Chapter 9. Gases
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1. Overview
3. Kinetic Molecular Theory of Gases
4. Physicochemical Analysis of Gases
5. Real Gases
1. Overview
⑴ Types : Elements that exist as gases at standard conditions (25 ℃, 1 atm)
① Monoatomic Gases : He, Ne, Ar, Kr, Xe, Rn (Noble gases, Group 8A elements)
② Diatomic Gases : H2, N2, O2, F2, Cl2, etc.
⑵ Physical Characteristics : Most gases have similar physical properties
① Represented by volume and shape of the container
② State of matter with the highest compressibility
③ Much lower density compared to liquids and solids
④ Follow a consistent set of laws regardless of the type of gas
2. History of Gas Studies
⑴ Torricelli Experiment
① Experiment Process : The height of the mercury column in an inverted tube is always constant.
② Application : Experiment to measure atmospheric pressure
⑵ Boyle’s Law
① Introduced by R. Boyle in 1662
② At a constant temperature, the volume of a fixed amount of gas is inversely proportional to its pressure (P ≪ 1)
⑶ Charles’s Law
① Introduced by J.A.C. Charles in 1787
② At a constant pressure, the volume of a gas is directly proportional to its temperature
③ Introduction of Absolute Temperature (Kelvin temperature)
○ Absolute zero is the point where the linear extension of the V - T curve intersects the x-axis.
○ In reality, as temperature drops, gases liquefy or vaporize, making it impossible to determine the extrapolation point.
○ However, since all gases have the same extrapolation point, this has special significance, leading to the introduction of absolute zero and absolute temperature concepts.
⑷ Avogadro’s Law : At constant temperature and pressure + (P ≪ 1)
① Volume of 1 mole of gas at standard pressure (1 bar) and standard temperature (0 ℃) is 22.4 L
⑸ Ideal Gas Equation : Applicable to all types of gases
① Gas constant : Represented as R
○ 8.31446 J / mol·K
○ 8.20574 × 10-2 L·atm / K·mol
○ 287 J / kg·K : Tip: Commonly used in industrial thermodynamics
② Limiting laws that apply only under certain conditions (pressure ≪ 1)
⑹ Dalton’s Law of Partial Pressures : The ratio of partial pressures is equal to the ratio of the number of gas molecules
3. Kinetic Molecular Theory of Gases: Describes the model for ideal gases
⑴ Rules of Kinetic Molecular Theory
① Rule 1: Gases undergo continuous random walk motion.
② Rule 2: Point mass : Gas molecules have infinitesimally small volume.
③ Rule 3: Gas particles undergo straight-line motion.
④ Rule 4: No interaction : Gases do not interact except during collisions.
⑤ Rule 5: Elastic collision : Collisions between gas molecules are perfectly elastic.
⑥ Rule 6: Energy Equipartition Law : Average kinetic energy of gas molecules is proportional to temperature.
○ Formulation
○ Note: Zeroth law of thermodynamics : Defines the equality of temperatures through thermal equilibrium.
○ Note: Experimental definition of temperature scale : Expansion of a thermometer due to heat
○ Mercury thermometer : Silver thermometer
○ Alcohol thermometer : Red thermometer. Appears red due to dye.
○ Physical significance : Defines the temperature scale such that the kinetic energy of gas molecules is linearly proportional to temperature.
○ Directly deduced in thermodynamics that internal energy is a function of temperature.
⑵ Physical Derivation of the Ideal Gas Equation
① Let’s consider calculating the pressure on the right side perpendicular to the x-axis.
② Keep in mind that gas molecules continuously undergo elastic collisions with walls like billiard balls.
③ Period : The time it takes for a gas molecule to collide with the right side.
④ Step 4 : Remember that a gas molecule moving towards the right side will collide, move away from the right side after the collision, and continue moving.
⑤ Force : The average force exerted by a gas molecule on the right side : Force = Fa = Rate of change of momentum = dp / dt
⑥ Leap of faith : Science sometimes makes risky assumptions, but if the results are beautiful, they are forgiven.
⑦ Pressure derivation : Using the definition of pressure P = F / A
⑧ Average number of colliding molecules per unit time on the wall
○ Average number of colliding molecules on one side per unit time
○ Average number of colliding molecules on the entire wall per unit time
4. Physicochemical Analysis of Gases
⑴ Graham’s Law
① Diffusion : Process where a substance mixes into another due to molecular collisions
② Effusion : Movement of particles from a high-pressure area to a low-pressure area (e.g., movement of particles in a vacuum)
③ Graham’s Law
○ Content : Effusion rate ∝ 1 / √M
○ Inference : Since they are at the same temperature, their kinetic energies are the same.
○ Graham’s law can also be applied to diffusion rates.
○ Applied in the Manhattan Project for uranium enrichment
④ Frequency of gas molecules colliding with the wall
○ Proportional to the effusion rate
○ Frequency of gas molecules colliding with the wall ∝ Moles / Volume × Average Velocity
⑵ Implications of the Kinetic Molecular Theory : Root Mean Square Velocity (vrms)
⑶ Maxwell-Boltzmann Speed Distribution
Figure. 1. Maxwell-Boltzmann Speed Distribution
① m : Mass of one molecule. T : Absolute temperature. kB : Boltzmann constant
② Meaning 1: Set to make the sum of all values equal to 1
③ Meaning 2: Form of the normal distribution function
④ Meaning 3: Setting to match the formula of root mean square velocity
⑤ Result 1: Root Mean Square Velocity (vrms)
⑥ Result 2: Average Velocity (vavg) : Can be calculated through integration
⑦ Result 3: Most Probable Velocity (vmp) : Can be found by differentiating to locate the maximum
⑧ Result 4: Relative Velocity (vrel)
⑨ As the molecular weight increases and the temperature decreases, the speed distribution fits well : (Note) Light molecules bounce quickly at high speeds.
5. Real Gases
⑴ Compressibility Factor (Z) : Ratio of the volume of real gases to that of ideal gases
① Definition
② Determining intermolecular interactions
Figure. 2. Relationship between Compressibility Factor and Pressure [Note: 2]
○ Z < 1 : Dominance of attraction forces. Conditions of high temperature and low pressure
○ Z > 1 : Dominance of repulsive forces. Conditions of low temperature and high pressure
○ Tip: As temperature increases, gases approach ideal behavior, making Z approach 1.
③ Hydrogen : Possesses very weak intermolecular forces
⑵ Van der Waals Equation of State : One model for the equation of state of real gases
① Constant a : Correction constant for pressure decrease due to molecular attraction
○ ‘Attraction’ coefficient a
○ Interactions involve the relationship between 2 molecules, related to (n / V)2 by nC2 = n(n-1) / 2
○ a > 0 : attractive, high polarity
○ a < 0 : repulsive, low polarity
② Constant b : Correction constant for the volume occupied by gas molecules
○ b is about twice the size of a gas molecule
③ Relationship with the pressure
④ (Note) Except for critical points in gas/liquid equilibrium, 3 volume corrections are present
⑤ (Note) An equation that has been forced to fit, surprisingly fits well
⑶ Virial Equation of State
① B, C, B’, C’, ··· : Virial coefficients (functions of temperature)
Input: 2018.12.27 20:34