avg}**= 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**, and the higher the temperature, the more disperse of the distribution. (Highest prob. decrease with temperature)_{rms}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

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