Expansion of an Ideal Gas

Uri Lachish, guma science

Contents:
Introduction
Elastic Collisions
The gas pressure
Drag
Expansion
Isothermal expansion
Adiabatic expansion
Expansion of a quantum well

1. Introduction
Gas particles in a box collide with its walls and transfer momentum to them during each collision. The gas pressure is equal to the momentum delivered to a unit area of a wall, during a unit time. Ideal gas particles do not collide with each other but only with the walls. A single particle moves arbitrarily along some direction until it strikes a wall. It then bounces back, changes direction and speed and moves towards another wall. The gas expansion equations are derived directly from the law of conservation of linear momentum and the law of conservation of energy.

2. Elastic collisions
Consider the elastic collision between two particles of mass m and M, moving toward each other along a straight line as shown in figure-1.

Figure-1: Elastic collision between two particles of mass m and M, moving toward each other along a straight line.

The velocities of the particles before the collisions are v and v_M, and after the collision they are v' and v_M'.
The law of conservation of linear momentum states that:

m v + M v_M = m v' + M v_M'       (1)

and the law of conservation of energy states that:

m v²+ M v_M² = m v'² + M v_M'²       (2)

Division of equation (2) by equation (1) yields:

v - v_M = -(v' - v_M')       (3)

That is, the relative velocity between the particles is conserved during the collision, and the direction is reversed.
Rearrangement of equations (1) - (2) gives:

(1 + m/M) v_M' - (1 - m/M) v_M = (m/M) 2v       (4)

(1 + m/M) v' + (1 - m/M) v = 2v_M       (5)

Assume that m is a microscopic particle and M is a macroscopic body. In the limit (m/M) --> 0:

v_M' = v_M       (6)

v' = -(v - 2v_M)       (7)

The direction of the microscopic particle is reversed and its velocity changes by 2v_M, two times the velocity of the macroscopic body. The velocity of the macroscopic body is almost not affected by the collision.

If the velocity of the macroscopic body is extremely slower than that of the microscopic particle, v_M << v, then the kinetic energy, ΔE_k, transferred from body to body will be, by equations (6) - (7):

ΔE_k = (m/2) (v'² - v²) = 2m v_M (v + v_M)       (8)

2m v_M v, the dominant term on the righthand side of equation (8), is a reversible energy transfer between the microscopic and macroscopic bodies. This energy is proportional to the velocity of the macroscopic body and is independent of its mass. In a front collision, when the two bodies move toward each other, the energy will transfer from the macroscopic to the microscopic body, and when they move at the same direction, the microscopic body will transfer energy to the macroscopic body. The other term, 2m v_M², is an irreversible transformation of kinetic energy into heat, for both cases. This term becomes important in processes of drag within an ideal gas (section 4).

3. The gas pressure
A particle moves with a velocity v at an angle φ to the perpendicular of a wall as shown in figure-2.

Figure-2: Elastic collision of a perticle with a wall.

It will collide with the wall during a time Δt, if its distance from the wall is less than |v| Cosφ Δt. The number ΔN_col of particle collisions with a unit area of the wall during Δt is:

ΔN_col = [n ∫f(v) |v| Cosφ d³v] Δt       (9)

n is the particle density, f(v) is the probability that the particle velocity is v, normalized by ∫4π v² f(v) dv = 1, and d³v = 4π v² dv Sinφ dφ . Substituting in equation (9):

ΔN_col = n (∫4π v² f(v) v dv) (∫Cosφ Sinφ dq) Δt       (10)

The three-dimensional average velocity <v> of particles that collide with the wall is
<v> = ∫4π v² f(v) v dv. <v> is not zero since the integration is only over positive values. The trigonometric integral over half the sphere is ∫Cosφ Sinφ dφ = 1/2, ( 0 < φ < π), therefore:

and particles that collide with the back surface lose a kinetic energy:

Δ^- = (m/2) ((2v_M - v_in Cos(β))² + v_in² Sin²(β) - v_in²)
= (m/2) 4v_M ( v_M - v_in Cos(β))            (19)

So that the average energy given to two particles by the disc is:

Δ⁺ + Δ^- = (m/2) 8v_M²       (20)

The overall energy given by the disc to the particles during a time interval Δt is ΔE = (Δ⁺ + Δ^-) A ΔN_col, where, by (11) and (15),
ΔN_col = (n <v>/2) Δt = n (2kT/πm)^1/2Δt is the number that collides with a unit front surface, and a similar number of particles collide with a unit back surface. Therefore, by equation (20), the energy transformed to the particles during this period is:

ΔE = (m/2) 8v_M² A n (2kT/πm)^1/2 Δt       (21)

The work of pushing the disc a distance Δx through the gas is p_ex A Δx = p_ex A v_M Δt, where p_ex is the pressure (force per unit area) that pushes the disc. This work is equal to the energy given to the particles, equation (21), therefore:

p_ex A v_M Δt = 4v_M² A n (2mkT/π)^1/2 Δt       (22)

Defining the drag coefficient γ by p_ex = γ v_M, its value will be:

γ = 4n (2mkT/π)^1/2       (23)

The faster particles, after colliding with the disc's front surface, and the slower particles, after colliding with its back surface, will thermalize with the other particles in the gas, and the drag energy will be transformed into heat. Thus, the isothermal movement of any surface within a gas, including pistons, is an irreversible thermodynamic process. It becomes reversible only in the limit of slow motion.

b. Drag of a tilted disc

A particle with a velocity v_in moves at an angle β toward a tilted moving disc. The disc's velocity is v_M
(figure-4 ). The tilt angle is α. The particle's velocity is first transformed from a static reference to coordinates that move together with the disc (a). Then the particle is mirror reflected in the moving coordinates (b). Finally, the particle's velocity is transformed back to the static reference (c).

Similarly, particles that collide with the back surface lose a kinetic energy:

Δ^- = (m/2) ((2v_M Cos(α) - v_in Cos(β))² + v_in² Sin²(β) - v_in²)

= (m/2) 4v_M Cos(α) (v_M Cos(α) - v_in Cos(β))            (25)

So that the average energy given to two particles by the disc is:

Δ⁺ + Δ^- = (m/2) 8v_M² Cos²(α)       (26)

so that, similarly to equations (21)-(23),

ΔE = Cos²(α) 4v_M² A n (2mkT/π)^1/2 Δt       (27)

p_ex A v_M Δt = Cos²(α) 4v_M² A n (2mkT/π)^1/2 Δt       (28)

and by p_ex = γ v_M, the drag coefficient is:

F = p_ex A = γ v_M A = (8n/3) (2mkT/π)^1/2 v_M (πR²)       (31)

where A = πR² is the sphere cross section. The drag coefficient is:

γ = (8n/3) (2mkT/π)^1/2       (32)

5. Expansion
When a wall of the box is a partition that moves at a velocity v_M, every particle that collides with the partition will transfer to it the energy 2m v Cosq v_M (equation (8)). The overall energy ΔE , transferred to a unit area of the moving partition, during a time Δt, is the sum of the energies delivered by all the colliding particles coming from all directions. The energy is calculated similarly to equation (12):

ΔE = v_M n 2m (∫4π v² f(v) v² dv) (∫Cos²q Sinq dq) Δt       (33)

By using equation (12):

ΔE = p v_M Δt = p ΔV       (34)

V Δt is the distance traveled by the partition and it is also the change of the volume ΔV of the box per unit area of the moving partition.

The calculation demonstrates that elastic particle collisions with a moving partition is the transformation mechanism of microscopic thermal energy of the particles into macroscopic mechanical work of the partition.

6. Isothermal expansion
The state equation (17), p = (N/V) kT, is inserted into the energy expression (34):

ΔE = N kT (ΔV/V)       (35)

n = N/V, V is the box volume and N is the number of particles in it. The energy is integrated in an isothermal system:

∫dE = N kT ∫dV/V       (36)

7. Adiabatic expansion of a mono-atomic gas
In an adiabatic system the particles do not exchange energy with the walls by inelastic collisions and the only energy transfer mechanism is via elastic collisions with a moving partition.

A single particle collides ΔN_col / N = (v/2) Δt / V times, equation (11), with the partition during Δt. During each collision the particle loses (or gains) a velocity 2v_M, equation (7), twice the partition velocity in the direction of its movement. A partition that moves in the x-direction will change the v_x component of the particle velocity. However, the velocity change will eventually spread to the v_y and v_z components. Therefore, the change of a particle velocity during Δt will be:

dv = -(1/3) 2v_M (v/2) Δt / V = -(v/3) dV/V       (37)

since v_MΔt = dV. The equation of adiabatic expansion of a mono-atomic gas, without internal degrees of freedom of gas particles, is calculated by integration of equation (37):

v/v₀ = (V₀/V)^1/3       (38)

and by equation (13):

p/p₀ = (V₀/V)^5/3       (39)

or, by the state equation (16), T/T₀ = (V₀/V)^2/3, or, p/p₀ = (T/T₀)^5/2.

8. Expansion of a quantum well
The ground state of a particle within an infinite square well potential that extends from x = 0 to x = L can be described as a combination of two travelling waves in opposite directions:

ψ(x, t) = (i/2L) (e^{-i(kx + wt)} - e^{+i(kx - wt)})       (40)

where k = π/L and w = hπ² / 2mL² . If the boundary at L is moving slowly with a velocity v_M , then the waves reflected from it will be Doppler shifted:

k' = k (1 - 2v_M/c)       (41)

where c = w/k is the velocity of the waves. During a time Δt each wave is reflected c Δt / 2L times by the moving boundary, so the overall shift of the wave number will be:

Δk = - k v_M Δt / L = - π ΔL/L²       (42)

When the well expands from L₁ to L₂ the wave number will shift, by integration of equation (42), from π/L₁ to π/L₂. Therefore, the ground state wave function related to L₁ will be shifted to the ground state function related to L₂.

See:
Boltzmann Transport Equation
Osmosis and Thermodynamics
van't Hoff's Evidence

Appendix: integrals

₀∫ ^∞xⁿ exp(-a²x²) dx = Γ((n+1)/2) / 2aⁿ⁺¹

Γ is the gamma function:

Γ(n) = (n – 1)!

Γ(1/2) = π^1/2

Γ(3/2) = (1/2) π^1/2

Γ(5/2) = (3/4) π^1/2

References
F.W. Sears, G.L. Salinger, Thermodynamics, Kinetic Theory, and Statistical Thermodynamics (Addison-Wesley, 3rd ed 1975) pp 254-266, 354-360.
U. Lachish, J. Chemical Education, vol 55(6), 369-371 (1978)

On the net: December 2002, section on drag added July 2006, drag of a sphere added June 2008.

By the author:

1. "Osmosis and thermodynamics", American Journal of Physics, Vol 75 (11), pp. 997-998, November (2007).
2. "van't Hoff's Evidence", http://urila.tripod.com/evidence.htm, October (2007).
3. "Osmosis and Thermodynamics", http://urila.tripod.com/osmotic.htm, January (2007).
4. "Expansion of an ideal gas", http://urila.tripod.com/expand.htm, December (2002).
5. "Optimizing the Efficiency of Reverse Osmosis Seawater Desalination", http://urila.tripod.com/Seawater.htm, May (2002).
6. "Boltzmann Transport Equation", http://urila.tripod.com/Boltzmann.htm, May (2002).
7. "Energy of Seawater Desalination", http://urila.tripod.com/desalination.htm, April (2000).
8. "Avogadro's number atomic and molecular weight", http://urila.tripod.com/mole.htm, April (2000).
9. "Vapor Pressure, Boiling and Freezing Temperatures of a Solution", http://urila.tripod.com/colligative.htm, December (1998).
10. "Osmosis Reverse Osmosis and Osmotic Pressure what they are", http://urila.tripod.com/index.htm, February (1998).
11. "Calculation of linear coefficients in irreversible processes by kinetic arguments", American Journal of Physics, Vol 46 (11), pp. 1163-1164, November (1978).
12. "Derivation of some basic properties of ideal gases and solutions from processes of elastic collisions", Journal of Chemical Education, Vol 55 (6), pp. 369-371, June (1978).

Links:

1. Thermodynamics Research Laboratory, http://www.uic.edu/~mansoori/Thermodynamics.Educational.Sites_html
2. Thermodynamik - Warmelehre, http://www.schulphysik.de/thermodyn.html
3. The Blind Men and the Elephant
4. My Spin on Lunacy, http://www.optics.arizona.edu/Palmer/moon/lunacy.htm
5. Five Weeks in a Balloon
6. The first man I saw
7. "Faster, Faster!"
8. Perfection can't be rushed
9. The man higher up
10. Brains
11. The First-Class Passenger
12. other