Study Notes on Gases from Chapter 10 of Chemistry: The Central Science

Characteristics of Gases

  • The physical properties of gases are consistent across various types.

  • Gases are primarily composed of nonmetallic elements which have simple molecular formulas and low molar masses.

  • Key distinguishing characteristics of gases include:

    • Expansion: Gases expand to fill their containers entirely.

    • Compression: Gases are highly compressible compared to liquids and solids.

    • Density: Gases have extremely low densities in comparison to liquids and solids.

  • When mixed, two or more gases form a homogeneous mixture.

Defining the State of a Gas Sample

To define the state of a gas sample, we utilize four main properties:

  1. Temperature (T)

  2. Pressure (P)

  3. Volume (V)

  4. Amount of gas: represented by the number of moles (n).

    • Previously discussed were temperature, volume, and amount, now we need a definition for pressure.

Pressure Definition

  • Pressure (P) is defined as the amount of force (F) applied to a specific area (A):

    P=FAP=AF​

  • Atmospheric Pressure is quantified as the weight of air per unit area.

Units of Pressure

Various units of pressure measurement include:

  • Pascal (Pa): 1 Pa=1 N/m21 Pa=1 N/m2 (the SI unit of pressure).

  • Bar: 1 bar=105 Pa=100 kPa1 bar=105 Pa=100 kPa.

  • mm Hg or torr: These units represent the height difference in mm of two connected mercury columns, often used in barometers.

  • Atmosphere (atm): 1 atm=760 torr=760 mm Hg=101.325 kPa1 atm=760 torr=760 mm Hg=101.325 kPa.

Torricelli's Barometer Analysis

  • Torricelli's Barometer works on the principle that the atmospheric pressure equals the pressure exerted by a column of mercury.

  • Observations and calculations include:

    • The pressure exerted by the mercury column is derived from the force of gravity acting on the mass of mercury.

    • Setting atmospheric pressure equal to mercury column pressure, we derive variable d=m/Vd=m/V, where d represents density.

    • Assuming the tube containing mercury has a constant cross-sectional area, one can calculate the density of mercury (dHgdHg​) using:

    Patm=PHg=dHghgPatm=PHg=dHg​hg

    • Substituting relevant values (760 mm height, atmospheric pressure =1.01×105 Pa=1.01×105 Pa), one can estimate the density of mercury to be reasonable.

Sample Exercise 10.1

  • Calculate the height of a mercury column under different conditions or different fluids such as water, given specific external pressures.

Manometer: Measuring Gas Pressure

  • Manometer is an instrument designed to measure the difference between atmospheric pressure and that of a gas contained in a vessel.

  • Sample Exercise 10.2 involves problem-solving for calculating gas pressure using a manometer, relying on pressure units and conditions.

Standard Pressure

  • The standard atmospheric pressure, particularly at sea level, is defined as:

    • 1.00 atm=760 torr (760 mmHg)=101.325 kPa1.00 atm=760 torr (760 mmHg)=101.325 kPa.

Gas Laws

Boyle's Law

  • States that the volume of a fixed quantity of gas at constant temperature is inversely proportional to the pressure:

    • Mathematically represented as:

    PV=constantPV=constant

    or in terms of two conditions:

    P1V1=P2V2P1V1=P2V2

  • Graphs of volume versus pressure are non-linear, but graphs of volume versus (1/P)(1/P) yield a linear relationship.

Charles’s Law

  • Relates volume directly to temperature at constant pressure:

    • Mathematically:

    V=kTV=kT

    where k is a constant (the relationship defines V1/T1=V2/T2V1/T1=V2/T2).

  • A graph of volume versus temperature at constant pressure is linear.

Avogadro’s Law

  • Establishes that the volume of a gas at constant temperature and pressure is directly proportional to the number of moles of gas:

    • Mathematically represented as:

    V=knV=kn

    • At STP, one mole of gas occupies 22.4 L22.4 L.

Ideal Gas Law

  • The combined relationships from the previous laws leads us to the Ideal-Gas Equation:

    PV=nRTPV=nRT

    where R is the ideal gas constant (0.0821 L⋅atm/(K⋅mol))(0.0821 L⋅atm/(K⋅mol)).

Density of Gases

  • The density of a gas can be calculated using the rearranged ideal gas equation:

    d=mPRTd=RTmP​

  • This allows the conversion of nn (moles) into mass (mm) using molecular mass:

    m=nMm=nM
    where M is the molar mass of the gas.

Kinetic-Molecular Theory

Main Tenets

  1. Gases consist of a large number of molecules in continuous and random motion.

  2. The combined volume of gas molecules is negligible compared to the container’s total volume.

  3. Intermolecular attractions and interactions are negligible.

  4. Energy transfer occurs during collisions, but the average kinetic energy remains constant if temperature is constant.

  5. The average kinetic energy is proportional to the absolute temperature.

Movement of Gas Molecules

  • Temperature corresponds to the average kinetic energy of Gas molecules, where individual molecules have varied speeds.

  • Different speeds include:

    • ump​: Most probable speed.

    • uav​: Average speed of molecules.

    • urms​: Root-mean-square speed associated with average kinetic energy.

Graham’s Law of Effusion and Diffusion

  • Graham’s Law relates molar mass with the effusion rates of gases:

    • Lighter gas molecules effuse faster than heavier ones.

Van der Waals Equation

  • The Van der Waals equation corrects for non-ideal behavior of gases accounting for intermolecular forces and molecular volume:

    (P+n2aV2)(V−nb)=nRT(P+V2n2a​)(V−nb)=nRT

Conclusion

  • Understanding these gas principles allows dissecting the behavior of real gases and how they deviate from the ideal models under various conditions. Gas laws and theories help in predicting responses in practical applications of chemistry, including reactions and environmental science interactions.

Here are the formulas and equations from the notes on gases:

Pressure Definition

  • Pressure (P) is defined as the amount of force (F) applied to a specific area (A):
    P = \frac{F}{A}

Units of Pressure

  • Pascal (Pa): 1 Pa = 1 N/m^2

  • Bar: 1 bar = 10^5 Pa = 100 kPa

  • Atmosphere (atm): 1 atm = 760 torr = 760 mmHg = 101.325 kPa

Torricelli's Barometer Analysis

  • The density formula: d = m/V

  • Atmospheric pressure equals mercury column pressure:
    Patm=PHg=dHg​hg

Gas Laws
Boyle's Law

  • PV = constant

  • For two conditions: P1V1=P2V2

Charles’s Law

  • V = kT

  • For two conditions: T1/V1​=T2/V2​

Avogadro’s Law

  • V = kn

Ideal Gas Law

  • The Ideal-Gas Equation: PV = nRT

  • Where R is the ideal gas constant: R = 0.0821 L⋅atm/(K⋅mol)

Density of Gases

  • The density of a gas can be calculated using the rearranged ideal gas equation (where M is the molar mass of the gas):
    d = \frac{PM}{RT}

  • Conversion of moles (n) into mass (m) using molar mass:
    m = nM

Van der Waals Equation

  • Corrects for non-ideal behavior of gases:
    (P + \frac{n^2a}{V^2})(V - nb) = nRT