Ideal Gas Law: Calculations and Assumptions in Chemistry

1. Chapter 14 The Ideal Gas Law and Kinetic Theory

2. Amedeo Avogadro (1776 -1856) Avogadro’s number • Mole – amount of substance containing a number of atoms (molecules) equal to the number of atoms in a 12 g sample of 12C • This number is known as Avogadro’s number (NA): NA = 6.02 x 1023 mol -1 • The number of moles in a sample N – total number of atoms (molecules) m – total mass of a sample, m0 – mass of a single atom (molecule); M – molar mass

3. Chapter 14 Problem 10 A cylindrical glass of water (H2O) has a radius of 4.50 cm and a height of 12.0 cm. The density of water is 1.00 g/cm3. How many moles of water are contained in the glass?

4. Ludwig Eduard Boltzmann (1844-1906) Ideal gases • Ideal gas – a gas obeying the ideal gas law: R – gas constant R = 8.31 J/mol ∙ K kB – Boltzmann constant kB = 1.38 x 1023 J/K

5. Chapter 14 Problem 29 One assumption of the ideal gas law is that the atoms or molecules themselves occupy a negligible volume. Verify that this assumption is reasonable by considering gaseous xenon (Xe). Xenon has an atomic radius of 2.0 x 10-10 m. For the standard temperature and pressure (STP) conditions, calculate the percentage of the total volume occupied by the atoms. Express your answer as a percentage with no units.

6. Ideal gases • The gas under consideration is a pure substance • All molecules are identical • Macroscopic properties of a gas: P, V, T • The number of molecules in the gas is large, and the average separation between the molecules is large compared with their dimensions – the molecules occupy a negligible volume within the container • The molecules obey Newton’s laws of motion, but as a whole they move randomly (any molecule can move in any direction with any speed)

7. Ideal gases • The molecules interact only by short-range forces during elastic collisions • The molecules make elastic collisions with the walls and these collisions lead to the macroscopic pressure on the walls of the container • At low pressures the behavior of molecular gases approximate that of ideal gases quite well

8. Ideal gases

9. Ideal gases • Root-mean-square (RMS) speed:

10. Translational kinetic energy • Average translational kinetic energy: • At a given temperature, ideal gas molecules have the same average translational kinetic energy • Temperature is proportional to the average translational kinetic energy of a gas

11. Internal energy • For the sample of n moles, the internal energy: • Internal energy of an ideal gas is a function of gas temperature only

12. Chapter 14 Problem 42 Compressed air can be pumped underground into huge caverns as a form of energy storage. The volume of a cavern is 5.6 × 105 m3, and the pressure of the air in it is 7.7 × 106 Pa. Assume that air is a diatomic ideal gas whose internal energy U is given by U = (5/2)nRT. If one home uses 30.0 kW h of energy per day, how many homes could this internal energy serve for one day?

13. James Clerk Maxwell (1831-1879) Distribution of molecular speeds • Not all the molecules have the same speed • Maxwell’s speed distribution law: NvΔv – fraction of molecules with speeds in the range from v to v + Δv

14. Distribution of molecular speeds • Average speed: • RMS speed: • Most probable speed:

15. Questions?