The Ideal Gas Flashcards
Ideal Gas Law (Both Equations)
Properties of low-density gases.
PV = nRT
R= 8.31 J/molK, n is number of moles of gas
PV = NkT
N = n * Na (number of molecules), k = R/Na
Barometric Equation
Models how the pressure (density) of the air changes with altitude.
dP= mg/A —> pVg/A —> dP/dz = -pg
dP/dz = -mg/kT P —> P = P(o) e^-mg/kT
How can you incorporate density in the ideal gas law equation?
PV = nRT —> PV = m/M RT —> PM = m/V RT —> PM = pRT
Microscopic Model of an Ideal Gas
Average Pressure Exerted on Piston over long time periods:
P = F on piston/A = -F on molecule/A = -m(Δv/Δt) / A
where Δt = 2L/v and Δv = -2v (bounces back)
so P = mv^2/V
What is the relationship between Pressure, Volume, Mass, and Velocity?
How can we rewrite this relationship? What does it give us?
From Microscopic model, we can conclude that
PV = mv^2
(this can be written as a sum for each v present in system)
Using Ideal Gas Law, we can rewrite this as:
kT = mv^2
If we multiple both sides by 1/2 and write as sum, we get the average translational kinetic energy.
K trans = 1/2mv^2 = 1/2kT
(for x, y, and z vector: = 3/2 kT)
Root-Mean Square Speed
Statistical average of the speeds of gas molecules at a given temperature.
v rms = (3kT/m)^1/2
What is a Degree of Freedom?
All forms of energy which formula is a quadratic function of a coordinate or velocity component.
e.g Translational Energy (1/2mv^2), Rotational Energy (1/2 Iw^2), Elastic Potential Energy 1/2kx^2
Equipartition Theorem:
At temperature T, the average energy of any quadratic degree of freedom is 1/2kT.
U thermal = N f 1/2 kT
(Average total thermal energy; NEVER total energy)
The Degrees of Freedom In Different Types of Gas:
Monoatomic Gas: f= 3 (translational motion only counts)
Diatomic Gas: f= 5 (molecule can rotate about two different axis (z & x))
Diatomic Gas molecules can vibrate too but they do not count because they do not contribute to the thermal energy. (frozen out at room temp)
