Ideal Gases

Question 1

Amount of substance is an __ ____ ________ with the base unit ___

Answer 1

SI base quantity, mol

Question 2

One mole of a substance is …

Answer 2

the amount containing a number of particles of that substance equal to the Avogadro constant NA

Question 3

A gas that obeys __ ∝ _ is known as an ideal gas

Answer 3

pV ∝ T, where T is the thermodynamic temperature

Question 4

Give the equations of state for an ideal gas

Answer 4

pV = nRT (n = amount of substance; the number of moles)
pV = NkT (N = number of molecules)

Question 5

What is the Boltzmann constant?

Answer 5

symbol: “k”
k = R / NA

Question 6

Derive the equation of state for an ideal gas

Answer 6

Boyle’s law: P ∝ 1/V
Charle’s law: V ∝ T (in Kelvin)
Pressure law: P ∝ T
PV = (constant) x T
PV = nRT (n = number of moles, R = universal gas constant)

Question 7

What are the basic assumptions of the kinetic theory of gases? (4)

Answer 7

1. intermolecular forces between the atoms is negligible
2. the collisions between atoms and the walls of the container are perfectly elastic
3. continuous random zig-zag (Brownian) motion
4. the actual volume of atoms compared to the volume occupied by the gas is negligible

Question 8

Explain how molecular movement causes the pressure exerted by a gas

Answer 8

1. molecules are in a state of continuous motion
2. they collide with the walls of the container elastically
3. they suffer change in momentum
4. according to Newton’s second law, the rate of change of the molecules’ momentum is the force they exert on the walls of the container
5. this total force, when divided by the area of the container can give a value for the pressure exerted on the walls of the container by the gas

Question 9

Derive the equation pV = (1/3)Nm<c²>

Answer 9

For a cube of length ‘l’, with ‘N’ being the number of atoms, ‘m’ being the mass of each atom and ‘t’ being the time taken to collide from one face to another:
P(Total) = P(1) + P(2) + P(3) + … + P(N)
If all acting in the same direction:
P(T) = mc(1)²/l³ + mc(2)²/l³ + mc(3)²/l³ + … + mc(N)²/l³ = (m/l³)[c(1)² + c(2)² + c(3)² + … + c(N)²]
{c(1)² + c(2)² + c(3)² + … + c(N)²}/N = <c²>
Since they are NOT all acting in the same direction:
P(A) = (1/3) x (m/l³) x N x <c²>
ρ = Nm / l³
P(A) = (1/3) x ρ x <c²>

Question 10

Give the root-mean-square speed c(r.m.s.)

Answer 10

√<c²>

Question 11

Deduce the average translational kinetic energy of a molecule

Answer 11

comparison of pV = (1/3)Nm<c²> and pV = NkT gives the average translational kinetic energy of a molecule as (3/2)kT

Question 12

Derive how C(r.m.s.) is related to T

Answer 12

(1/2)m<c²> = (3/2)kT
<c²> = 3kT / m
C(rms) = √<c²> = √(3kT/m)
C(rms) ∝ √T

Question 13

Derive how KE is related to T

Answer 13

For 1 mole:
PV = RT (because n=1)
[ (1/3) x (M/V) x <c²> ] x V = R x T
(1/3) M <c²> = R T
(1/2 M <c²> = (3/2) R T
KE(avg) = (3/2) R T
For 1 atom:
(1/2)Nm<c²> = (3/2)RT
(1/2)m<c²> = (3/2) x (R/N) x T
(1/2)m<c²> = (3/2)kT (where k represents the Boltzmann constant, R/N)
KE ∝ T