Chapter 5: Detailed Study Notes on Gases and the Properties of Gases

General Properties and Pressure

• Nature of Gases: The properties and behaviors of gases are significantly easier to understand compared to other phases of matter like solids or liquids. While gases may exhibit very different chemical properties from one another, they share very similar physical properties.

• Fundamental Variables: Four variables are interrelated in the study of gases: Pressure ($P$), Volume ($V$), Temperature ($T$), and amount ($n$).

• Definition of Pressure ($P$): Pressure is defined as the force ($F$) exerted per unit area ($A$). The mathematical formula is:   $P = \frac{F}{A}$

• SI Units of Pressure: The standard SI unit for pressure is the Pascal ($Pa$). It is defined as:   $1\,Pa = 1\,kg \cdot m^{-1} \cdot s^{-2} = 1\,kg/(m \cdot s^2)$   $1\,Pa = 1\,N/m^2$

• Pressure Example: An area of $0.0616\,m^2$ exerts a pressure of $1 \times 10^5\,Pa$.

• Measuring Pressure (The Barometer): A barometer is used to measure atmospheric pressure. Mercury ($Hg$) is typically used in the column. At standard atmospheric pressure, the height of the mercury column is $760.\,mm\,Hg$.

• Standard Atmospheric Pressure Equivalents:   $1\,atm = 760.\,mm\,Hg = 760.\,torr = 1.01325 \times 10^5\,Pa$

The Empirical Gas Laws

• General Behavior: All gases behave simply regarding pressure, temperature, volume, and molar amount. By holding two of these properties constant, scientists identified simple relationships between the remaining two. These studies took place from the mid-17th to mid-19th centuries.

• Boyle’s Law (Volume and Pressure):     * Definition: The volume of a sample of gas at a constant temperature varies inversely with the applied pressure.     * Observations: At $1\,atm$, volume is $100\,mL$. When pressure is doubled ($2\,atm$), volume halves to $50\,mL$. When pressure is tripled, volume decreases to one-third ($33\,mL$).     * Relationships:       $V \propto \frac{1}{P}$       $PV = \text{constant}$       $P_1V_1 = P_2V_2$

• Charles’s Law (Volume and Temperature):     * Absolute Zero: The temperature $-273.15^{\circ}C$ is known as absolute zero. It is the hypothetical temperature at which the volume of a gas would be zero. This is the foundation of the Kelvin ($K$) absolute temperature scale.     * Definition: The volume of a sample of gas at constant pressure is directly proportional to the absolute temperature in Kelvin.     * Relationships:       $V \propto T$       $\frac{V}{T} = \text{constant}$       $\frac{V_1}{T_1} = \frac{V_2}{T_2}$

• Avogadro’s Law (Volume and Molar Amount):     * Definition: Equal volumes of any two gases at the same temperature and pressure contain the same number of molecules.     * Molecular Comparison at STP ($1\,atm, 0^{\circ}C$):         * Argon ($Ar$): Volume $1\,L$, Mass $1.783\,g$, Molecules $2.688 \times 10^{22}$.         * Oxygen ($O_2$): Volume $1\,L$, Mass $1.428\,g$, Molecules $2.688 \times 10^{22}$.         * Carbon Dioxide ($CO_2$): Volume $1\,L$, Mass $1.964\,g$, Molecules $2.688 \times 10^{22}$.     * Relationship:       $V \propto n$

Standard Temperature and Pressure (STP)

• Reference Conditions: Chosen by convention to be exactly $0^{\circ}C$ ($273.15\,K$) and $1\,atm$ pressure.

• Molar Volume ($V_m$): The volume occupied by one mole of an ideal gas at STP is exactly $22.4\,L/mol$. For visual reference, a basketball is smaller than a cube of $22.4\,L$.

The Ideal Gas Law

• Derivation: By combining the empirical laws ($V \propto 1/P$, $V \propto T$, and $V \propto n$), we get the combined proportionality:   $V \propto \frac{nT}{P}$

• The Equation: By introducing the gas constant ($R$), the Ideal Gas Law is established:   $PV = nRT$

• Ideal Gas Definition: A gas that perfectly follows this equation under all conditions.

• The Ideal Gas Constant ($R$): The value commonly used for variables in $atm$, $L$, and $K$ is:   $R = 0.08206\,L \cdot atm \cdot mol^{-1} \cdot K^{-1}$

• Calculating Mass (Nitrogen Cylinder Example):     * Given: $V = 50.0\,L$, $P = 17.1\,atm$, $T = 23^{\circ}C = 296\,K$.     * Calculate moles: $n = \frac{PV}{RT} = \frac{(17.1\,atm)(50.0\,L)}{(0.08206\,L \cdot atm \cdot mol^{-1} \cdot K^{-1})(296\,K)} = 35.2\,mol$.     * Calculate mass: $mass = 35.2\,mol \times 28.02\,g/mol = 986\,g$.

Derived Gas Laws and Combined Gas Law

• Deriving Empirical Laws from Ideal Gas Law: Because $PV/nT = R$, and $R$ is constant, we can set ratios for two states equal to each other:   $\frac{P_1V_1}{n_1T_1} = \frac{P_2V_2}{n_2T_2}$     * Boyle’s Law (constant $n, T$): $P_1V_1 = P_2V_2$     * Charles’s Law (constant $n, P$): $\frac{V_1}{T_1} = \frac{V_2}{T_2}$

• Combined Gas Law: Used when pressure, volume, and temperature all change for a fixed amount of gas:   $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$     * Ascending Balloon Example:         * Initial: $V_1 = 6.0\,L$, $P_1 = 1.0\,atm$, $T_1 = 25.0^{\circ}C = 298\,K$.         * Final: $P_2 = 0.23\,atm$, $T_2 = -23.0^{\circ}C = 250.\,K$.         * Calculation: $\frac{(1.0\,atm)(6.0\,L)}{298\,K} = \frac{(0.23\,atm)(V_2)}{250.\,K}$         * Result: $V_2 = 22\,L$.

Gas Density and Molar Mass

• Molar Concentration: From $PV = nRT$, isolating $n/V$ gives molarity:   $\frac{n}{V} = \frac{P}{RT}$

• Molar Mass ($M_m$) and Density ($d$): Multiplying molarity ($mol/L$) by molar mass ($g/mol$) yields density ($g/L$):   $d = \frac{PM_m}{RT}$   $M_m = \frac{dRT}{P}$

• Methane Density Example:     * Given: $CH_4$ ($M_m = 16.04\,g/mol$), $T = 125^{\circ}C = 398\,K$, $P = 3.50\,atm$.     * Calculation: $d = \frac{(3.50\,atm)(16.04\,g/mol)}{(0.08206\,L \cdot atm \cdot mol^{-1} \cdot K^{-1})(398\,K)} = 1.72\,g/L$.

Gas Stoichiometry

• Process: Use the Ideal Gas Law to convert between gas properties ($P, V, T$) and moles ($n$) for use in balanced equations.

• Neutralization Example (HCl spill):     * Reaction: $CaCO_3(s) + 2HCl(aq) \rightarrow CaCl_2(aq) + H_2O(l) + CO_2(g)$     * Given: $1.20\,kg$ of $CaCO_3$; conditions for $CO_2$: $735\,mm\,Hg$, $20.^{\circ}C$.     * Moles of $CO_2$: $1200\,g\,CaCO_3 \times (1\,mol/100.09\,g) \times (1\,mol\,CO_2/1\,mol\,CaCO_3) = 11.99\,mol$.     * Conditions: $P = 735/760 = 0.967\,atm$, $T = 293\,K$.     * Volume: $V = \frac{(11.99\,mol)(0.08206)(293\,K)}{0.967\,atm} = 2.98 \times 10^2\,L$.

Gas Mixtures and Partial Pressures

• Dalton’s Law of Partial Pressures: In a mixture of unreactive gases, each gas exerts a pressure independent of the other gases. The total pressure ($P_{total}$) is the sum of the partial pressures ($P_i$):   $P_{total} = P_A + P_B + P_C + \dots$

• Partial Pressure and Mole Fraction ($X$):     * The mole fraction is the ratio of moles of a specific component to the total moles:       $X_1 = \frac{n_1}{n_{total}}$     * The partial pressure can be related to the total pressure via the mole fraction:       $P_1 = X_1 \cdot P_{total}$     * Mole percent is calculated as $X_1 \times 100$.

Collecting Gas Over Water

• Principle: When gas is collected over water, it becomes saturated with water vapor. The total pressure measured is the sum of the desired gas pressure and the vapor pressure of water:   $P_{total} = P_{gas} + P_{H_2O}$

• Vapor Pressure: $P_{H_2O}$ depends solely on temperature and is looked up in standard reference tables (e.g., Table 5.6).

Kinetic-Molecular Theory (KMT)

• The Postulates:     1. Negligible Volume: The volume of gas molecules is very small compared to the container volume.     2. Random Motion: Gases consist of a large number of molecules in continuous, random motion at various speeds.     3. No Interactions: Attractive or repulsive forces between molecules are insignificant except during collisions.     4. Elastic Collisions: Collisions are perfectly elastic; average kinetic energy remains constant if temperature is constant.     5. Temperature Proportionality: The average kinetic energy of a gas is directly proportional to the absolute temperature ($K$).

• Relationship to Gas Laws:     * Boyle’s Law: Increasing volume at constant $T$ means molecules travel longer distances between walls, resulting in fewer collisions per unit time and thus lower pressure.     * Charles’s Law: Increasing temperature increases average speed and kinetic energy. This leads to more frequent and stronger collisions with walls, increasing pressure.

Molecular Speeds and Graham’s Law

• Root-Mean Square (RMS) Speed ($u_{rms}$): The speed of a molecule possessing the average kinetic energy.   $u_{rms} = \sqrt{\frac{3RT}{M_m}}$     * In this formula, $R = 8.314\,J/(mol \cdot K)$, and $M_m$ must be in $kg/mol$.

• Diffusion: The process where gas spreads throughout a volume or substance to occupy space uniformly.

• Effusion: The flow of a gas through a small hole (e.g., a pinprick in a balloon).

• Graham’s Law of Effusion: The rate of effusion is inversely proportional to the square root of the gas's molar mass:   $\frac{Rate_1}{Rate_2} = \sqrt{\frac{M_{m2}}{M_{m1}}}$

Real Gases and Van der Waals Equation

• Deviations from Ideality: Real gases do not follow $PV = nRT$ perfectly, especially at high pressures and low temperatures. Deviations are largest near the boiling point.

• Reasons for Non-Ideal Behavior:     1. Finite Volume: Gas molecules themselves occupy physical space; they are not points.     2. Intermolecular Attractions: Molecules attract each other, slowing them down before they hit the container walls, which reduces the actual pressure compared to ideal predictions.

• Van der Waals Equation: Accounts for deviations with constants $a$ (attraction correction) and $b$ (volume correction):   $\left(P + \frac{n^2a}{V^2}\right)(V - nb) = nRT$   * The term $\frac{n^2a}{V^2}$ corrects for intermolecular forces.   * The term $nb$ corrects for the finite volume of the gas molecules.