# Properties of Gases | General Chemistry 2

Properties of gases are studied in this chapter: gas pressure, gas laws, ideal gas equation, relationship between molar volume, molar mass and density, Dalton’s law of partial pressures, kinetic theory and speed distribution, Graham’s law of effusion, mean free path, Van der Waals equation

## Gas Properties

Gas: variable volume and shape ⇒ it expands to fill entire container
Most of the volume is empty space

Properties:

• Gas volume changes significantly with pressure and temperature
• Gases have relatively low densities
• Gases flow very freely
• Gases are miscible with each other ⇒ form solution in any proportions
• Gases exert pressure on the walls of any surface they touch

Pressure:
Pressure is defined as a force per unit area:

P = $\frac{\mathrm{F}}{\mathrm{A}}$

F = force (in Newton = kg.m.s-2)
A = area (in m2)

Pressure is measured in a variety of units:

• atm and torr are most commonly used in gas law problems
• Pa is used if SI units are needed

Conversion factor: 1 atm = 760 torr = 1.013 bar = 1.013 x 105 Pa

## Gas Laws

4 important variables:

• Pressure P (in Pa)
• Temperature T (in K)
• Volume V (in L)
• Number of moles n (in mol)

Ideal gas: gas that exhibits linear relationships among these variables

Boyle’s Law (P vs. V):
At constant temperature: P and V are inversely proportional

P1 V1 = P2 V2    (at T = cst)

Charles’s Law (V vs T):
At constant pressure: V and T are directly proportional

$\frac{{\mathrm{V}}_{1}}{{\mathrm{T}}_{1}}$ = $\frac{{\mathrm{V}}_{2}}{{\mathrm{T}}_{2}}$   (at P = cst)

Gay-Lussac’s Law (P vs T):
At constant volume: P and T are directly proportional

$\frac{{\mathrm{P}}_{1}}{{\mathrm{T}}_{1}}$ = $\frac{{\mathrm{P}}_{2}}{{\mathrm{T}}_{2}}$   (at V = cst)

At constant temperature and pressure: V and n are directly proportional

$\frac{{\mathrm{V}}_{1}}{{\mathrm{n}}_{1}}$ = $\frac{{\mathrm{V}}_{2}}{{\mathrm{n}}_{2}}$   (at T and P = cst)

## Ideal Gas Equation

PV = nRT

P = pressure (in Pa)
V = volume (in m3)
n = number of moles (in mol)
R = ideal gas constant = 8.314 (in J.K-1.mol-1)
T = temperature (in K)

This equation is true for ideal gases, or gases behaving ideally
Better conditions for a gas to behave like an ideal gas: monoatomic gas, low pressure, high temperature

## Ideal Gas Law – Molar Volume, Molar Mass and Density

Molar Volume Vm (in L.mol-1) in the Ideal Gas Law:

Vm$\frac{\mathrm{V}}{\mathrm{n}}$$\frac{\mathrm{RT}}{\mathrm{P}}$

R = 8.314 J.mol-1.K-1
T = temperature (in K)
P = pressure (in Pa)

At standard temperature and pressure (STP): Vm = 22.41 L.mol-1

Molar Mass M (in g.mol-1) in the Ideal Gas Law:

M = $\frac{\mathrm{m}}{\mathrm{n}}$ = $\frac{\mathrm{mRT}}{\mathrm{PV}}$

m = mass (in g)
R = 8.314 J.mol-1.K-1
T = temperature (in K)
P = pressure (in Pa)
V = volume (in m3)

Density ρ (in g.m-3) in the Ideal Gas Law:

ρ = $\frac{\mathrm{m}}{\mathrm{V}}$ = $\frac{\mathrm{PM}}{\mathrm{RT}}$

P = pressure (in Pa)
M = molar mass (in g.mol-1)
R = 8.314 J.mol-1.K-1
T = temperature (in K)

## Dalton’s Law of Partial Pressures

Dalton’s Law: at low pressures, the total pressure of a mixture of ideal gases is equal to the sum of the partial pressures of all the gases in the mixture

Ptot = PA + PB + PC + … + Pi

## Kinetic Theory and Speed Distribution

Kinetic energy of a gas molecule (in J):

Ek = $\frac{1}{2}$mv2

m = mass of the molecule (in kg)
v = velocity (in m.s-1)

Kinetic energy of a mole of gas molecules (in J.mol-1):

Ek = $\frac{1}{2}$Mkgv2

Mkg = molar mass (in kg.mol-1)
v = velocity (in m.s-1)

Kinetic theory of gases postulates:

• The gas molecules are constantly moving in random directions with a distribution of speeds

They collide randomly with one another and the walls of the container

• All collisions of molecules with the walls of the container are elastic ⇒ no energy is lost during a collision
• The average distance between the molecules in a gas is much larger than the size of the molecules ⇒ most of the gas is empty space
• The molecules exert no attractive or repulsive forces on each other except during collisions ⇒ between collisions they move in straight lines
• The mean kinetic energy of the gas molecules $\overline{){E}_{k}}$ is proportional to the temperature

$\overline{){E}_{k}}$ = $\frac{1}{2}$Mkg$\overline{{\mathrm{v}}^{2}}$ = αT

A detailed analysis of the collisions shows:

α = $\frac{3}{2}$R   ⇒    $\overline{{\mathrm{v}}^{2}}$ = $\frac{3\mathrm{RT}}{{\mathrm{M}}_{\mathrm{kg}}}$

Root-mean-square speed vrms (in m.s-1): square root of the mean of the square of the molecular speeds

vrms $\sqrt{\left(\overline{{\mathrm{v}}^{2}}\right)}$ ​​​​​= $\sqrt{\frac{3\mathrm{RT}}{{\mathrm{M}}_{\mathrm{kg}}}}$

R = molar gas constant = 8.314 J.K-1.mol-1
T = temperature (in K)
Mkg = molar mass of the gas (in kg.mol-1)

## Graham’s Law of Effusion

Diffusion: gradual dispersal of one gas through another (equal pressure process)
Effusion: escape of a gas through one or more small holes from a region of high pressure to a region of lower pressure

Graham’s Law: for 2 gases at the same temp. and pressure, the rate of effusion is directly proportional to vrms:

rate = α vrms = α $\sqrt{\frac{3\mathrm{RT}}{{\mathrm{M}}_{\mathrm{kg}}}}$

α = proportionality coefficient

The formula mass of a gas can be determined by using effusion:

$\frac{{\mathrm{rate}}_{\mathrm{A}}}{{\mathrm{rate}}_{\mathrm{B}}}$ = $\sqrt{\frac{{\mathrm{M}}_{\mathrm{B}}}{{\mathrm{M}}_{\mathrm{A}}}}$

MA = molar mass of the gas A (in kg.mol-1)
MB = molar mass of the gas B (in kg.mol-1)

## Mean Free Path

Mean Free Path l (in m): average distance a molecule travels between collisions

I =

R = molar gas constant = 8.314 J.K-1.mol-1
T = temperature (in K)
d = diameter of the gas molecule (in m)
NA = Avogadro’s number = 6.022 x 1023
P = pressure (in Pa)

Collision frequency z (in collisions.s-1): number of collisions that a molecule undergoes per second

z = $\frac{{\mathrm{v}}_{\mathrm{rms}}}{\mathrm{I}}$

vrms = root-mean-square speed (in m.s-1)
l = mean free path (in m)

## Van der Waals Equation

Equation for non-ideal gas ⇒ it accounts for deviations from Gas Ideality

In the ideal-gas equation:

• V is substituted by V – nb, where b = cst
• P is substituted by P + a , where a = cst

The 2 constants a and b depend upon the particular gas and are called van der Waals constants

Van der Waals equation:

(P + a $\frac{{\mathrm{n}}^{2}}{{\mathrm{V}}^{2}}$)(V - nb) = nRT

P = pressure (in Pa)
V = volume (in m3)
n = number of moles (in mol)
R = molar gas constant = 8.314 J.K-1.mol-1
T = temperature (in K)