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Chapter 9. Gases

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1. Overview

2. History of Gas Studies

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

PressureLaw

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

Rule 1: Gases undergo continuous random walk motion.

Rule 2: Point mass : Gas molecules have infinitesimally small volume.

Rule 3: Gas particles undergo straight-line motion.

Rule 4: No interaction : Gases do not interact except during collisions.

Rule 5: Elastic collision : Collisions between gas molecules are perfectly elastic.

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.

Period : The time it takes for a gas molecule to collide with the right side.

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.

Force : The average force exerted by a gas molecule on the right side : Force = Fa = Rate of change of momentum = dp / dt

Leap of faith : Science sometimes makes risky assumptions, but if the results are beautiful, they are forgiven.

Pressure derivation : Using the definition of pressure P = F / A

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

Diffusion : Process where a substance mixes into another due to molecular collisions

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

Meaning 1: Set to make the sum of all values equal to 1

Meaning 2: Form of the normal distribution function

Meaning 3: Setting to match the formula of root mean square velocity

Result 1: Root Mean Square Velocity (vrms)

Result 2: Average Velocity (vavg) : Can be calculated through integration

Result 3: Most Probable Velocity (vmp) : Can be found by differentiating to locate the maximum

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

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