# Properties of Gases | General Chemistry 2

## 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 à 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 m^{2})

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 10^{5} Pa

## The 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

P_{1 }V_{1} = P_{2} V_{2} (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)

**Avogadro’s Law (V vs n):**

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 m^{3})

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.

Best ideal gas: monatomic, low pressure, high temperature.

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

Molar Volume V_{m} (in L.mol^{-1}) in the Ideal Gas Law:

V_{m} = $\frac{\mathrm{V}}{\mathrm{n}}$ = $\frac{\mathrm{RT}}{\mathrm{P}}$

R = 8.314 J.mol^{-1}.K^{-1}

T = temperature (in K)

P = pressure (in Pa)

At standard temperature and pressure (STP): V_{m} = 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 m^{3})

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 the sum of the partial pressures of all the gases in the mixture.

P_{tot} = P_{A} + P_{B} + P_{C} + … + P_{i} = $\sum _{}{P}_{i}$

## Kinetic Theory and Speed Distribution

Kinetic energy of a gas molecule (in J):

E_{k} = $\frac{1}{2}$mv^{2}

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}):

E_{k} = $\frac{1}{2}$M_{kg}v^{2}

M_{kg} = 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

$\overline{){E}_{k}}$ = $\frac{1}{2}$M_{kg}$\overline{{\mathrm{v}}^{2}}$ = 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 v_{rms} (in m.s^{-1}): square root of the mean of the square of the molecular speeds

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

R = molar gas constant = 8.314 J.K^{-1}.mol^{-1}

T = temperature (in K)

M_{kg} = 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 v_{rms}:

rate = α v_{rms} = 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:

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

M_{A} = molar mass of the gas A (in kg.mol^{-1})

M_{B} = 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 = $\frac{\mathrm{RT}}{\mathrm{\pi}\sqrt{2}{\mathrm{d}}^{2}{\mathrm{N}}_{\mathrm{A}}\mathrm{P}}$

R = molar gas constant = 8.314 J.K^{-1}.mol^{-1}

T = temperature (in K)

d = diameter of the gas molecule (in m)

N_{A} = Avogadro’s number = 6.022 x 10^{23}

P = pressure (in Pa)

Collision frequency z (in collisions.s^{-1}):

number of collisions that a molecule undergoes per second.

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

v_{rms} = 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 m^{3})

n = number of moles (in mol)

R = molar gas constant = 8.314 J.K^{-1}.mol^{-1}

T = temperature (in K)