Properties of Gases Vocabulary

Chapter Overview: Properties of Gases

• Scope of Study: The chapter covers the fundamental properties of gases, the kinetic molecular theory, atmospheric behavior, various gas laws (Bole's, Charles's, Avogadro's, Amontons's, and the Combined Gas Law), the Ideal Gas Law, gas densities, stoichiometry in chemical reactions, mixtures (Dalton's Law), and the behavior of real gases.

• Key Sections:     * 9.1 An Invisible Necessity: The Properties of Gases     * 9.2 Gas Diffusion and the Kinetic Molecular Theory of Gases     * 9.3 The Air We Breathe     * 9.4 Relating P, T, and V: The Gas Laws     * 9.5 The Combined Gas Law     * 9.6 Ideal Gases and the Ideal Gas Law     * 9.7 Densities of Gases     * 9.8 Gases in Chemical Reactions     * 9.9 Mixtures of Gases     * 9.10 Real Gases

Properties of Gases: An Invisible Necessity

• Compressibility: Gases are highly compressible, whereas liquids and solids are generally not.

• Fundamental Proportionalities and Relationships:     * Volume and Pressure: Volume is inversely proportional to pressure ($V \propto \frac{1}{P}$).     * Pressure and Temperature: Pressure is directly proportional to temperature ($P \propto T$).     * Pressure and Quantity: Pressure is directly proportional to the quantity (moles) of gas ($P \propto n$).

• Miscibility: Gases are completely miscible with one another, forming homogeneous mixtures regardless of the identity of the gases.

• Density: The density of a gas is directly proportional to its molar mass.

• Expansion: Gases expand spontaneously to occupy the entire volume of their container.

• Effusion: Gases effuse at rates that are inversely proportional to their molar mass ($\text{rate} \propto \frac{1}{\sqrt{\mathcal{M}}}$).

Diffusion and the Kinetic Molecular Theory (KMT)

• Diffusion: Defined as the spread of one substance through another.

• Mean Free Path: The average distance that a particle can travel through air or any gas before colliding with another particle.     * At a pressure of $1\,bar$, the mean free path of a molecule in air is approximately $6.8 \times 10^{-8}\,m$.     * This frequency translates to approximately $10^{10}$ collisions per second.

• Foundational Assumptions of Kinetic Molecular Theory:     1. Negligible Molecular Volume: Gas molecules have volumes that are tiny and negligible compared with the total volume the gas occupies.     2. Lack of Interaction: Gas particles do not interact with each other (no attractive or repulsive forces).     3. Elastic Collisions: Collisions between gas molecules are perfectly elastic, meaning there is no net loss of kinetic energy.     4. Kinetic Energy and Temperature: The average kinetic energy of the molecules in a gas is directly proportional to the absolute temperature (measured in Kelvin).

Root-Mean-Square Speed and Kinetic Energy

• Definition: The root-mean-square speed ($u_{rms}$) is the speed of a particle whose kinetic energy is exactly the same as the average kinetic energy of all the particles in a gas.

• Mathematical Models:     * Average Kinetic Energy ($KE_{avg}$): $KE_{avg} = \frac{1}{2} m u_{rms}^2$     * Relationship to Temperature: $KE_{avg} = \frac{3}{2} k_B T$     * Root-Mean-Square Speed Formula: $u_{rms} = \sqrt{\frac{3RT}{\mathcal{M}}}$

• Variables and Constants:     * $\mathcal{M}$: Molar Mass     * $T$: Temperature in Kelvin ($K$)     * $R$: Universal gas constant, valued at $8.314\,kg \cdot m^2 / s^2 \cdot mol \cdot K$     * $k_B$: Boltzmann constant

• Temperature Effects:     * Root-mean-square speed increases as temperature increases.     * In a distribution plot, the most probable speeds are indicated by dashed lines, while the $u_{rms}$ represents the average energy-equivalent speed.

Graham’s Law and Relative Diffusion Rates

• Mathematical Representation: The law states that if particles of gas $A$ have higher speeds than particles of gas $B$, gas $A$ particles will collide more frequently with the walls, leading to increased effusion/diffusion.

• Formula: $\frac{\text{diffusion rate}_A}{\text{diffusion rate}_B} = \sqrt{\frac{\mathcal{M}_B}{\mathcal{M}_A}}$

• Numerical Example (Odorous Gas): An odorous gas from a hot spring diffuses through air at $0.342$ times the rate of helium ($He$). This ratio can be used to calculate the molar mass of the emitted gas using the inverse square root relationship.

Pressure and Atmospheric Measurement

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

• Barometers: Atmospheric pressure is measured using a barometer. This device measures the height of a mercury column supported by atmospheric pressure against a vacuum.

• Pressure Units and Conversions:     * $1\,atm = 1.01325 \times 10^5\,Pa$ (Pascal)     * $1\,atm = 101.325\,kPa$ (Kilopascal)     * $1\,atm = 760\,mmHg$ (Millimeters of mercury)     * $1\,atm = 760\,torr$.     * $1\,atm = 1.01325\,bar$.     * $1\,atm = 1013.25\,mbar$ (Millibar/mb).     * $1\,atm = 14.7\,psi$ (Pounds per square inch).     * $1\,atm = 29.92\,\text{inches of Hg}$.

• Altitude and Pressure Variation:     * At Sea Level ($0\,m$): Pressure is approximately $1.0\,atm$.     * Denver: Pressure is notably lower than $1.0\,atm$.     * La Paz, Bolivia ($3640\,m$): Pressure is significantly reduced.     * Mt. Everest ($8848\,m$): Pressure is approximately between $0.3$ and $0.4\,atm$.     * At $10,000\,m$: Pressure is near $0.2\,atm$.

The Fundamental Gas Laws

• Boyle’s Law (Pressure and Volume): At constant temperature, volume is inversely proportional to pressure.     * $PV = \text{constant}$     * $P_1 V_1 = P_2 V_2$

• Charles’s Law (Volume and Temperature): At constant pressure, volume is directly proportional to absolute temperature.     * $\frac{V}{T} = \text{constant}$     * $\frac{V_1}{T_1} = \frac{V_2}{T_2}$     * Absolute Zero: Extrapolating volume-temperature curves to $V = 0$ yields a temperature of $-273.15^\circ C$ or $0\,K$.

• Avogadro’s Law (Volume/Pressure and Quantity):     * In Non-rigid Containers: Volume is proportional to moles ($n$) at constant $P$ and $T$. $\frac{V}{n} = \text{constant}$.     * In Rigid Containers: Pressure is proportional to moles ($n$) at constant $V$ and $T$. $\frac{P}{n} = \text{constant}$. Increasing particles increases collision frequency, thus increasing pressure.

• Amontons’s Law (Pressure and Temperature): In a rigid container ($V$ is constant), pressure is proportional to absolute temperature.     * $\frac{P_1}{T_1} = \frac{P_2}{T_2}$

Combined and Ideal Gas Laws

• The Combined Gas Law: Merges Boyle’s and Amontons’s laws for situations where $P$, $V$ , and $T$ vary.     * $\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$

• The Ideal Gas Law:     * $PV = nRT$     * Assumptions: Individual gas particle volumes are insignificant; particles do not interact.

• Standard Conditions (STP):     * Standard Temperature: $0^\circ C$     * Standard Pressure: $1\,atm$     * Molar Volume at STP:         * Using $1\,atm$: $22.4\,L$         * Using $1\,bar$: $22.7\,L$

Densities of Gases

• The density ($d$) of any gas can be determined by its molar mass ($\mathcal{M}$), pressure ($P$), and temperature ($T$).

• Formula: $d = \frac{\mathcal{M}P}{RT}$

• At STP, density is simply molar mass divided by molar volume.

Gas Mixtures and Dalton’s Law

• Independent Behavior: In a mixture of ideal gases, each gas behaves independently.

• Dalton’s Law of Partial Pressures: The total pressure ($P_{total}$) of a mixture is the sum of the individual partial pressures ($P_i$) of each gas.     * $P_{total} = P_1 + P_2 + P_3 + \dots$

• Mole Fraction ($x_i$): The ratio of the moles of a specific component to the total moles in the mixture.     * $x_i = \frac{n_i}{n_{total}}$

• Partial Pressure Calculation: The partial pressure exerted by a gas is proportional to its mole fraction.     * $P_i = x_i P_{total}$

Gas Collection and Real Gases

• Water Displacement: Used to collect gases that do not dissolve in water (e.g., $O_2$ from the decomposition of $KClO_3$).     * Formula for total pressure: $P_{total} = P_{atm} = P_{gas} + P_{H_2O}$     * $P_{H_2O}$ is the vapor pressure of water at the specific collection temperature.

• Real Gases and Nonideality: Gases deviate from ideal behavior at low temperatures and high pressures.     * Intermolecular Forces: At low temperatures, molecules lack the kinetic energy to overcome attractions, leading to lower pressure than predicted as particles collide less forcefully with container walls.     * Particle Volume: At high pressures, gas molecules are forced into small volumes, making the physical volume of the particles significant (gases are not infinitely compressible).

The van der Waals Equation

• Purpose: Describes real gases by accounting for particle volume (incompressibility) and intermolecular interactions.

• Formula: $(P + a(\frac{n}{V})^2)(V - nb) = nRT$     * $a$: Accounts for attractive forces between particles.     * $b$: Accounts for the volume occupied by the gas molecules themselves.

Quantitative Problems and Scenarios

• Root-Mean-Square Speed Calculation: Determine the speed of nitrogen ($N_2$) molecules at $25^\circ C$.

• Weather Balloon Scenario: A balloon released at $798\,mmHg$, $131\,L$, and $20^\circ C$. Calculate the new volume at an altitude where pressure is $235\,mmHg$ and temperature is $-52^\circ C$.

• Water Displacement Molar Mass calculation: Decomposition of $KClO_3$: $2KClO_3(s) \rightarrow 2KCl(s) + 3O_2(g)$. $92.0\,mL$ of gas collected at $25.0^\circ C$ and $756\,mmHg$. (Water vapor pressure at $25.0^\circ C = 23.8\,mmHg$). Find the mass of $O_2$.

• Compressed Oxygen Tank Comparison: A $2.24\,L$ tank contains $0.500\,kg$ of $O_2$ at $20^\circ C$. Compare pressure calculated via the Ideal Gas Law versus the van der Waals equation.

Properties of Gases: An Invisible Necessity

• Compressibility: Highly compressible compared to liquids and solids.

• Fundamental Proportionalities and Relationships:
    - Volume and Pressure: Inversely proportional ($V ext{ } op \frac{1}{P}$ ).   - Pressure and Temperature: Directly proportional ($P ext{ } op T$ ).   - Pressure and Quantity: Directly proportional ($P ext{ } op n$ ).

• Density: Directly proportional to molar mass.

• Expansion: Gases occupy the entire volume of their container.

• Effusion: Rates inversely proportional to molar mass ($ext{rate} ext{ } op \frac{1}{ ext{sqrt}{ ext{M}}}$ ).

Diffusion and the Kinetic Molecular Theory (KMT)

• Diffusion: Spread of substances through one another.

• Mean Free Path: Average distance a particle travels before colliding. Approximately $6.8 imes 10^{-8} ext{ m}$ in air.

• Assumptions of KMT
    1. Negligible molecular volume.
    2. No interactions between gas particles.
    3. Perfectly elastic collisions.
    4. Kinetic energy directly proportional to temperature.

Root-Mean-Square Speed and Kinetic Energy

• Definition: Speed of a particle with the same kinetic energy as the average.

• Mathematical Models:
    - $KE_{avg} = \frac{1}{2} m {u_{rms}}^2$
    - $KE_{avg} = \frac{3}{2} k_B T$
    - $u_{rms} = ext{sqrt}{\frac{3RT}{M}}$

Graham’s Law and Relative Diffusion Rates

• Formula:.

Pressure and Atmospheric Measurement

• Definition of Pressure: $P = \frac{F}{A}$

• Barometers: Measure atmospheric pressure using mercury.

• Pressure Units and Conversions:
    - $1 ext{ atm} = 101.325 kPa$
  
    -$1 ext{ atm} = 760 mmHg$

The Fundamental Gas Laws

• Boyle’s Law: $P_1 V_1 = P_2 V_2$ at constant temperature.

• Charles’s Law: $\frac{V}{T} = ext{constant}$ at constant pressure.

• Avogadro’s Law: $\frac{V}{n} = ext{constant}$ in non-rigid containers.

Combined and Ideal Gas Laws

1. Combined Gas Law: $\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$

2. Ideal Gas Law: $PV = nRT$

Densities of Gases

• Formula: $d = \frac{M P}{RT}$

Gas Mixtures and Dalton’s Law

• Dalton’s Law: $P_{total} = P_1 + P_2 + P_3 + ext{…}$

Gas Collection and Real Gases

• Water Displacement: $P_{total} = P_{gas} + P_{H2O}$

• Real Gases: Deviate from ideal behavior under low temperatures and high pressures.