Chapter 15. Ideal Gases

Question 1

Define the mole

Answer 1

The mole is the amount of substance which contains 6.02214076 * 10²³ elementary entities, usually atoms or molecules but could also be ions, electrons, protons, neutrons, or other particles.

Question 2

What is Avogadro’s constant

Answer 2

The number of elementary entities in one mole of any substance

Question 3

Define molar mass

Answer 3

The mass of one mole of a substance, in gmol^-1

Question 4

Define relative molecular mass

Answer 4

The mass of a molecule of substance as compared to one twelfth the mass of an atom of the ¹²C isotope.

Question 5

Define atomic mass unit

Answer 5

One twelfth the mass of a ¹²C atom.

Question 6

State Boyle’s law

Answer 6

For a given fixed mass of a gas, the volume of the gas is inversely proportional to the pressure at constant temperature. so P₁V₁ = P₂V₂

Question 7

State Charles’ law

Answer 7

For a given fixed mass of a gas, the volume of the gas is directly proportional to the temperature provide pressure stays constant.

Question 8

State Gay-Lussac’s law

Answer 8

For a given fixed mass of a gas, the pressure of the gas is directly proportional to the temperature provided volume stays constant.

Question 9

Define an ideal gas

Answer 9

A gas that obeys the ideal gas equation PV = nRT at all thermodynamic temperatures, volumes and pressures.

Question 10

What is R in the ideal gas equation / equation of state for an ideal gas?

Answer 10

The molar gas constant / universal gas constant = 8.31 JK^-1mol^-1

Question 11

What is the Boltzmann constant?

Answer 11

k = R/N_A where N_A is the Avogadro constant

Question 12

What are the assumptions of an ideal gas in the kinetic theory of matter?

Answer 12

1. All molecules behave as identical, hard, perfectly elastic spheres
2. There are no forces of attraction or repulsion between the molecules
3. The volume of the molecules is negligible compared with the volume of the containing vessel
4. The molecules are a lot and in constant random motion.

Question 13

Derive the formula for the pressure of an ideal gas from its density

Answer 13

1. Take the container to a cube of sides L
2. The velocity of a particle, c can be resolved into the directions x, y, z, but lets take c_x, the time the particle takes to travel between two walls is 2L/c_x
3. The change in momentum when it hits the wall, Δp_x = 2mc_x, where m is the mass of the molecule
4. The rate of change of this momentum will be 2mc_x / (2L/c_x) = mc_x²/L, this is the average force exerted by the particle on the wall, by Newton’s second law
5. There are N particles of the gas and pressure is force over area so p = Nmc_x²/L³ so p = Nmc_x²/V
6. From an extension of the pythagoras theorem, c² = c_x² + c_y² + c_z² and since we are dealing with huge numbers of molecules, taking the averages will give <c_x²> = <c_y²> = <c_z²> so <c_x²> = <c²>/3, this would then give us the formula, pV = (1/3)Nm<c²>
7. N is the number of total molecules in the container and Nm/V will be its density hence p = (1/3)ρ<c²>

Question 14

Derive the relationship between the average kinetic energy of a molecule in the gas and the thermodynamic temperature of the gas

Answer 14

1. pV = (1/3)Nm<c²> and <E_k> = (1/2)m<c²> so if we rewrite the first formula we can get,
   pV = (2/3)N((1/2)m<c²>) = (2/3)NE_k
2. pV is also equal to NkT so <E_k> = (3/2)kT

Question 15

Derive the root-mean-square speed of the molecules

Answer 15

1. <E_k> = (3/2)kT = (1/2)m<c²>
2. So <c²> = (3kT/m) square root everything to get the r.m.s speed of the molecules
   N.B This is not exactly the average speed of the molecules, the actual average is about 0.92 of the r.m.s speed.