Ideal Gas Law and Gas Properties

The Ideal Gas Law

The most important equation in gas chemistry:

$PV = nRT$

Where:

• $P$ = pressure (usually atm, but can be kPa, mmHg, torr)
• $V$ = volume (usually L)
• $n$ = number of moles
• $R$ = ideal gas constant
• $T$ = temperature (must be in Kelvin!)

Gas Constant (R) Values

Most common:

$R = 0.0821 \frac{\text{L·atm}}{\text{mol·K}}$

Other values (less common):

$R = 8.314 \frac{\text{J}}{\text{mol·K}} = 8.314 \frac{\text{kPa·L}}{\text{mol·K}}$

$R = 62.4 \frac{\text{L·mmHg}}{\text{mol·K}}$

Choose R based on units of pressure and volume!

Standard Temperature and Pressure (STP)

Definition:

• Temperature: 0°C = 273.15 K
• Pressure: 1 atm = 101.3 kPa = 760 mmHg = 760 torr

At STP, one mole of any ideal gas occupies:

$V = 22.4 \text{ L}$

This is a useful conversion factor!

Using PV = nRT

The ideal gas law connects all four variables

If you know any 3, you can find the 4th:

Example problems:

1. Know P, V, T → find n (moles)
2. Know n, T, P → find V (volume)
3. Know n, V, P → find T (temperature)
4. Know n, V, T → find P (pressure)

Key conversions:

• Temperature: °C + 273.15 = K
• Pressure: 1 atm = 760 mmHg = 101.3 kPa
• Volume: Usually given in L (if not, convert!)

Gas Density and Molar Mass

Density of a gas can be related to molar mass

Starting with PV = nRT:

$n = \frac{m}{M}$

Where:

• $m$ = mass (g)
• $M$ = molar mass (g/mol)

Substitute into ideal gas law:

$PV = \frac{m}{M}RT$

Rearrange for density:

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

$d = \frac{PM}{RT}$

Key relationships:

• Higher molar mass → Higher density (at same T and P)
• Higher temperature → Lower density (at same P and M)
• Higher pressure → Higher density (at same T and M)

Why this is useful:

• Calculate molar mass from density measurements
• Predict relative densities of different gases
• Explain why hot air rises (lower density)

Combined Gas Law

When amount of gas (n) is constant:

$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$

This combines:

• Boyle's Law: $P_1V_1 = P_2V_2$ (constant T and n)
• Charles's Law: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$ (constant P and n)
• Gay-Lussac's Law: $\frac{P_1}{T_1} = \frac{P_2}{T_2}$ (constant V and n)

Use when:

• Gas undergoes change in conditions
• Amount of gas stays the same
• Need to find final state from initial state

Special cases:

Constant temperature (isothermal):

$P_1V_1 = P_2V_2$ (Boyle's Law)

Constant pressure (isobaric):

$\frac{V_1}{T_1} = \frac{V_2}{T_2}$ (Charles's Law)

Constant volume (isochoric):

$\frac{P_1}{T_1} = \frac{P_2}{T_2}$ (Gay-Lussac's Law)

Dalton's Law of Partial Pressures

In a mixture of gases, each gas exerts pressure independently

Total pressure = Sum of partial pressures:

$P_{total} = P_1 + P_2 + P_3 + ...$

For each gas in the mixture:

$P_i = \chi_i \cdot P_{total}$

Where:

• $P_i$ = partial pressure of gas i
• $\chi_i$ = mole fraction of gas i
• $P_{total}$ = total pressure

Mole fraction:

$\chi_i = \frac{n_i}{n_{total}}$

Key concept: Each gas behaves as if it alone occupies the entire volume

Application: Collecting Gas Over Water

Common lab technique:

• Gas produced in reaction bubbles through water
• Collected gas is mixture of desired gas + water vapor

Pressure relationship:

$P_{total} = P_{gas} + P_{H_2O}$

Where:

• $P_{total}$ = atmospheric pressure
• $P_{gas}$ = partial pressure of collected gas
• $P_{H_2O}$ = vapor pressure of water (depends on temperature)

To find amount of dry gas:

$P_{gas} = P_{total} - P_{H_2O}$

Then use ideal gas law with $P_{gas}$:

$n_{gas} = \frac{P_{gas}V}{RT}$

Water vapor pressure at common temperatures:

| Temp (°C) | P(H₂O) mmHg | |-----------|-------------| | 20 | 17.5 | | 25 | 23.8 | | 30 | 31.8 | | 35 | 42.2 |

Kinetic Molecular Theory (KMT)

Model explaining ideal gas behavior

Five Postulates

1. Gas particles are in constant, random motion

• Particles move in straight lines until collision
• Collisions are perfectly elastic (no energy lost)

2. Gas particles have negligible volume

• Volume of particles << volume of container
• Most of gas volume is empty space

3. No attractive or repulsive forces between particles

• Particles don't interact except during collisions
• Move independently

4. Average kinetic energy proportional to absolute temperature

$KE_{avg} = \frac{3}{2}RT$

Or for a single molecule:

$KE = \frac{1}{2}mv^2 = \frac{3}{2}k_BT$

Where:

• $m$ = mass of particle
• $v$ = velocity
• $k_B$ = Boltzmann constant = $R/N_A$

Higher temperature → Higher average KE → Faster particles

5. Collisions are perfectly elastic

• Total kinetic energy conserved
• Energy transferred between particles
• No energy lost to heat, sound, etc.

Implications of KMT

Temperature and KE:

• At same T, all gases have same average KE
• Lighter gases move faster (same KE, less mass)

Root-mean-square speed:

$v_{rms} = \sqrt{\frac{3RT}{M}}$

Where:

• $R$ = 8.314 J/(mol·K)
• $T$ = temperature (K)
• $M$ = molar mass (kg/mol)

Relationships:

• $v_{rms} \propto \sqrt{T}$ (higher T → faster)
• $v_{rms} \propto \frac{1}{\sqrt{M}}$ (lighter → faster)

At same temperature:

• He atoms move faster than O₂ molecules
• All have same average KE

Maxwell-Boltzmann Distribution

Not all particles have same speed!

Distribution of molecular speeds:

• Some particles very slow
• Most particles near average
• Some particles very fast
• Shape depends on temperature and molar mass

Effects of temperature:

• Higher T → distribution shifts right (faster average)
• Higher T → distribution broadens (wider range)
• Peak becomes lower and flatter

Effects of molar mass:

• Lower M → distribution shifts right (faster)
• Lower M → distribution broadens

Effusion and Diffusion

Graham's Law of Effusion

Effusion: Gas escapes through tiny hole into vacuum

Rate of effusion inversely proportional to square root of molar mass:

$\frac{\text{rate}_1}{\text{rate}_2} = \sqrt{\frac{M_2}{M_1}}$

Lighter gases effuse faster

Example: He effuses faster than N₂

$\frac{\text{rate}_{He}}{\text{rate}_{N_2}} = \sqrt{\frac{28}{4}} = \sqrt{7} \approx 2.65$

Helium effuses 2.65 times faster than nitrogen

Diffusion

Diffusion: Gas spreads through space or through another gas

Also follows Graham's Law (approximately):

• Lighter gases diffuse faster
• Same mathematical relationship

Example: Smell of perfume spreading across room

Why faster with lighter gases?

• Lighter → higher average velocity (from KMT)
• Higher velocity → spreads faster

Real Gases vs. Ideal Gases

When Do Real Gases Deviate from Ideal Behavior?

Ideal gas law assumes:

1. No volume (particles are points)
2. No intermolecular forces

Real gases deviate when:

1. High pressure

• Gas particles forced close together
• Volume of particles becomes significant
• $V_{real} > V_{ideal}$

2. Low temperature

• Particles move slowly
• IMFs have more effect
• Particles attracted to each other
• $P_{real} < P_{ideal}$

General rule:

Real gases behave most ideally at:

• Low pressure (particles far apart)
• High temperature (high KE overcomes IMFs)

Real gases deviate most at:

• High pressure (particles close)
• Low temperature (IMFs significant)

Van der Waals Equation

Correction to ideal gas law for real gases:

$\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT$

Pressure correction: $\frac{an^2}{V^2}$

• Accounts for attractive forces
• Reduces effective pressure
• $a$ = strength of IMFs (larger for polar molecules)

Volume correction: $nb$

• Accounts for volume of particles
• Reduces available volume
• $b$ = size of particles (larger for bigger molecules)

Constants a and b:

• Specific to each gas
• Larger a → stronger IMFs
• Larger b → larger molecules

When van der Waals ≈ Ideal:

• Low pressure (correction terms become small)
• High temperature (KE >> IMFs)

Gas Stoichiometry

Using ideal gas law in chemical reactions

Mole Relationships

For reactions involving gases:

$\ce{aA(g) + bB(g) -> cC(g) + dD(g)}$

Mole ratios from balanced equation:

• At same T and P, volume ratios = mole ratios
• This is Avogadro's Law

Example:

$\ce{2H2(g) + O2(g) -> 2H2O(g)}$

At same T and P:

• 2 volumes H₂ + 1 volume O₂ → 2 volumes H₂O
• 2 moles H₂ + 1 mole O₂ → 2 moles H₂O

Problem-Solving Strategy

Given: Conditions of gas (P, V, T) and reaction

Steps:

1. Use PV = nRT to find moles of given gas
2. Use stoichiometry (mole ratios) to find moles of desired gas
3. Use PV = nRT to find conditions of desired gas

Or use directly:

$\frac{P_1V_1}{n_1T_1} = \frac{P_2V_2}{n_2T_2}$

With $\frac{n_1}{n_2}$ from stoichiometry

Summary of Gas Laws

| Law | Equation | Constant | Relationship | |-----|----------|----------|--------------| | Boyle's | $P_1V_1 = P_2V_2$ | T, n | Inverse (P↑V↓) | | Charles's | $\frac{V_1}{T_1} = \frac{V_2}{T_2}$ | P, n | Direct (T↑V↑) | | Gay-Lussac's | $\frac{P_1}{T_1} = \frac{P_2}{T_2}$ | V, n | Direct (T↑P↑) | | Avogadro's | $\frac{V_1}{n_1} = \frac{V_2}{n_2}$ | P, T | Direct (n↑V↑) | | Combined | $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$ | n | - | | Ideal | $PV = nRT$ | - | All variables | | Dalton's | $P_T = P_1 + P_2 + ...$ | - | Partial pressures | | Graham's | $\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}$ | - | Effusion rate |

Key Problem-Solving Tips

Always convert:

• Temperature to Kelvin (K = °C + 273.15)
• Pressure to match R (usually atm)
• Volume to match R (usually L)

Choose correct R:

• Match units to pressure and volume
• Most common: 0.0821 L·atm/(mol·K)

Partial pressures:

• Use mole fractions
• Account for water vapor when collecting gas

Real vs ideal:

• Real gases → use at low P, high T
• Deviations at high P, low T

Gas stoiometry:

• Convert to moles first
• Use mole ratios
• Convert back to gas conditions