Kinetic theory of gases

1. Kinetic theory of gases

2. 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

3. Internal energy • Internal energy: due to disordered motion of molecules • Glass of water on microscopic scale: • kinetic energy (molecules in motion) • potential energy (attraction between molecules)

4. Temperature measures translational kinetic energy (so T1 = T2does not imply U1 = U2!) Temperature

5. State variables • state 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

6. Empirical gas laws • V  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

7. Avogadro’s number • NA= the number of atoms in 12 g of 12C. Value: 6.0221023 mol-1 • A mole of molecular species has NA molecules • Rewrite: • R = 8.3145 J mol-1 K-1

8. Kinetic theory of gases • Ideal gas: • neglect intermolecular attractions • all collisions perfectly elastic • dilute gas, volume occupied is negligible • Pressure due to collisions with wall • Newton’s Second Law:

9. Kinetic theory of gases II • Forcedue to collisions with wall • Works because total momentum is conserved in molecular collisions

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

11. Kinetic Theory IV • So: • Not all molecules have same vx: use • Substitute:

12. Kinetic Theory V • Compare with empirical ideal gas law: • For ideal monatomic gases this translational kinetic energy is the only form of energy:

13. Kinetic Theory – Summary • Using 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

14. 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

15. Question 2 • Container 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

16. Question 3 • Container 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

17. Question 4 • Container 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

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

19. Attractive forces • Molecules near the wall are only attracted by other molecules from the other side • The gas is less dense near the wall • We won’t derive this, but: the average velocity is the same throughout the gas

20. Van der Waals equation • This leads to an improved formula • Not as easy to use but agrees better with experiment at high densities, near phase transitions, etc.

21. Van der Waals gas & ideal gas I • Consider 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

22. Van der Waals gas & ideal gas II • Consider 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

23. Van der Waals gas & ideal gas III • An 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

24. Van der Waals gas & ideal gas IV • An 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

25. PS225 – Thermal Physics topics • The atomic hypothesis • Heat and heat transfer • Kinetic theory • The Boltzmann factor • The First Law of Thermodynamics • Specific Heat • Entropy • Heat engines • Phase transitions