Study Notes on Gases

Chapter 8: Gases

Properties of Gases

• Gases in the atmosphere:

  • Oxygen ($O_2$): Comprises approximately 21% of Earth's atmosphere; essential for aerobic respiration and combustion processes.

  • Nitrogen ($N_2$): The most abundant gas, making up about 78% of the atmosphere; relatively unreactive but crucial for forming biologically important compounds via nitrogen fixation.

  • Ozone ($O<em>3$): Formed in the stratosphere when ultraviolet (UV) light interacts with oxygen molecules ($O</em>2$); forms a protective layer that absorbs most harmful UV radiation from the sun.

  • Other gases: Include Argon (a noble gas), Carbon Dioxide ($CO<em>2$ (a greenhouse gas vital for photosynthesis and a product of respiration/combustion), and water vapor ($H</em>2O$ (a variable component crucial for weather and climate).

• General Characteristics: Gases lack a fixed shape or volume, taking on the shape and volume of their container. They are highly compressible and have much lower densities compared to liquids and solids due to the large distances between particles.

Learning Goal

• Describe the kinetic molecular theory of gases and the units of measurement used for gases.

Kinetic Molecular Theory

• The Kinetic Molecular Theory (KMT) provides a model to explain the behavior of ideal gases. A gas consists of small particles (atoms or molecules) that:

  • Move rapidly and randomly in straight lines, constantly colliding with each other and the container walls. This continuous motion is responsible for gas pressure.

  • Experience essentially no attractive or repulsive forces between particles. This assumption implies that gas particles act independently of one another.

  • Are very far apart from each other, meaning the volume occupied by the gas particles themselves is negligible compared to the total volume of the container they occupy. This explains why gases are so compressible.

  • Have very small volumes compared to the overall volume of the container. The empty space between particles accounts for the low density of gases.

  • Have average kinetic energies that are directly proportional to the absolute temperature (in Kelvin). This means kinetic energy increases with an increase in temperature, leading to more frequent and forceful collisions.

Properties That Describe a Gas

• Gases are described in terms of four interdependent properties:

  1. Pressure (P)

  2. Volume (V)

  3. Temperature (T)

  4. Amount (n)

Volume

• The volume of a gas:

  • Is identical to the volume of the container it completely fills, because gas particles move freely and spread out to occupy all available space.

  • Is typically measured in liters (L), milliliters (mL), or cubic meters ($m^3$). Common conversions include $1 \text{ L} = 1000 \text{ mL} = 1 \text{ dm}^3$ and $1 \text{ mL} = 1 \text{ cm}^3$.

  • Tends to increase with an increase in temperature at a constant pressure (Charles's Law).

Temperature

• The temperature of a gas is directly related to the average kinetic energy of its molecules and is always measured in Kelvin (K) for gas law calculations. The Kelvin scale is an absolute temperature scale where $0 \text{ K}$ (absolute zero) represents the theoretical point at which all atomic motion ceases.

• When the temperature of a gas is decreased:

  • The molecules move more slowly, resulting in fewer and less forceful collisions with the container walls, leading to lower pressure (if volume is constant).

• When the temperature is increased:

  • The molecules move faster, leading to more frequent and energetic collisions, resulting in higher pressure (if volume is constant).

• Conversion: $K = \text{°C} + 273.15$

Pressure

• Pressure is defined as the force exerted by gas particles as they collide with the sides of a container per unit area ($Pressure = \frac{\text{Force}}{\text{Area}}$). It is measured in various units:

  • Millimeters of mercury (mmHg): Commonly used in medical and meteorological contexts.

  • Torr: Equivalent to mmHg ($1 \text{ Torr} = 1 \text{ mmHg}$).

  • Atmospheres (atm): A standard unit, approximately equal to the average atmospheric pressure at sea level.

  • Pascals (Pa) or kilopascals (kPa): The SI unit of pressure ($1 \text{ kPa} = 1000 \text{ Pa}$).

  • Pounds per square inch (psi): Often used in engineering for tire pressure or industrial applications.

• Atmospheric pressure is the pressure exerted by the column of air particles in the Earth's atmosphere. At sea level, it is approximately 1 atm ($760 \text{ mmHg}$ or $101.325 \text{ kPa}$).

Barometers

• A barometer is an instrument that measures the pressure exerted by the gases in the atmosphere. It typically works by indicating the height of a mercury column that the atmospheric pressure can support.

• At standard atmospheric pressure (1 atm), the mercury column in a barometer is exactly 760 mm high.

• This gives rise to the key equivalences: $1 \text{ atm} = 760 \text{ mmHg} = 760 \text{ Torr} = 101.325 \text{ kPa} = 14.7 \text{ psi}$.

Atmospheric Pressure

• Atmospheric pressure:

  • Is the pressure exerted by the immense column of air extending from the top of the atmosphere down to the surface of Earth. This pressure is a consequence of the weight of the air above.

  • Decreases as altitude increases because there is less air above to exert pressure. For example, at sea level, atmospheric pressure is approximately 1 atm, but it is significantly lower at higher elevations like mountain peaks.

Variations in Atmospheric Pressure

• Atmospheric pressure changes with weather conditions:

  • Higher atmospheric pressure is generally associated with clear, sunny weather due to the presence of a dense, descending air mass, causing the mercury column in a barometer to rise.

  • Lower atmospheric pressure is often linked to stormy or overcast weather as lighter, ascending air masses are present, causing the mercury column to fall.

Learning Check: Pressure Conversions

1. What is 475 mmHg expressed in atmospheres?

   • Calculation: $475 \text{ mmHg} \times \frac{1 \text{ atm}}{760 \text{ mmHg}} = 0.625 \text{ atm}$

   • Answer: B. 0.625 atm.

2. The pressure in a tire is 2.00 atm. What is this pressure in millimeters of mercury?

   • Calculation: $2.00 \text{ atm} \times \frac{760 \text{ mmHg}}{1 \text{ atm}} = 1520 \text{ mmHg}$

   • Answer: B. 1520 mmHg.

Boyle's Law

• Boyle's Law describes the inverse relationship between the pressure and volume of a gas when the temperature (T) and the amount of gas (n) are kept constant. This means that as one quantity increases, the other decreases proportionally.

• The volume increases as pressure decreases, and conversely, the volume decreases as pressure increases.

• Mathematical Relationship: For a given amount of gas at constant temperature, the product of pressure and volume is constant.
  $P<em>1 V</em>1 = P<em>2 V</em>2$

• Where $P<em>1$ and $V</em>1$ are the initial pressure and volume, and $P<em>2$ and $V</em>2$ are the final pressure and volume.

• Rearranged to solve for $V<em>2$: $V</em>2 = V<em>1 \times \frac{P</em>1}{P_2}$

• Rearranged to solve for $P<em>2$: $P</em>2 = P<em>1 \times \frac{V</em>1}{V_2}$

• Key Observations: If volume decreases, the gas particles are confined to a smaller space, leading to more frequent collisions with the container walls, thus increasing the pressure. The opposite occurs if volume increases.

Application of Boyle's Law

• Boyle’s Law demonstrates relevance in biological systems, most notably in the mechanics of breathing:

  • During inhalation, the diaphragm contracts and moves downward, and the rib muscles pull the rib cage up and out. This increases the lung volume (V), which, according to Boyle's Law, causes a decrease in the internal pressure within the lungs ($P<em>{\text{internal}}$) below atmospheric pressure ($P</em>{\text{atm}}$). Air then flows from the higher atmospheric pressure to the lower internal lung pressure, drawing air in.

  • During exhalation, the diaphragm relaxes and moves upward, and the rib cage moves down and inward. This decreases the lung volume (V), which increases the internal pressure within the lungs ($P<em>{\text{internal}}$) above atmospheric pressure ($P</em>{\text{atm}}$). Air then flows from the higher internal lung pressure to the lower atmospheric pressure, pushing air out.

Calculation Example Using Boyle's Law

• If an 8.0 L sample of Freon gas is at 550 mmHg and the pressure is changed to 2200 mmHg, what is the new volume?

  • Given:

    • Initial Pressure $(P_1)$ = 550 mmHg

    • Initial Volume $(V_1)$ = 8.0 L

    • Final Pressure $(P_2)$ = 2200 mmHg

  • Using: The Boyle's Law equation: $P<em>1 V</em>1 = P<em>2 V</em>2$

  • Solve for $V<em>2$: $V</em>2 = V<em>1 \times \frac{P</em>1}{P_2} = 8.0 \text{ L} \times \frac{550 \text{ mmHg}}{2200 \text{ mmHg}} = 2.0 \text{ L}$

  • Answer: The new volume is 2.0 L. Notice that as pressure increased (550 mmHg to 2200 mmHg), the volume decreased (8.0 L to 2.0 L), consistent with Boyle's Law.

Charles's Law

• Charles's Law describes the direct relationship between the temperature and volume of a gas when the pressure (P) and moles (n) are kept constant. This means that as one quantity increases, the other increases proportionally.

• When the temperature increases, the volume increases, assuming constant pressure and amount of gas.

• Mathematical Relationship: For a given amount of gas at constant pressure, the ratio of volume to absolute temperature is constant.
  $\frac{V<em>1}{T</em>1} = \frac{V<em>2}{T</em>2}$

• Where $V<em>1$ and $T</em>1$ are the initial volume and absolute temperature, and $V<em>2$ and $T</em>2$ are the final volume and absolute temperature.

• Rearranged to solve for $V<em>2$: $V</em>2 = V<em>1 \times \frac{T</em>2}{T_1}$

• Rearranged to solve for $T<em>2$: $T</em>2 = T<em>1 \times \frac{V</em>2}{V_1}$

• It is crucial to use the Kelvin scale when using temperature ($T$) in all gas law calculations, as $T$ represents the absolute kinetic energy, and Celsius or Fahrenheit scales would lead to incorrect results (e.g., negative temperatures).

Applications of Charles's Law

• Temperature effects on gas volume can be observed in everyday scenarios:

  • An inflated balloon left in a cold environment (e.g., outdoors in winter) will shrink as the gas inside cools and contracts. Conversely, if moved to a warmer environment, the gas will expand, and the balloon will reinflate.

  • Hot air balloons operate on Charles's Law: heating the air inside the balloon increases its volume (making it less dense than the surrounding cooler air), generating lift.

Gay-Lussac's Law

• Gay-Lussac's law states that the pressure of a gas is directly related to its Kelvin temperature when the volume (V) and moles (n) are constant. This means that as temperature increases, pressure also increases proportionally.

• If the temperature of a gas in a rigid container increases, the gas particles move faster and collide more forcefully and frequently with the container walls, leading to an increase in pressure.

• Mathematical Relationship: For a given amount of gas at constant volume, the ratio of pressure to absolute temperature is constant.
  $\frac{P<em>1}{T</em>1} = \frac{P<em>2}{T</em>2}$

• Rearranged to solve for $P<em>2$: $P</em>2 = P<em>1 \times \frac{T</em>2}{T_1}$

• Rearranged to solve for $T<em>2$: $T</em>2 = T<em>1 \times \frac{P</em>2}{P_1}$

Combined Gas Law

• The combined gas law integrates Boyle's, Charles's, and Gay-Lussac's laws into a single equation. It is applicable when the amount of gas (n) remains constant, but pressure, volume, and temperature might all change simultaneously.
  $\frac{P<em>1 V</em>1}{T<em>1} = \frac{P</em>2 V<em>2}{T</em>2}$

• This law is particularly useful for calculations where two or three of the gas properties (P, V, T) change between an initial and a final state, while the amount of gas stays the same.

Ideal Gas Law

• The ideal gas law combines aspects of all previous gas laws into one comprehensive equation, relating all four state variables of a gas (P, V, n, T).

• Formulation: $PV = nRT$

• Where:

  • P = Pressure of the gas, usually in atmospheres (atm).

  • V = Volume of the gas, usually in liters (L).

  • n = Amount of gas, in moles (mol).

  • R = Ideal gas constant, its value depends on the units used for pressure and volume. The most common value is $0.0821 \frac{\text{L atm}}{\text{K mol}}$.

  • T = Absolute temperature, always in Kelvin (K).

• This law is fundamental for solving for any one of the four variables if the other three are known for an ideal gas.

Molar Volume at STP

• Standard Temperature and Pressure (STP) is a set of standard conditions for experimental measurements, established to allow comparisons between data.

• Standard Temperature is $0 \text{ °C}$, which is $273.15 \text{ K}$.

• Standard Pressure is $1 \text{ atm}$ ($760 \text{ mmHg}$).

• At STP, the volume occupied by exactly 1 mole of any ideal gas is the molar volume, which is $22.4 \text{ L}$. This is a crucial conversion factor for relating moles of gas to volume under standard conditions.

Dalton’s Law of Partial Pressures

• Dalton’s law of partial pressures states that in a mixture of non-reacting gases, the total pressure exerted is the sum of the partial pressures of each individual gas in the mixture.
  $P<em>T = P</em>1 + P<em>2 + P</em>3 + \ldots$

• Where $P<em>T$ is the total pressure and $P</em>1, P<em>2, P</em>3$ are the partial pressures of the individual gases.

• Each gas in the mixture exerts pressure independently of the others because gas particles are far apart and do not significantly attract or repel each other.

Real Gas Behavior

• The gas laws and Kinetic Molecular Theory describe the behavior of ideal gases, which are theoretical. Real gases deviate from ideal behavior under certain conditions because their particles:

  • Have finite volume (not negligible at high pressures where particles are closer).

  • Experience intermolecular forces (attractions or repulsions become significant at low temperatures where particles move slower and are closer).

• Deviations are most significant at high pressures and low temperatures.

Diffusion and Effusion

• Diffusion is the process by which gas molecules spread out from an area of higher concentration to an area of lower concentration, often through another medium or into a vacuum. This explains how odors travel through a room.

• Effusion is the process where gas escapes through a tiny pinhole opening into a vacuum. The rate of effusion is measured by how quickly the gas escapes.

• Graham's Law of Effusion quantifies this: The rate of effusion of a gas is inversely proportional to the square root of its molar mass.

• Mathematical Representation of Graham's Law:
  $\frac{Rate<em>1}{Rate</em>2} = \sqrt{\frac{MM<em>2}{MM</em>1}}$

• This implies that lighter gases effuse faster than heavier gases at the same temperature, because lighter molecules have higher average speeds at a given temperature.

Density of a Gas

• Density ($d$) is a fundamental physical property defined as the ratio of mass to volume:
  $Density = \frac{\text{mass}}{\text{volume}}$

• For gases, density is highly dependent on temperature