Lecture 9

The Gaseous State

Shaun Williams, PhD

The Behavior of Gases

Properties of Gas Molecules

• Gases consist of particles that are relatively far apart (as compared to liquids and solids).
  • This results in observed properties:
    • Lower densities
    • Compressible by applying external pressure
• Gas particles move about rapidly.
• Gas particles have little effect on one another unless they collide.
  • When they collide, gas particles simply bounce off one another.
• Gases expand to fill containers.
  • Take volume and shape of their containers

Effect of Temperature and Density

• All gases expand if heated and contract if cooled.
  • Heat increases the kinetic energy of gas particles, making them move faster and farther apart.
• Since the gas particles move farther apart, there are fewer particles within a given volume.
  • Therefore, warm gases have lower densities.

Physical Effect of Temperature and Density

Pressure

• Is amount of force applied per unit area \[ P=\frac{\chem{force}}{\chem{area}} \]
• For a gas in a container: \[ P=\frac{\text{force of gas particles}}{\text{area of container}} \]

Crushing a Can

Pressure

• Pressure is measured using a barometer.
• Units of pressure
  • \( 1\, atm = 760\, mm\, Hg \) (mm Hg and torr are the same)
  • \( 1\, atm = 14.7\,\bfrac{lb}{in^2} \) (psi)
  • \( 1\, atm = 101325\, Pa \)

Factors That Affect the Properties of Gases

• Volume
  • Measured in liters (L)
• Pressure
  • Measured in atmospheres (atm)
• Temperature
  • Measured in Kelvin (K)
• Amount of particles
  • Measured in moles
• An ideal gas is a gas that behaves according to predicted linear relationships

Volume vs. Pressure

Boyle's Law

• For a given mass of gas at constant temperature, volume varies inversely with pressure. \[ V \propto \frac{1}{P} \] \[ P_1V_1 = P_2V_2 \]
• As volume increases, pressure decreases.

Volume vs. Temperature

Charles' Law

• For a given mass of gas at constant pressure, volume is directly proportional to temperature on an absolute (kelvin) scale. \[ T_K=T_{{}^\circ C} + 273.15 \] \[ V \propto T_K \] \[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]
• As volume increases, temperature increases.

Combined Gas Law

• For a constant amount of gas, volume is proportional to absolute temperature divided by pressure. \[ V \propto \frac{T}{P} \] \[ \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} \]

Combining Volumes

• Gay-Lussac's law of combining volumes
  • Gases combine in simple whole-number volume proportions at constant temperature and pressure.

Avogadro's Hypothesis

• The volume occupied by a gas at a given temperature and pressure is directly proportional to the number of gas particles and thus to the moles of gas. \[ V \propto n \] \[ \frac{V_1}{n_1} = \frac{V_2}{n_2} \]
• As volume increases, the moles of gas increase.

Avogadro's Hypothesis (cont.)

• At a given pressure and temperature, equal volumes of all gases contain equal numbers of moles (or particles).
• This hypothesis was measured at standard temperature and pressure (STP): \( 0^\circ C \) and \( 1\, atm \)
• The volume of an ideal gas at STP is called its molar volume: 1 mole of gas = 22.414 L

Summary of The Gas Laws

• Boyle's law \[ V \propto \frac{1}{P} \text{ at constant T, n} \]
• Charles's law \[ V \propto T \text{ at constant P, n} \]
• Avogadro's hypothesis \[ V \propto n \text{ at constant T, P} \]
• Combining them all into one proportionality: \[ V \propto \frac{nT}{P} \]

The Ideal Gas Law

• We can express the previous statement as an equality: \[ V = \text{constant}\times \frac{nT}{P} \]
• This constant is called the ideal gas constant, \(R\).
• This gives us the Ideal Gas Law: \[ V=R\times \frac{nT}{P} \xrightarrow{\text{algebra}} PV=nRT \]
• \(R\) is calculated at STP using the molar volume of a gas: \[ R=0.08206\,\frac{\chem{L\cdot atm}}{\chem{K\cdot mol}} \]

Density of a Gas

\[ \text{Density}=\frac{\text{mass}}{\text{volume}} \]

• We can solve for density in the Ideal Gas Law using substitution: \[ PV=nRT \] \[ n=\bfrac{m}{MM} \] \[ PV=\frac{mRT}{MM} \] \[ \frac{P\left(MM\right)}{RT} =\frac{m}{V} = d \]

Dalton's Law of Partial Pressures

• Gases in a mixture behave independently and exert the same pressure they would exert if they were in a container alone. \[ P_\text{total} = P_A+P_B+P_C+\cdots \] \[ P_\text{total} = P_\text{dry air} + P_\text{water} \]

Vapor Pressure of Water at Various Temperatures

Kinetic-Molecular Theory of Gases

• A model that explains experimental observations about gases under normal temperature and pressure conditions that we encounter in our environment
• Has 5 postulates:
  1. Gases are composed of small and widely separated particles (molecules or atoms).
     • Low volumes of particles
     • Low densities of gas
     • High compressibility

Kinetic-Molecular Theory of Gases (cont.)

• Postulates continued:
  2. Particles of a gas behave independently of one another.
     • Independent movement, unless two particles collide.
     • No forces of attraction or repulsion operate between and among gas particles.
  3. Each particle in a gas is in rapid, straight-line motion, until it collides with another molecule or with its container.
     • When collisions occur, the collisions are perfectly elastic.
     • Energy transferred from one particle to another with no net loss of energy.

Kinetic-Molecular Theory of Gases (cont. again)

• Postulates continued:
  4. The pressure of a gas arises from the sum of the collisions of the particles with the walls of the container.
     • The smaller the container, the more collisions between the gas particles and the walls of the container, resulting in higher pressures.
     • Predicts pressure should be proportional to the number of gas particles.

Kinetic-Molecular Theory of Gases (concluded)

• Postulates continued:
  5. The average kinetic energy of gas particles depends on the absolute temperature.
     • Relationship between kinetic energy (\(KE\)) and velocity (\(v\)) of the gas particles: \[ KE_{av} = \frac{1}{2} mv_{av}^2 \]
     • The average velocity for gas particles is greater at higher temperatures.
     • Thus, if the particles move faster, they hit the walls of the container more often, resulting in higher pressure.

The 5^th Postulate

Effect of the Kinetic-Molecular Theory of Gases

Diffusion

• The movement of gas particles from regions of high concentration to regions of low concentration. \[ KE_{av} = \frac{1}{2}mv_{av}^2 \]
• Gases have different average velocities even though they all have the same average kinetic energy.
• Light molecules or atoms move faster than heavy molecules or atoms.

Effusion

• The passage of a gas through a small opening.
• Smaller particles effuse faster than larger ones.

Gases and Chemical Reactions

• We can convert from volume to moles or moles to volume using the Ideal Gas Law as our conversion factor. \[ \text{Vol A} \xrightarrow{PV=nRT} \text{Mol A} \xrightarrow{\text{Mole ratio}} \text{Mol B} \xrightarrow{PV=nRT} \text{Vol B} \]
• We can put mass into our conversion \[ \text{Mass A} \xrightarrow{\text{Molar Mass}} \text{Mol A} \xrightarrow{\text{Mole ratio}} \text{Mol B} \xrightarrow{\text{Molar Mass}} \text{Mass B} \] \[ \text{Vol A} \xrightarrow{PV=nRT} \text{Mol A} \xrightarrow{\text{Mole ratio}} \text{Mol B} \xrightarrow{\text{Molar Mass}} \text{Mass B} \]