Thermal & Kinetic Lecture 9 Diffusion and thermal conductivity of a gas

1. Thermal & Kinetic Lecture 9 Diffusion and thermal conductivity of a gas LECTURE 9 OVERVIEW Recap…. Diffusion and the diffusion equation Milk drops and the ‘arrow of time’ Thermal conductivity of a gas

2. Last time…. Diffusion – random walks. Coefficient of diffusion – relates flux of molecules to the concentration gradient

3. Consider thin slab of cross-sectional area A that extends from x to x + dx. The concentration at x at time t is n’(x,t). dx The diffusion equation (in 1D) We want to determine the relationship between the rate of change of the number density (i.e. concentration) of the molecules at a point to the spatial variation of the concentration at that point. J(x,t) J(x+dx,t) The number of molecules that enter the slab per unit time is J(x,t)A. Rate of increase of concentration inside slab due to flux into left hand side of slab:

4. ? What is the rate of change of concentration inside the slab due to the flux leaving the right hand side of slab? ? Therefore the net rate of change of concentration inside the slab is…..? The diffusion equation (in 1D) Rate of change of concentration inside slab due to flux into left hand side of slab: J(x,t) J(x+dx,t) dx

5. Using Fick’s law, we can rewrite the equation above as: The diffusion equation The diffusion equation (in 1D) This is a 2nd order (partial) differential equation. NB Pay particular attention to your GM2 notes regarding ODEs – you’ll be seeing (and solving) a lot of ODEs and PDEs during your degree! To solve DE we need to know initial (or boundary) conditions.

6. Solution to diffusion equation Gaussian function. n’(x,t) t1> t0 t = 0 t2> t1 t3> t2 x The diffusion equation (in 1D) Our intial condition is that at x=0 and t=0 we have a certain concentration n’(0,0). We release molecules which occupy a thin slab at x = 0 at time t = 0 and let them diffuse. Molecules spread out as a function of time – Gaussian curve broadens.

7. Note that <x2>  t. RMS displacement varies as t. The mean square displacement is related to the diffusion coefficient, D, as follows: <x2> = 2Dt (CW 5). What does D represent microscopically? Reconsider random walker. We’ll return to the ideas of diffusion and diffusion coefficient when we come to consider the thermal properties of solids.

8. ? Imagine we film the simulation. Will we observe strong differences in the evolution of the state of the drop if we run the film backwards or forwards? ? Imagine we film the motion of a single isolated moleculeperforming a random walk. How different will this motion appear if we play the film backwards? ? Is it possible that the molecules could “reorganise” themselves into an ordered drop? Diffusion, milk drops and Newton’s laws Consider diffusion of particles in 2D. Initial state is (somewhat artificial) square lattice of milk molecules on coffee. Assume only diffusion – no convection. Milk molecules spread out as a function of time. ……why?

9. Diffusion, milk drops and Newton’s laws ? If there is nothing in the laws of motion forcing the molecules to “spread out” and become more disordered, why do they?! (What defines the ‘arrow of time’?) ANS: Entropy.

10. k is the coefficient of thermal conductivity and A is the cross-sectional area. dT/dz is the temperature gradient. Rate of heat flow (heat current) Thermal conductivity NB Here we consider conduction of heat through a gas via intermolecular collisions – convection is not involved. If the gas at the top of a container is hotter than the gas at the bottom heat will flow from the top to the bottom. Transfer of heat from the hotter to the colder gas is by intermolecular collisions. To determine the flow of thermal energy we need to consider net energy flow across a given plane.

11. Thermal conductivity of a gas z + l Similar model as for derivation of diffusioncoefficient. A B z - l No. of molecules on average which cross unit area of plane AB per second and come from a plane at position z + l = NB This is the mean speed at (z+l) No. of molecules on average which cross unit area of AB per second and come from a plane at position z – l = Molecules from different layers have different mean energies (remember: there’s a temperature gradient). This is now the most important effect: neglect difference in n’ above and below AB  n’(z+l) = n’(z-l) = n’.

12. ? Write down an expression for the average energy carried in unit time by the molecules from z – l. Net flow of energy across AB (in +z direction) per unit time: Thermal conductivity of a gas Molecules at z+lhave higher average energy than molecules at z-l (due to dT/dz). Average energy carried by molecules from z+l in unit time = z + l A B z - l We now make the assumption that <v>(z+l) = <v>(z-l) (G&P do not make this assumption (p.484))

13. Substituting into the equation highlighted in yellow above: Thermal conductivity of a gas z + l 2l A B z - l From the diagram above:

14. Compare this with earlier expression for heat current: Coefficient of thermal conductivity for a gas: NB is simply cV, the heat capacity per molecule at constant volume ? Write down cv for a monatomic ideal gas. Thermal conductivity of a gas Therefore, ANS: 3k/2

15. ? Write down an expression for k in terms of the molar specific heat capacity, CV. Coefficient of thermal conductivity for a gas: ? How can we get an expression for <v>? Thermal conductivity of a gas We can substitute the value of the heat capacity into the expression for k: and get an expression for k in terms of the molar specific heat capacity, Cv ANS: Equipartition of energy/ Maxwell-Boltzmann distribution

16. Thermal conductivity of a gas is independent of n’! ..but what if n’ gets close to 0 and there are very few gas molecules? How can the thermal conductivity NOT depend on n’?! Independent of pressure ONLY when the mean free path is << dimensions of container. If the gas number density is so low that a molecule has a good chance of crossing the container without colliding with another molecule then none of the above holds. Thermal conductivity of a gas

17. Convection I previously mentioned that we were neglecting convection in determining the thermal conductivity. What is convection? If the temperature of a given mass of air increases at constant pressure, volume must increase (PV=RT). Volume increases  density decreases and buoyancy increases. Warm air expands and rises Cooler air sinks