Study Notes: Properties and Laws of Gases (General Chemistry, Chapter 7)

Gases: Properties and Basic Concepts

Composition and Nature of Gases

Gases are one of the fundamental states of matter, characterized by their ability to expand and fill any container. The Earth's atmosphere is primarily composed of nitrogen and oxygen, with smaller amounts of other gases.

• Major components of air: 78% N2, 21% O2, and 1% other gases.

• Common diatomic molecules: N2, O2, Cl2, F2, H2.

• Group 8 elements (noble gases): Exist as monoatomic gases (e.g., He, Ne, Ar).

Physical Characteristics of Gases

• Gases assume the volume and shape of their containers.

• They are the most compressible state of matter.

• Gases mix evenly and completely when confined together.

• They have much lower densities than liquids and solids.

• Gas molecules are in constant motion and exert pressure on surfaces they contact.

Gas Pressure and Measurement

Definition and Units of Pressure

Pressure is the force exerted per unit area by gas molecules colliding with surfaces.

• SI unit: Pascal (Pa), where 1 Pa = 1 N/m2

• Other common units: 1 atm = 760 mmHg = 760 torr = 101,325 Pa

Atmospheric Pressure and Barometers

• Atmospheric pressure: The pressure exerted by Earth's atmosphere.

• Barometer: Instrument used to measure atmospheric pressure.

• Standard atmospheric pressure is the pressure that supports a column of mercury 760 mm high at 0°C.

Manometers

Manometers are devices used to measure the pressure of gases other than the atmosphere.

• Closed-end manometer: Measures pressures below atmospheric pressure.

• Open-end manometer: Can measure both above and below atmospheric pressure by comparing the height difference (h) of mercury columns.

Type	Purpose	Pressure Calculation
Closed-end	Below atmospheric	$P_{gas} = h$
Open-end (Pgas < Patm)	Below atmospheric	$P_{gas} = P_{atm} - h$
Open-end (Pgas > Patm)	Above atmospheric	$P_{gas} = P_{atm} + h$

Gas Laws

Boyle’s Law: Pressure-Volume Relationship

At constant temperature, the volume of a fixed amount of gas is inversely proportional to its pressure.

• Mathematical form: $P \propto \frac{1}{V}$ or $PV = k_1$

• For two conditions: $P_1V_1 = P_2V_2$

• Example: If 946 mL of Cl2 at 726 mmHg is compressed to 154 mL at constant temperature, $P_2 = \frac{726 \times 946}{154} = 4460$ mmHg.

Charles’s Law: Volume-Temperature Relationship

At constant pressure, the volume of a fixed amount of gas is directly proportional to its absolute temperature (in Kelvin).

• Mathematical form: $V \propto T$ or $\frac{V}{T} = k_2$

• For two conditions: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$

• Example: 3.2 L CO at 125°C (398 K) will occupy 1.54 L at $T_2 = 191.5$ K if pressure is constant.

Avogadro’s Law: Volume-Mole Relationship

At constant temperature and pressure, the volume of a gas is directly proportional to the number of moles present.

• Mathematical form: $V \propto n$ or $\frac{V}{n} = k_3$

Combined Gas Law

The combined gas law relates pressure, volume, and temperature for a fixed amount of gas.

• Equation: $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$

• Use when the amount of gas is constant but conditions change.

Ideal Gas Law

The ideal gas law combines Boyle’s, Charles’s, and Avogadro’s laws into a single equation.

• Equation: $PV = nRT$

• R (ideal gas constant): $0.08206$ L·atm/(mol·K) or $8.314$ J/(mol·K)

• Standard Temperature and Pressure (STP): 0°C (273.15 K) and 1 atm; 1 mol of ideal gas occupies 22.4 L at STP.

• Example: Calculate the volume occupied by 2.12 mol NO at 6.54 atm and 76°C: $V = \frac{nRT}{P} = \frac{2.12 \times 0.08206 \times (273+76)}{6.54} = 9.3$ L

Density and Molar Mass of Gases

The ideal gas law can be rearranged to calculate the density or molar mass of a gas.

• Density: $d = \frac{PM}{RT}$, where M is molar mass.

• Example: Density of UF6 at 779 mmHg and 62°C: Convert pressure to atm: $779/760 = 1.025$ atm Molar mass: $238 + (19 \times 6) = 352$ g/mol Temperature: $62 + 273 = 335$ K $d = \frac{1.025 \times 352}{0.08206 \times 335} = 13.1$ g/L

Stoichiometry Involving Gases

Gas volumes in chemical reactions can be related using the ideal gas law and stoichiometric coefficients.

• At constant T and P, the volume ratios of gases are the same as the mole ratios in the balanced equation.

• Example: $2C_4H_{10} + 13O_2 \rightarrow 8CO_2 + 10H_2O$ To combust 14.9 L of butane, $14.9 \times \frac{13}{2} = 96.85$ L of $O_2$ required.

Mixtures of Gases and Partial Pressures

Dalton’s Law of Partial Pressures

In a mixture, each gas exerts a pressure as if it were alone; the total pressure is the sum of partial pressures.

• Equation: $P_{total} = P_1 + P_2 + ... + P_n$

• Partial pressure: $P_i = X_i P_{total}$, where $X_i$ is the mole fraction of component $i$.

• Mole fraction: $X_i = \frac{n_i}{n_{total}}$

• Example: If a sample contains 8.24 mol CH4, 0.421 mol C2H6, and 0.116 mol C3H8 at 1.37 atm, calculate each partial pressure using mole fractions.

Kinetic Molecular Theory of Gases

This theory explains the behavior of gases at the molecular level.

• Gases consist of small particles in constant, random motion.

• Collisions between particles and with container walls are elastic (no loss of kinetic energy).

• Gas particles are very small compared to the distances between them.

• The average kinetic energy of gas particles is proportional to the absolute temperature (Kelvin).

Implications of Kinetic Theory

• Compressibility: Gases are compressible because of the large spaces between particles.

• Boyle’s Law: Pressure increases as volume decreases due to more frequent collisions.

• Charles’s Law: As temperature increases, kinetic energy and collision frequency increase, expanding volume.

• Avogadro’s Law: More particles (moles) increase the number of collisions, raising pressure or volume.

Root-Mean-Square (rms) Speed

The rms speed is a measure of the average speed of gas molecules.

• Equation: $u_{rms} = \sqrt{\frac{3RT}{M}}$ where M is molar mass in kg/mol.

• Example: At -23°C (250 K), $u_{rms}$ for N2 ($M = 0.028$ kg/mol): $u_{rms} = \sqrt{\frac{3 \times 8.314 \times 250}{0.028}} = 472$ m/s

Summary Table: Gas Laws and Their Relationships

Law	Relationship	Equation	Constant
Boyle's Law	P vs. V (T, n constant)	$PV = k$	T, n
Charles's Law	V vs. T (P, n constant)	$\frac{V}{T} = k$	P, n
Avogadro's Law	V vs. n (P, T constant)	$\frac{V}{n} = k$	P, T
Combined Gas Law	P, V, T (n constant)	$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$	n
Ideal Gas Law	P, V, n, T	$PV = nRT$	R

Key Terms

• Pressure (P): Force per unit area exerted by gas molecules.

• Barometer: Device to measure atmospheric pressure.

• Manometer: Device to measure pressure of a gas sample.

• Partial Pressure: Pressure exerted by a single component in a mixture.

• Mole Fraction (X): Ratio of moles of one component to total moles.

• Root-mean-square speed (urms): Measure of average molecular speed.

Additional info: Some examples and explanations have been expanded for clarity and completeness. The summary table and key terms were added for self-contained review.