Physics of Gases Lecture Notes PDF

A Review of the Physics of Gases: PHYS3280 Note: This note generally follows the treatment in the course text Prolss: Physics of the Earth’s Space Environment1. Required Reading Sections 2.1 - 2.3 Part I Notes on Gas Physics Question: How do we describe gases and what do these parameters mean?...

A Review of the Physics of Gases: PHYS3280 Note: This note generally follows the treatment in the course text Prolss: Physics of the Earth’s Space Environment1. Required Reading Sections 2.1 - 2.3 Part I Notes on Gas Physics Question: How do we describe gases and what do these parameters mean? We are used to describing gases by their macroscopic physical properties. Properties like: Pressure Temperature Mass Density etc. How are these macroscopic properties connected to the microscopic properties of the gas? You may look at Table 2.1 in the course text for example numerical values of gas parameters on the Eart’s surface and in the upper atmosphere. How do we connect macroscopic and microscopic properties? We need the mi- croscopic properties to describe some important phenomenon like diffusion. 1 Prolss, G. W. (2004). Physics of the Earth’s space environment: an introduction. Springer, Berlin. 1  Figure 1: Figure from the course text. 1 Particle Number Density Particle number density at any point with spatial and temporal coordinates (~r, t) is given by ✓ ◆ N n(~r, t) = lim V !dV V ~r,t where N is the number of particles in the volume V. The above expression only makes sense if the volume element under consid- eration (dV) has many particles but is not as large as the macroscopic density variations. So what are these distances and how can we calculate them? Consider a cubic volume with sides of length a (See Figure 1). Assume the particles are uniformly spaced within the volume element. The spacing is in a cubic geometry with distance d between the particles. The number of particles along each edge is given by a/d. The number of particles in the cube is: a3 N= d3 a d= p 3 N but N = na3 so 1 d⇡ p 3 n 2  d is an estimate of the mean distance between the particles. So our volume element dV d3. How big does it need to be? Let’s say we want a volume element that has variations in the number of particles in the volume element less than 0.1%. Statistical physics would point us to the variation in the number of particles being 1 N⇡p N so for our 0.1% variation, we require 1 million particles in our volume. For 106 m 3 this would require a volume of 1m3. Notice from Table 2.1 in the text that macroscopic variations are ~kms (scale heights). 2 Gas Velocities A gas can be characterized by a number of velocities: Individual particle velocities ~v macroscopic flow (wind or bulk) velocity ~u random velocities due to thermal motion ~c How do these velocities relate to each other? The flow velocity is the overall bulk motion of the gas and may be related to the individual particle velocities by ~u(~r, t) =< ~v >~r,t In other words, the mean motion of the individual gas molecules is equivalent to the flow velocity. The thermal motion is the motion left once this bulk component is removed. ~c = ~v ~u < ~c >=< ~v ~u >=< ~v > ~u = 0 3 Collision Frequency In order to understand the collision dynamics in a gas we define the collision frequency as: ⌧ # of collisions collision frequency = =⌫ unit time 3 Figure 2: Figure from the course text. 4 4 Mean Free Path Let’s consider a gas of particle type 1 with radius r1 and particle i of this gas has a velocity c1,i. This particle is in a second gas (type 2) with radius r2 and velocity c2,j. If we are interested only in collisions, we can simplify the problem to using the relative velocities. ~ c1,2 =< |c1 ~ c~2 | > So we can consider the problem to be particle 1 moving with the above velocity in a sea of stationary particle 2’s. We can likewise simplify the geometry by considering particle 1 to have a radius of r = r 1 + r2 Particle 1 now is present in a sea of point particles. Consider the volume of particles encountered in a time interval by the now larger particle 1. ⇡(r1 + r2 )2 c~1,2 n t ⌫1,2 = t so the collision frequency is given by ⌫1,2 = 1,2 c1,2 n2 where the cross section is defined as 1,2 = ⇡(r1 + r2 )2 Here the cross section only depends on the species but in reality there will be energy and velocity dependencies. What we really need is a calculation for c~1,2. Let’s approximate by assuming that the particles all move in one of 6 direc- tions: ±x, ±y, ±z in equal proportions. ~c1,2 =< |~c2 ~c1 | > or q 1 1 4 ~c1,2 = |~c1 ~c2 | + |~c1 + ~c2 | + c21 + c22 6 6 6 r c2 2 ⇡ c2 1+( ) c1 Now as we will see later on, the relationship between the mean random particle velocity and temperature is 5  r 8kT c̄ = ⇡m s ✓ ◆ 8k T1 T2 c1,2 = + ⇡ m1 m2 We can further simplify these expressions through the use of an effective (or reduced) mass and temperature. m1 m2 m1,2 = m1 + m2 and m2 T1 + m1 T2 T1,2 = m1 + m2 therefore s 8kT1,2 ⌫1,2 = 1,2 n2 ⇡m1,2 The mean free path is c¯1 1 1 l1,2 = = q = p ⌫1,2 2 1,2 n2 1 + (m1 T2 )/(m2 T1 ) 1,2 n2 1 + (c¯2/c¯1 ) when the particles are all the same this simplifies to: 2 p p 1,1 = 4⇡r , ⌫1,1 = 4 1,1 n kT/⇡m and l 1,1 = /( 2n 1 1,1 ) This result is important for later considerations. 5 Flux What do we mean by flux? The definition of flux is: The net amount of a scalar quantity transported per unit area and per unit time. This concept is applicable to any scalar quantity such as photons, particles, heat, charge or a component of a vector. The previous defnitions of particle density and flow velcoity become impor- tant. The flux in the x -direction can be calculated by: The number of particles that passes an area given by A is # = ux t A n 6  so the flux is given by ux t A n x = = nux tA or in three dimensions ~ = n~u 6 Pressure In order to understand what pressure means we need to understand the mo- mentum transferred to the walls of a pressure vessel by the flux of particles that impact those walls. How do we calculate the momentum of a particle flux? Remembering that the momentum of single particle is m~v , the net momentum in the x direction is given by the net particle velocity in that direction, the number of particles and the mass. The text uses the the following notation for momentum flux I(u) x,x = mnu2x So the thermodynamic pressure can be understood to be the cartesian aver- age of the momentum flux. 1 ⇣ I(u) I(u) I(u) ⌘ p= x,x + y,y + z,z 3 We really need the velocity distribution to calculate the pressure. Given that momentum is a linear fuction of velocity, it is equivalent that we know the mean velocity in order to calculate the momentum. Going back to the simplified model, where we have 1/6 of the particles travelling in the +x direction, 1/6 in the -x direction, etc., we have a momentum flux impinging on the wall of our cubic volume given by I(u) 1 x,x = mnū2 6 remembering that our velocity after impacting and bouncing off the wall will be in the opposite direction, the transfered momentum is actually twice the above value. 1 p= mnū2 3 7 Temperature As we have discussed in class, temperature is a measure of the kinetic energy due to the random motions of particles. An important consideration of this discussion is the number of degrees of freedom available to a system. Fo a 7