Ideal Gas Law

1. Ideal Gas Law The combination of these three laws gives the ideal gas law which is a special form of an equation of state, i.e., an equation relating the variables that characterize a gas (pressure, volume, temperature, density, ….). The ideal gas law is applicable to low-density gases.

2. Statistical Mechanics and Thermodynmaics • In thermodynamics a small number of macroscopic variables completely describes the state of the system (N,P,T,V,…) • In statistical mechanics, the dynamical state of the system is described by stating the position and momenta of every particle in the system • Since there are so many particles (≈1023) these can then be treated statistically.

3. Statistical Mechanics & Kinetic Theory of Gases • The dynamical state of the system is compleletely described by 6N variable where N is the number of particles. 3N variables for positions and 3N variables for momenta. • The energy of the system is described by the Hamiltonian The dynamical state of the system then evolves according to Newton’s laws of motion.

4. Kinetic theory of an Ideal Gas • Consider a large number N indistinguishable particles • With mass m that move with random velocities (each direction is equally probable and a distribution of speeds from zero to infinity) • Confined to a cubic box of length l • The particles interact elastically with the box walls (kinetic energy is conserved in the interaction) • They do not interact with eachother hence they occupy no volume

5. v m v x Fx <Fx> t Pressure • The force exerted by the particles on the walls of the container is the time-averaged momentum transfer due to collisions between the particles and the walls: • For a single collision: (the x-component changes sign) • If the time between such collisions = Dt, then the average force on the wall due to this particle is:

6. Area A vx d Pressure and Kinetic Energy of an Ideal Gas Now consider collisions of gas molecules in a closed volume. • Time between collisions with wall: • round-trip time (depends on speed) • Average force: • (one molecule) • Net average force: • (N molecules) PRESSURE: • We can relate this to the average translational KE of each molecule: • The pressure from molecular collisions is proportional • to the average translational kinetic energy of the gas: which • If combined with equipartition gives the ideal gas law.

7. The Ideal Gas Law • The Pressure-Energy relation we derived: • Together give us the Ideal Gas Law: • Comparing to the Physically derived Ideal Gas law we can • Get the relationship between Temperature and <KE>: = 3/2 kbT

8. Effect of Temperature on a Gas PV Relationships EOS

9. Distribution of Molecular Speeds Atoms or molecules in a gas do not all travel at the same speed. There is a distribution of speeds, with some travelling slow, others fast, and the majority peaked about a value called the most probable speed. A plot of the number of molecules travelling at a given speed versus speed, at a fixed temperature is called the MAXWELL-BOLTZMANN distribution.

10. Velocity Analyzer d Determine intensity as function of angular speed This gives speed distribution function and confirms Maxwell

13. Internal Energy of an ideal gas • U = KE + PE • Since the atoms do not interact, the energy of an ideal gas is just from Kinetic energy • This comes from translational energy and rotational energy. • Equipartition (a result from stat mech.) says that each quadratic degree of freedom has ½ kT of energy.

14. = U 3 Nk T Internal Energy of a Classical ideal gas • “Classical” means Equipartition Principle applies: each molecule has average energy ½ kT per quadratic modein thermal equilibrium. At room temperature, for most gases: • monatomic gas (He, Ne, Ar, …) • 3 translational modes (x, y, z) U= 3/2 NkT diatomic molecules (N2, O2, CO, …) 3 translational modes (x, y, z) + 2 rotational modes (wx, wy) U= 5/2 NkT • non-linear molecules (H2O, NH3, …) • 3 translational modes (x, y, z) • + 3 rotational modes (wx, wy, wz) • In general, for a classical ideal gas: • (a depends on the type of molecule)

15. Microscopic Physics of Pressure: Thought experiments • Collision frequency of atoms with walls • Momenta transfer per collision How does changing each of the macroscopic variables that Pressure depends on (n,V,T) change each of these? • N • V • T

16. Thought Experiments Gedankenexperiment. • If the number of molecules is increased… • What happens to collision frequency? • What happens to the momentum transfer per collision? • What happens to the mechanical equilibrium of the system if the temperature is increased in (b) on the left side?

17. Thought Experiments Gedankenexperiment. • If the volume is decreased… • What happens to collision frequency? • What happens to the momentum transfer per collision? • What happens to the mechanical equilibrium of the system if the temperature is increased in (b) on the left side?

18. Thought Experiments Gedankenexperiment. • If the Temperature is increased… • What happens to collision frequency? • What happens to the momentum transfer per collision? • What happens to the mechanical equilibrium of the system if the temperature is increased in (b) on the left side?

19. Thought Experiments Gedankenexperiment. • At the same temperature, pressure and volume what are the differences between gases made of atoms of different masses? • Number of molecules in each system? • Speed of the molecules? • Collision Frequency? • Momentum transfer per collision?

20. Homework Question? • How does the internal energy of the molecules in this room change with Temperature? • First consider does the volume change? • Does the pressure change? • Do the number of Molecules change in the room change?

21. 1 • If you doubleT, how many times as often will a particular atom hit the container walls? • (A)´1 (B)´1.4 (C)´ 2 (D)´ 4 The collision rate is proportional to the velocity, which is proportional to the square root of the energy and temperature. Rate increases by Ideal gas behavior Consider a fixed volume of an ideal gas. • If you double either T or N, p goes up a factor of 2. • ( pV = NkT )

22. The molecules are assumed to be non-interacting except for elastic collisions so the motion of any individual does not depend on the others. 2 • If you doubleN, how many times as often will a particular atom hit the container walls? • (A)´1 (B)´1.4 (C)´ 2 (D)´ 4 The total number of collisions scales with N, but the collision rate for one molecule is independent of N. Ideal gas behavior Consider a fixed volume of an ideal gas. • If you double either T or N, p goes up a factor of 2. • ( pV = NkT )

23. 1 • If you doubleT, how many times as often will a particular atom hit the container walls? • (A)´1 (B)´1.4 (C)´ 2 (D)´ 4 2 • If you doubleN, how many times as often will a particular atom hit the container walls? • (A)´1 (B)´1.4 (C)´ 2 (D)´ 4 Ideal gas behavior Consider a fixed volume of an ideal gas. • If you double either T or N, p goes up a factor of 2. • ( pV = NkT )

24. Thought Experiments Gedankenexperiment. • What happens when you decrease the volume of a gas (leaving n,T constant)? • From ideal gas law at same n,T what would the pressure be? • How does changing the Volume change the length of the box? • How does it change the Area of a side? • What happens to the Collision Frequency? • Momentum transfer per collision?

25. Ideal Gas Law or Avogadro’s number NA 6.0220451023 • The assumptions made in the kinetic theory of gases for an ideal gas were that: • The particles were non-interacting • No repulsive forces means that each particle takes up no volume. isotherms The P-V diagram – the projection of the surface of the equation of state onto the P-V plane. Note the isothermals and isobars

26. van der Waals equation EOS Real Gases Under many conditions, real gases do not follow the ideal gas law ... -- Intermolecular forces of attraction cause the measured pressure of a real gas to be less than expected -- When molecules are close together, the volume of the molecules themselves becomes a significant fraction of the total volume of a gas

27. The van der Waals model of real gases For real gases – both quantitative and qualitative deviations from the ideal gas model U(r) U(r) U(r) Electric interactions between electro-neutral molecules : r van der Waals attraction repulsion attraction The van der Waals equation of state for real gases

28. EOS van der Waals Equation Corrections for real gas behavior are made using the parameters a and b a – accounts for intermolecular attractions in real gases b – accounts for the real volumes of gases