Chapter 6: Gases – Properties, Laws, and Applications

Chapter 6: Gases

1. Review of Gases

Gases are one of the fundamental states of matter, characterized by their unique physical properties and behaviors. Understanding gases is essential for explaining phenomena ranging from atmospheric pressure to chemical reactions in the air.

• Characteristics of Gases:

  • Gases have no fixed shape or volume; they expand to fill their container.

  • They are highly compressible compared to solids and liquids.

  • Gas molecules move rapidly and randomly.

  • Gases mix evenly and completely when confined to the same container.

  • Low density compared to solids and liquids.

• Common Gas Molecules: Examples include H2, He, Ne, O2, F2, Cl2, CO2, NO, N2O, NO2, HCl, HCN, CO, NH3.

• Pressure: Pressure is a key parameter for gases, defined as the force exerted per unit area by gas molecules colliding with surfaces.

  • Pressure is created by the constant, random motion of gas molecules striking the walls of their container.

2. Measuring Pressure

Pressure is measured as the force exerted by gas molecules per unit area. Several devices are used to measure gas pressure:

• Barometer (Closed): Measures atmospheric pressure using a column of mercury or another liquid.

• Manometer (Open): Measures the pressure of a gas in a container relative to atmospheric pressure.

• Units of Pressure: Common units include atmospheres (atm), millimeters of mercury (mmHg or torr), and pascals (Pa).

  • 1 atm = 760 mmHg = 101,325 Pa

Example: If the height of the mercury column in a barometer decreases, the atmospheric pressure is lower. In a manometer, the difference in liquid height indicates the pressure of the gas relative to the atmosphere.

3. Standard Temperature and Pressure (STP) and Units

STP is a reference point for gas measurements, defined as 0°C (273.15 K) and 1 atm pressure. It is used for comparing gas volumes and calculations.

• STP Conditions: 0°C (273.15 K), 1 atm

• Conversions: 1 atm = 760 mmHg = 101.325 kPa

4. Basic Gas Relationships (Gas Laws) – Creating PV = nRT

The behavior of gases can be described by several empirical laws, which are combined into the Ideal Gas Law.

• Boyle's Law: At constant temperature (T) and amount of gas (n), the pressure (P) and volume (V) of a gas are inversely proportional.

  • As P increases, V decreases (T, n constant).

  • Mathematically: $P_1V_1 = P_2V_2$

• Charles's Law: At constant pressure (P) and amount of gas (n), the volume (V) of a gas is directly proportional to its temperature (T, in Kelvin).

  • As T increases, V increases (P, n constant).

  • Mathematically: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$

• Avogadro's Law: At constant temperature and pressure, the volume (V) of a gas is directly proportional to the number of moles (n).

  • As n increases, V increases (P, T constant).

  • Mathematically: $\frac{V_1}{n_1} = \frac{V_2}{n_2}$

• Ideal Gas Law: Combines the above relationships into a single equation:

  • $PV = nRT$

  • Where P = pressure, V = volume, n = moles, R = gas constant (0.0821 L·atm·mol−1·K−1), T = temperature in Kelvin.

Example: If you double the temperature of a gas at constant pressure, its volume will also double (Charles's Law).

5. Ideal Gas Assumptions

The Ideal Gas Law is based on several assumptions about the nature of gases:

• Gas particles have negligible volume compared to the container.

• Gas particles experience no intermolecular attractions or repulsions.

• Gas particles are in constant, random motion.

• Collisions between gas particles and with the container walls are perfectly elastic.

6. Applications of the Ideal Gas Law

• Density of a Gas: The density (d) can be derived from the ideal gas law:

  • $d = \frac{PM}{RT}$

  • Where M = molar mass of the gas.

• Molar Mass of a Gas: Rearranging the ideal gas law allows calculation of molar mass:

  • $M = \frac{dRT}{P}$

• Gas Stoichiometry: The ideal gas law can be used to relate volumes of gases in chemical reactions, especially at STP (1 mol gas = 22.4 L at STP).

7. Problems Using Gas Law Equations

Gas law equations are used to solve a variety of problems, such as calculating the mass, volume, or identity of a gas under specific conditions.

• Identify which variables are constant and which are changing to select the appropriate equation (e.g., $PV = nRT$, $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$).

• Common problem types include:

  • Finding the mass of a gas at given P, V, T.

  • Calculating the new volume of a gas after a change in pressure or temperature.

  • Determining the identity of a gas from its density and conditions.

  • Calculating the molecular formula from percent composition and density.

  • Finding the total volume of products in a reaction at given conditions.

8. Dalton's Law of Partial Pressures

Dalton's Law states that in a mixture of non-reacting gases, the total pressure is the sum of the partial pressures of each individual gas.

• Partial Pressure: The pressure exerted by each gas in a mixture as if it were alone in the container.

• Mathematically: $P_{total} = P_1 + P_2 + P_3 + \ldots$

• Mole Fraction (X): The ratio of the number of moles of a component to the total number of moles in the mixture.

  • $X_A = \frac{n_A}{n_{total}}$

  • $P_A = X_A \times P_{total}$

Example: If a mixture contains 2 moles of N2 and 3 moles of H2, the mole fraction of N2 is 2/5.

9. Non-Ideal Gases: Deviations from Ideality

Real gases deviate from ideal behavior at high pressures and low temperatures due to intermolecular forces and the finite volume of gas particles.

• Best Conditions for Ideal Behavior: High temperature and low pressure.

• Deviations:

  • At high pressure, gas particles are closer together, so their volume and intermolecular attractions become significant.

  • At low temperature, attractive forces cause gases to condense.

• Compressibility Factor (Z): $Z = \frac{PV}{nRT}$; for an ideal gas, Z = 1.

10. The van der Waals Equation

The van der Waals equation corrects the ideal gas law for intermolecular attractions and the finite volume of gas particles:

• Equation:

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

  • a = correction for intermolecular attractions

  • b = correction for finite volume of particles

• Significance of a and b:

  • As 'a' increases, intermolecular attractions are stronger.

  • As 'b' increases, the size of the gas molecules is larger.

Gas	a (L2·atm/mol2)	b (L/mol)
He	0.034	0.0237
O2	1.36	0.0318
CO2	3.59	0.0427
NH3	4.17	0.0371
CH4	2.25	0.0428
H2	0.244	0.0266
N2	1.39	0.0391

Example: Calculate the pressure of a real gas using both the ideal gas law and the van der Waals equation to compare results.

11. Kinetic Molecular Theory (KMT)

KMT explains the behavior of gases based on the motion of their particles.

• Gas particles are in constant, random motion.

• Collisions are perfectly elastic.

• No attractive or repulsive forces between particles.

• Average kinetic energy is proportional to temperature (in Kelvin):

  • $KE_{avg} = \frac{3}{2}RT$ (per mole)

• Root Mean Square Speed:

  • $u_{rms} = \sqrt{\frac{3RT}{M}}$

  • Where M = molar mass in kg/mol, R = 8.314 J·mol−1·K−1, T = temperature in K.

Diffusion: The movement of gas molecules from high to low concentration. Effusion: The escape of gas molecules through a small hole.

• Graham's Law of Effusion:

  • Rate of effusion is inversely proportional to the square root of molar mass:

  • $\text{Rate}_1/\text{Rate}_2 = \sqrt{M_2/M_1}$

12. Practice and Applications

• Practice problems involve manipulating the gas laws to solve for unknowns, interpreting graphs, and applying concepts to real-world scenarios.

• Stoichiometry with gases often uses the ideal gas law to relate moles and volumes.

• Dalton's Law is used to calculate partial pressures and mole fractions in mixtures.

• Non-ideal behavior is analyzed using the van der Waals equation and compressibility factors.

Summary Table: Key Gas Laws and Equations

Law	Equation	Variables Held Constant
Boyle's Law	$P_1V_1 = P_2V_2$	n, T
Charles's Law	$\frac{V_1}{T_1} = \frac{V_2}{T_2}$	n, P
Avogadro's Law	$\frac{V_1}{n_1} = \frac{V_2}{n_2}$	P, T
Ideal Gas Law	$PV = nRT$	None
Dalton's Law	$P_{total} = \sum P_i$	V, T
Graham's Law	$\text{Rate}_1/\text{Rate}_2 = \sqrt{M_2/M_1}$	P, T
van der Waals	$(P + a\frac{n^2}{V^2})(V - nb) = nRT$	None

Additional info: These notes include both conceptual explanations and practical applications, with emphasis on the relationships between pressure, volume, temperature, and amount of gas. The van der Waals equation and kinetic molecular theory provide insight into real gas behavior and molecular motion, respectively.