Kinetic theory of gases

A glass of water (again)A glass of water can have potential energy (because I lift it from the table)It can have kinetic energy (because I drop it)These are bulk properties

Look at the water in detail: disordered motion (thermal motion) of the molecules. Energy associated with it

Internal energyInternal energy: due to disordered motion of moleculesGlass of water on microscopic scale:kinetic energy (molecules in motion)potential energy (attraction between molecules)

TemperatureTemperature measures translational kinetic energy (so T1 = T2 does not imply U1 = U2!)

State variablesstate variable: precisely measurable physical property which characterizes the state of a system, independently of how the system was brought to that state

Examples: p, V, T, U

Any property that is a combination of state variables is a state variable itself

Empirical gas lawsV N when p,T constant (Avogadro)p T when N,V constant (Charles/Gay-Lussac)p 1/V when N,T constant (Boyle)

Ideal gas law:

k is the same for all gases: 1.381 10-23 J K-1

Avogadros numberNA= the number of atoms in 12 g of 12C. Value: 6.0221023 mol-1A mole of molecular species has NA molecules

Rewrite:

R = 8.3145 J mol-1 K-1

Kinetic theory of gasesIdeal gas:neglect intermolecular attractionsall collisions perfectly elasticdilute gas, volume occupied is negligiblePressure due to collisions with wallNewtons Second Law:

Kinetic theory of gases IIForce due to collisions with wall

Works because total momentum is conserved in molecular collisions

Kinetic theory IIIxvvyvcollision 1: t = 0collision 2: t = 2L/vxL

Kinetic Theory IVSo:

Not all molecules have same vx: use

Substitute:

Kinetic Theory VCompare with empirical ideal gas law:

For ideal monatomic gases this translational kinetic energy is the only form of energy:

Kinetic Theory SummaryUsing Newtonian mechanics we have established:the relationship between p, N/V, T;the universality of the gas constant;the relationship between temperature and K.E.the internal energy of a monatomic ideal gas

Question time!Consider a fixed volume of gas. When N or T is doubled the pressure doubles since pV=NkT

T is doubled: what happens to the rate at which a molecule hits a wall? (a) 1 (b) 2 (c) 2

N is doubled: what happens to the rate at which a molecule hits a wall? (a) 1 (b) 2 (c) 2

Question 2Container A contains 1 l of helium at 10 C, container B contains 1 l of argon at 10 C.

a) A and B have the same internal energyb) A has more internal energy than Bc) A has less internal energy than B

Question 3Container A contains 1 l of helium at 10 C, container B contains 1 l of argon at 10 C.

a) The argon and helium atoms have the same average velocityb) The argon atoms are on average faster than the helium atomsc) The argon atoms are on average slower than the helium atoms

Question 4Container A contains 1 l of helium at 10 C, container B contains 1 l of helium at 20 C.

a) The average speeds are the sameb) The average speed in A is only a little higher c) The average speed in A is about 2 higherd) The average speed in A is about twice as high

Van der Waals gasesTwo phenomena that we have neglected so far can easily be includedmolecules are not point particles molecules attract each other

Volume occupied: replace V by V-Nbb is about 4 times the spatial volume occupied by a molecule (b depends on the distance at which they feel each other)

Attractive forcesMolecules near the wall are only attracted by other molecules from the other sideThe gas is less dense near the wall

We wont derive this, but: the average velocity is the same throughout the gas

Van der Waals equationThis leads to an improved formula

Not as easy to use but agrees better with experiment at high densities, near phase transitions, etc.

Van der Waals gas & ideal gas IConsider two equal amounts of gas at identical temperature. One can be treated as an ideal gas, the other is a Van der Waals gas.

a) The internal energies are the sameb) The Van der Waals gas has more internal energyc) The Van der Waals gas has less internal energyd) We cant be sure

Van der Waals gas & ideal gas IIConsider two equal amounts of gas at identical temperature. One can be treated as an ideal gas, the other is a Van der Waals gas. The specific heat at constant volume is

a) The same for both gasesb) Higher for the Van der Waals gas c) Lower for the Van der Waals gasd) We cant be sure

Van der Waals gas & ideal gas IIIAn ideal gas and a Van der Waals gas at the same temperature expand isothermally by the same amount. The work done is

a) The same for both gasesb) Higher for the Van der Waals gas c) Lower for the Van der Waals gasd) We cant be sure

Van der Waals gas & ideal gas IVAn ideal gas and a Van der Waals gas at the same temperature expand isothermally by the same amount. The heat added is

a) The same for both gasesb) Higher for the Van der Waals gas c) Lower for the Van der Waals gasd) We cant be sure

PS225 Thermal Physics topicsThe atomic hypothesisHeat and heat transferKinetic theoryThe Boltzmann factorThe First Law of ThermodynamicsSpecific HeatEntropyHeat enginesPhase transitions