Kinetic theory of gases

A glass of water (again) zA glass of water can have potential energy (because I lift it from the table) zIt can have kinetic energy (because I drop it) zThese are “bulk properties” zLook at the water in detail: disordered motion (“thermal motion”) of the molecules. Energy associated with it

Internal energy zInternal energy: due to disordered motion of molecules zGlass of water on microscopic scale: ykinetic energy (molecules in motion) ypotential energy (attraction between molecules)

Temperature zTemperature measures translational kinetic energy (so T 1 = T 2 does not imply U 1 = U 2 !)

State variables zstate variable: precisely measurable physical property which characterizes the state of a system, independently of how the system was brought to that state zExamples: p, V, T, U zAny property that is a combination of state variables is a state variable itself

Empirical gas laws zV  N when p,T constant (Avogadro) zp  T when N,V constant (Charles/Gay-Lussac) zp  1/V when N,T constant (Boyle) zIdeal gas law: zk is the same for all gases:  J K -1

Avogadro’s number zN A = the number of atoms in 12 g of 12 C. Value:  mol -1 zA mole of molecular species has N A molecules zRewrite: zR = J mol -1 K -1

Kinetic theory of gases zIdeal gas: yneglect intermolecular attractions yall collisions perfectly elastic ydilute gas, volume occupied is negligible zPressure due to collisions with wall zNewton’s Second Law:

Kinetic theory of gases II zForce due to collisions with wall zWorks because total momentum is conserved in molecular collisions

Kinetic theory III x v v y v collision 1: t = 0 collision 2: t = 2L/v x L

Kinetic Theory IV zSo: zNot all molecules have same v x : use zSubstitute:

Kinetic Theory V zCompare with empirical ideal gas law: zFor ideal monatomic gases this translational kinetic energy is the only form of energy:

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

Question time! zConsider a fixed volume of gas. When N or T is doubled the pressure doubles since pV=NkT yT is doubled: what happens to the rate at which a molecule hits a wall? (a)  1 (b)  2 (c)  2 yN is doubled: what happens to the rate at which a molecule hits a wall? (a)  1 (b)  2 (c)  2

Question 2 zContainer 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 energy b) A has more internal energy than B c) A has less internal energy than B

Question 3 zContainer 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 velocity b) The argon atoms are on average faster than the helium atoms c) The argon atoms are on average slower than the helium atoms

Question 4 zContainer 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 same b) The average speed in A is only a little higher c) The average speed in A is about  2 higher d) The average speed in A is about twice as high

Van der Waals gases zTwo phenomena that we have neglected so far can easily be included ymolecules are not point particles ymolecules attract each other zVolume occupied: replace V by V-Nb yb is about 4 times the spatial volume occupied by a molecule (b depends on the distance at which they “feel” each other)

Attractive forces zMolecules near the wall are only attracted by other molecules from the other side zThe gas is less dense near the wall zWe won’t derive this, but: the average velocity is the same throughout the gas

Van der Waals equation zThis leads to an improved formula zNot as easy to use but agrees better with experiment at high densities, near phase transitions, etc.

Van der Waals gas & ideal gas I zConsider 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 same b) The Van der Waals gas has more internal energy c) The Van der Waals gas has less internal energy d) We can’t be sure

Van der Waals gas & ideal gas II zConsider 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 gases b) Higher for the Van der Waals gas c) Lower for the Van der Waals gas d) We can’t be sure

Van der Waals gas & ideal gas III zAn 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 gases b) Higher for the Van der Waals gas c) Lower for the Van der Waals gas d) We can’t be sure

Van der Waals gas & ideal gas IV zAn 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 gases b) Higher for the Van der Waals gas c) Lower for the Van der Waals gas d) We can’t be sure

PS225 – Thermal Physics topics zThe atomic hypothesisThe atomic hypothesis zHeat and heat transferHeat and heat transfer zKinetic theoryKinetic theory zThe Boltzmann factorThe Boltzmann factor zThe First Law of ThermodynamicsThe First Law of Thermodynamics zSpecific HeatSpecific Heat zEntropyEntropy zHeat enginesHeat engines zPhase transitionsPhase transitions