Dr Roger Bennett Rm. 23 Xtn Lecture 4

Kinetic Theory of Gases The Maxwell-Boltzmann Velocity Distribution –We assume an ideal gas (non interacting massive point particles undergoing elastic collisions). Imagine a box of gas with atoms bouncing around inside. Each one with a velocity (u=v x,v=v y,w=v z ) in Cartesian coordinates. The probability that a molecule has its x component of velocity in the range du about u is defined as –P u = f(u)du where f(u) is (the currently unknown) probability density function. Similarly P v = f(v)dv and P w = f(w)dw

Kinetic Theory of Gases The probability that u will be in a range du and v will be in a range dv and w will be in a range dw is –P uvw = P u P v P w = f(u)f(v)f(w) du dv dw Now the probability distribution (f(u)f(v)f(w)) must be spherically symmetric as we have no preferred direction:- u v w

Kinetic Theory of Gases The spherical symmetry of the surface means that if we move around on a contour of constant probability we don’t change the probability: d(f(u)f(v)f(w)) = 0 –Using the product rule again f’(u)du f(v)f(w) + f’(v)dv f(u)f(w) + f’(w)dw f(u)f(v) = 0 Equation 4.1 –But we have a spherical surface so u 2 + v 2 + w 2 = constant udu + vdv + wdw = 0 Equation 4.2 –For arbitrary du and dv we can rearrange to: dw = (-udu – vdv) / w Equation 4.3

Kinetic Theory of Gases Substuting dw in eqn. 4.1 with dw from eqn. 4.3 and rearranging gives: {f’(u)/f(u) – (u/w) f’(w)/f(w))}du + {f’(v)/f(v) – (v/w) f’(w)/f(w))}dv = 0 As du and dv are arbitrary the terms in brackets must be zero: –f’(u)/f(u) – (u/w) f’(w)/f(w) = 0 –f’(v)/f(v) – (v/w) f’(w)/f(w) = 0 So –f’(u)/(uf(u)) = f’(w)/(wf(w)) = f’(v)/(vf(v)) = -B – B is an unknown constant –f’(u) = - Buf(u) (and similarly for v and w)

Kinetic Theory of Gases f’(u) = - Buf(u) (and similarly for v and w) f(u) = Ae -1/2Bu 2 This gives the shape of the probability distribution for the velocity in each direction f(v) = Ae -1/2Bv 2 and f(w) = Ae -1/2Bw 2 –A is an arbitrary scaling constant

Kinetic Theory of Gases f(u) = Ae -1/2Bu 2 Find A by normalising

Kinetic Theory of Gases A more useful quantity is the speed distribution. P uvw =f(u)f(v)f(w)dudvdw is the probability of finding an atom in du at u and in dv at v and in dw at w.

Kinetic Theory of Gases

This is the Maxwell-Boltzman Distribution for the total speed c in a gas. We still need to understand B.

Kinetic Theory of Gases To find B we need to relate our microscopic understanding to the macroscopic. We know PV = (2/3) U = nkT = (2/3) n –Average kinetic energy per molecule = 3/2kT –What is the average speed and kinetic energy of the Maxwell-Boltzmann distribution? –(3/2)kT = (1/2)m = (3/2) m/B –B = m/kT

Maxwell-Boltzman Distribution for the speed c.

Dr Roger Bennett Rm. 23 Xtn Lecture 5

Equations of State We have found that PV = nkT = NRT = (2/3) U –When three of the five state variables P,V, N (or n), T or U are specified then the remaining two are determined and the state of the system is known. –If the variables do not change with time it is an equilibrium state. –We are most often interested in changes of state as a result of external action such as compressing a gas, stretching a rubber band, cooling etc. Such actions are processes.

Equations of State For the ideal Gas: PV = nkT = NRT = (2/3) U –We can sketch this equation of state as a surface:

Changes of State We have already looked at one change of state – adiabatic compression. On a P-V diagram this can be visualised as:- dw = -PdV V dV A The blue dots define initial and final states with unique values of P and V and hence T (or U).

Changes of State If we do this carefully and slowly through a succession of intermediate equilibrium states we can show that the work done on the system is area under curve.

Changes of State The succession of intermediate equilibrium states has a special name – a quasistatic process. It is important as it means we always lie on the equation of state during the process. If we moved from initial to final states rapidly pressure and temperature gradients would occur, and the system would not be uniquely described by the equation of state.

Changes of State If we don’t do this carefully and slowly what happens? Consider a compression where the wall moves quite fast, with velocity u B. A molecule with:- Recoils with u  u+2u B V dV A

Changes of State Thus the molecule has gained excess energy: If u B <<u then we recover our standard result  ke=u B (2mu)= u B (momenta to wall per impact)  U(total/area/sec)= u B (total momenta/area/sec)  U(total/area A/time dt) = u B PAdt = -PdV = dW If u B <<u is not true then 2mu B 2 additionally increases the internal energy U as u B 2 is always positive irrespective of the sign of u B. This additional term is physically manifested as the flow of heat.

Changes of State Thus in general we write the conservation of energy as: dU = đW + đQ This is the First Law of Thermodynamics in differential form Note the subtle difference between the d in dU and đ in đW. dU is a perfect differential because U is a state function and uniquely determined. đW and đQ are imperfect differentials because W and Q are path functions that depend on the path taken.

Changes of State – Thermodynamic Processes Adiabatic a process with no heat transfer into or out of the system. Therefore, the system may have work done on it or do work itself. Isochoric a process undertaken at constant volume. If the volume is constant then the system can do no work on its surroundings đW = 0. Isobaric a process undertaken at constant pressure. Q, U and W can all vary but finding W is easy as W = -P(V 2 -V 1 ).

Changes of State – Thermodynamic Processes Isothermal processes are undertaken at constant temperature. This is achieved by coupling the system to a reservoir or heat bath. Heat may flow in or out of the system at will but the temperature is fixed by the bath. In general isothermal processes U, Q and W can all vary. For the special cases, such as the ideal gas, where U only depends on the temperature the heat entering the system must equal the work done by the system.

Changes of State – Thermodynamic Processes These processes for a fixed quantity of ideal gas can be shown on a single P-V indicator diagram.