Gas Law and Kinetic Theory
Pressure is defined as perpendicular force per unit area. Mathematically P=F/A. It’s a
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scalar, and has unit Nm or Pascal (Pa), others including atm, mmHg or psi.
The pressure due to gas in a container is unique, and gas under higher pressure is pushed
into region of lower pressure (since F=A*ΔP). Pressure can be measured by a Bourdon gauge.
Atmospheric pressure is the pressure due to the atmospheric environment (gas). It’s about
101kPa (1atm)
Gas law – concerning P (pressure), V (volume), T (temperature), n (mass/amount of gas)
Note that T concerned is the absolute temperature, which implies the Kelvin scale. If a
variable is directly proportional to absolute temperature, then it varies linearly with the
temperature in Celsius scale.
1) Boyle’s law: P inversely proportional to V under fixed n, T. Mathematically P1V1=P2V2.
2) Pressure law: P directly proportional to T under fixed n, V, Mathematically P1/T1=P2/T2.
3) Charles’ law: V directly proportional to T under fixed n, P, Mathematically V1/T1=V2/T2.
4) General gas law : P1V1/T1=P2V2/T2. The gas is an ideal gas if it obeys this law for all
temperature. Under room conditions, O, He, Ne behaves almost like an ideal gas. Gas
behaves like ideal under very high T and very low P.
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5) Ideal gas law: PV=nRT, where n is number of moles of the gas, and R=8.31Jmol K
which is known as universal gas constant. There are different forms related to this law,
such as PV=NkT, where N is number of molecules and k=R/NA, where NA is the
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Avogadro’s number, 6.02*10 mol , and k=1.38*10 JK .
Kinetic theory
Properties related to kinetic theory:
- Temperature of gas is the average KE due to random motion of the gas molecules.
- Volume of gas spreads freely throughout the container.
- Gas pressure is caused by collision of gas on the walls of container.
- Particles moves in zigzag paths (Brownian motion), since they always collide and
change direction.
Kinetic theory model
- Boyle’s law: weight added to piston (P↑), T unchanged, speed of molecules unchanged, it
collides with the wall more frequently, exerting larger force, balancing weight of piston.
- Pressure law: Voltage supply increase (T↑), speed of molecules↑, it hits the walls more
vigorously and frequently, exerting a larger force on the piston. In order to maintain the
same volume (height of piston), a weight is added to balance the force exerted by the gas.
- Charles’ law: Voltage supply increase (T↑), speed of molecules↑, it hits the walls more
vigorously and frequently, exerting a larger force on the piston. P unchanged, therefore the
weight of piston unchanged, then the piston moves upward to reduce frequency of
collision, and reduce force exerted by the gas and balance the force.
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
3
(0.0224m ) 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.
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Derivation of PV=Nmc /3(Note: c 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, vy, vz 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 = -2mvx.
Time between successive collisions on the wall is given by Δt = 2L/vx.
2
Force exerted on the gas molecule Fg = Δp/Δt = -mvx /L, then force exerted on the wall = Fw =
2 2 2 2
-Fg = mvx /L. Total force exerted on the wall = ΣFi = m(Σvix )/L = Nmvx /L (vx is the sum of
square root of the components of velocity of the molecules)
2 2 2 3 2
Px = F/A = (Nmvx /L)/L = Nmvx /L = Nmvx /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:
2 2 2 2 2 2
P = (Px+Py+Pz)/3 = (Nmvx /V+ Nmvy /V+ Nmvz /V)/3 = Nm(vx + vy + vz )/3V
When the three components add up it’s the original velocity. Therefore arranging the terms
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gives PV=Nmc , considering density ρ=Nm/V, P=ρc /3.
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Taking square root on c gives the root-mean-square speed of the gas molecules, written as
crms. It’s inversely proportional to square root of molar mass of the gas.
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Considering the average transitional kinetic energy, KEavg=mc /2, putting into PV=Nmc /3,
2
PV=NkT= Nmc /3 = N(KEavg)(2/3), therefore KEavg = 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) crms, 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 KEavg = 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.