Gases

States of Matter & Gaseous Bulk Properties

Matter exists in four fundamental physical states: solid, liquid, gas, and plasma. BUMHS 2024

Among the physical states:

Plasma represents over 99% of the visible universe (stars, interstellar nebulas, and fusion cores). It consists of an ionized thermal mixture of positive ions, free conduction electrons, and neutral atoms.
The gaseous state is the structurally simplest state of matter. Gases lack definite shape and definite volume, expanding spontaneously to fill the total volume of their container.
Over 99.9% of a gas volume is empty intermolecular space, leaving only ~0.1% occupied by actual molecules. This immense void makes gases extraordinarily compressible under mechanical pressure.

Macroscopic Characteristics of Gases

Spontaneous Diffusion: Molecules disperse continuously from regions of high concentration to low concentration.
Effusion: Gas particles escape through pinhole orifices into vacuum without molecular collisions.
High Thermal Expansion: Gases expand uniformly when heated at constant pressure with high cubic expansion coefficients.
Isotropic Pressure: Exert equal pressure in all directions against container boundaries due to non-directional particle collisions.

Crucial Physiological Boundary

Is plasma a simple chemical compound like ether?

No! Plasma is the fourth state of matter consisting of ionized gas with free electrons. Ether (diethyl ether) is an organic liquid solvent.

Provincial Board Variance

BUMHS: BUMHS past papers test that plasma forms the vast majority of the visible cosmos, whereas solids and liquids are cosmically rare.

STB: STB notes that intermolecular forces in gases are negligible at standard room temperature and atmospheric pressure.

Atmospheric Pressure & Unit Conversions

Pressure ($P$) is defined as normal force exerted per unit surface area:

$$P = \frac{F}{A}$$

Atmospheric air pressure is measured using a mercury barometer (invented by Torricelli), while the pressure of an enclosed laboratory gas sample is quantified using a manometer.

The SI coherent unit of pressure is the Pascal ($\text{Pa}$), defined as one Newton per square meter ($\text{N/m}^2$). NUMS 2023

Standard Atmospheric Pressure Equivalences at Sea Level

1 atmosphere (atm)
101,325 Pa or 101.325 kPa
760 mm Hg or 760 torr
76 cm Hg (76 cm of mercury, not 760 cm!) ETEA 2023
1.01325 bar
14.7 psi (pounds per square inch)

Crucial Physiological Boundary

Does 1 atm equal 760 cm of mercury?

No! 1 atm equals 76 cm of mercury or 760 mm of mercury. Confusing centimeters with millimeters is a frequent exam pitfall!

Provincial Board Variance

PTB: PTB defines 1 torr as exactly equal to 1 mm of mercury column height at 0 degrees Celsius.

FTB: FTB provides engineering conversion: 1 bar equals $10^5\text{ Pa}$, so 1 atm = 1.01325 bar.

Kinetic Molecular Theory & Speed Distributions

Formulated by Bernoulli and advanced by Clausius, Maxwell, and Boltzmann, the Kinetic Molecular Theory (KMT) bridges microscopic molecular dynamics with macroscopic gas laws:

Gases consist of tiny, discrete particles (molecules or noble gas atoms).
Molecules undergo continuous, rapid, random rectilinear motion, colliding elastically with each other and container walls. NUMS 2023
Actual volume of gas molecules is strictly negligible compared to total container volume.
Intermolecular attractive and repulsive forces between ideal gas molecules are strictly zero. UHS 2024
Gas collisions are completely elastic: total kinetic energy is conserved ($Q = 0$), preventing molecules from settling under gravity. SZABMU 2022
Average translational kinetic energy of gas molecules is directly proportional to absolute Kelvin temperature:

$$\overline{\text{KE}} = \frac{3RT}{2N_A} \propto T$$ KMU 2026

At identical temperature, all gases share the exact same average translational kinetic energy regardless of molar mass. UHS 2023

Clausius derived the fundamental kinetic pressure equation:

$$PV = \frac{1}{3} m N \overline{c^2}$$

Molecular Speed Formulations

Root-Mean-Square Speed ($c_{\text{rms}} Robertson$):
$$c_{\text{rms}} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3PV}{M}} = \sqrt{\frac{3P}{d}}$$ NUMS 2024
Average Speed ($c_{\text{avg}} Robertson$):
$$c_{\text{avg}} = \sqrt{\frac{8RT}{\pi M}} \approx 0.921 \, c_{\text{rms}}$$
Most Probable Speed ($c_{\text{mp}} Robertson$):
$$c_{\text{mp}} = \sqrt{\frac{2RT}{M}} \approx 0.816 \, c_{\text{rms}}$$
Speed Hierarchy:
$$c_{\text{rms}} > c_{\text{avg}} > c_{\text{mp}}$$
Ratio: $\sqrt{3} : \sqrt{\frac{8}{\pi}} : \sqrt{2} \approx 1.224 : 1.128 : 1.000$

Crucial Physiological Boundary

Do lighter gas molecules have higher kinetic energy than heavier molecules at equal temperature?

No! Average kinetic energy depends strictly on absolute temperature alone. Lighter molecules move with higher root-mean-square speed, but their average kinetic energy is identical!

Provincial Board Variance

KTB: KTB explicitly calculates the speed ratio $c_{\text{mp}} : c_{\text{avg}} : c_{\text{rms}} = 1 : 1.128 : 1.224$.

PTB: PTB derives root-mean-square speed from the kinetic pressure equation by substituting ideal gas relations.

Boyle's Law & Isothermal Compression

Robert Boyle (1662) established the inverse relationship between pressure and volume for a fixed quantity of gas at constant thermal temperature:

$$V \propto \frac{1}{P} \iff P_1 V_1 = P_2 V_2 = k \quad (T, n = \text{constant})$$

A plot of pressure ($P$) versus volume ($V$) at constant temperature produces a smooth rectangular hyperbola termed an isotherm. SZABMU 2024

Boyle's Law Graphical Archetypes

$P$ vs $V$ Graph: Rectangular hyperbola asymptotic to both axes. At higher temperature ($T_2 > T_1$), the isotherm shifts outward away from origin. DUHS 2023
$V$ vs $1/P$ Graph: Straight line passing through the coordinate origin. Slope equals constant $k = nRT$.
$PV$ vs $P$ Graph: Perfectly horizontal straight line parallel to the pressure axis, verifying that product $PV$ remains constant.

Crucial Physiological Boundary

Does Boyle's law hold true if gas temperature changes during compression?

No! Boyle's law is strictly isothermal. If mechanical work heats the gas during rapid compression, pressure will rise faster than predicted by Boyle's law.

Provincial Board Variance

BUMHS: BUMHS notes atmospheric air density decreases at high mountain altitudes because reduced barometric pressure allows gas volume expansion.

PTB: PTB emphasizes that plotting PV against P yields a zero-slope horizontal line for ideal gases.

Charles's Law & Absolute Zero Extrapolation

Jacques Charles (1787) formulated the direct proportional coupling between gas volume and absolute Kelvin temperature at constant isobaric pressure:

$$V \propto T \iff \frac{V_1}{T_1} = \frac{V_2}{T_2} = k \quad (P, n = \text{constant})$$

For every $1^\circ\text{C}$ rise or fall in temperature, the volume of a gas expands or contracts by a constant fraction:

$$\Delta V = V_0 \left(1 + \frac{t}{273.15}\right)$$

where $V_0$ is volume at $0^\circ\text{C}$ and $t$ is Celsius temperature.

A plot of volume versus temperature at constant pressure yields a straight line termed an isobar. Extrapolating the isobar to zero volume ($V = 0$) intercepts the temperature axis at:

$$\text{Absolute Zero} = -273.15^\circ\text{C} = 0\text{ K}$$ KMU 2026, DUHS 2024

Physical Nature of Absolute Zero

Absolute zero is the lowest theoretical temperature at which all molecular translational motion ceases.
Real gases condense into liquids and freeze into solids long before reaching $-273.15^\circ\text{C}$, preventing experimental attainment of true zero volume. BUMHS 2022
Temperature conversions:
$$T(\text{K}) = t(^\circ\text{C}) + 273.15$$
$$t(^\circ\text{C}) = \frac{5}{9}\left[t(^\circ\text{F}) - 32\right]$$ BUMHS 2024

Crucial Physiological Boundary

Can a gas actually achieve zero volume at absolute zero?

No! Real gas molecules possess finite incompressible molecular volume (van der Waals b). Gases liquefy into condensed phases before reaching absolute zero.

Provincial Board Variance

STB: STB records absolute zero as -273.16 degrees Celsius in its reference tables, while other provincial boards record -273.15 degrees Celsius.

KTB: KTB notes calculations round absolute zero addition to 273 for standard entrance examination problems.

Avogadro's Law & Molar Volume at STP

Amedeo Avogadro (1811) established that equal volumes of all ideal gases under identical conditions of temperature and pressure contain equal numbers of particles:

$$V \propto n \iff \frac{V_1}{n_1} = \frac{V_2}{n_2} \quad (T, P = \text{constant})$$

Standard Molar Constants at STP

Standard Temperature & Pressure (STP): $T = 0^\circ\text{C}$ ($273.15\text{ K}$) and $P = 1.00\text{ atm}$ ($101.325\text{ kPa}$).
Molar Volume ($V_m$): One mole of any ideal gas occupies:
$$V_m = 22.414\text{ dm}^3 = 22,414\text{ cm}^3 = 0.022414\text{ m}^3$$
Avogadro's Number ($N_A$): One mole of gas contains:
$$N_A = 6.022 \times 10^{23}\text{ molecules}$$
Room Temperature & Pressure (RTP): $25^\circ\text{C}$ ($298\text{ K}$) and $1\text{ atm}$, where molar volume expands to 24.45 dm^3.

Crucial Physiological Boundary

Do equal volumes of different gases at STP have equal masses?

No! Equal volumes at STP contain equal molecule counts (6.022 x 10^23), but their masses differ according to their molecular weights: 22.414 dm^3 of O2 weighs 32 g, while 22.414 dm^3 of H2 weighs only 2 g!

Provincial Board Variance

PTB: PTB highlights that 1 mole of H2 (2.016 g) and 1 mole of He (4.00 g) occupy identical 22.414 dm^3 volumes at STP.

BTB: BTB notes that real gas molar volumes deviate slightly from 22.414 dm^3 (e.g. CO2 is 22.25 dm^3 at STP).

Ideal Gas Equation & Universal Gas Constant R

Synthesizing Boyle's, Charles's, and Avogadro's laws yields the Ideal Gas Equation:

$$PV = nRT$$ UHS 2024, NUMS 2022

For a fixed mass of gas transitioning between thermodynamic states, the Combined Gas Law applies:

$$\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$$ DUHS 2023

Molar Mass & Gas Density Expressions

Moles ($n = \frac{m}{M}$) substitution yields molar mass:
$$M = \frac{mRT}{PV}$$
Gas Density ($d = \frac{m}{V}$) formulation:
$$d = \frac{PM}{RT}$$ UHS 2023
Gas density is directly proportional to pressure and molar mass, and inversely proportional to absolute temperature.

The numerical magnitude of the Universal Gas Constant ($R$) depends strictly upon the units chosen for pressure and volume:

Universal Gas Constant R Across Physical Units

Atmospheric Unit:
$$R = 0.0821\text{ atm}\cdot\text{dm}^3\cdot\text{K}^{-1}\cdot\text{mol}^{-1}$$
SI Metric Unit:
$$R = 8.314\text{ J}\cdot\text{K}^{-1}\cdot\text{mol}^{-1} = 8.314\text{ N}\cdot\text{m}\cdot\text{K}^{-1}\cdot\text{mol}^{-1}$$
Millimeter Mercury / Torr Unit:
$$R = 62.4\text{ dm}^3\cdot\text{torr}\cdot\text{K}^{-1}\cdot\text{mol}^{-1} = 62,400\text{ cm}^3\cdot\text{torr}\cdot\text{K}^{-1}\cdot\text{mol}^{-1}$$
Calorie Unit:
$$R = 1.987\text{ cal}\cdot\text{K}^{-1}\cdot\text{mol}^{-1} \approx 2\text{ cal}\cdot\text{K}^{-1}\cdot\text{mol}^{-1}$$

Crucial Physiological Boundary

Does doubling both absolute temperature and volume change gas pressure?

No! By PV = nRT, P = nRT/V. If T and V both double simultaneously, the factor 2/2 cancels, leaving gas pressure completely unchanged!

Provincial Board Variance

STB: STB records R in calories as 1.99 cal/mol K.

KTB: KTB records R as 1.987 cal/mol K and 8.313 J/mol K.

BTB: BTB records R as 1.986 cal/mol K and 0.08206 dm^3 atm/mol K.

FTB: FTB records R as 62.4 dm^3 torr/mol K and 8.3143 J/mol K.

PTB: PTB adopts standard values R = 0.0821 dm^3 atm/mol K and 8.314 J/mol K.

Real Gas Non-Ideality & Van der Waals Mechanics

Real gases deviate from ideal behavior at High Pressure and Low Temperature:

High Pressure: Gas molecules are forced close together; the actual volume of molecules becomes a significant fraction of total volume, making real gases less compressible than predicted.
Low Temperature: Molecular kinetic energy decreases, allowing intermolecular attractive forces (van der Waals forces) to pull molecules together, reducing wall collision pressure.

The Compressibility Factor (Z)

$$Z = \frac{PV}{nRT} = \frac{V_{\text{real}}}{V_{\text{ideal}}}$$
Ideal Gas: $Z = 1$ at all temperatures and pressures.
$Z < 1$ (Negative Deviation): Intermolecular attractive forces dominate, compressing the gas more than predicted (observed in $CH_4, CO_2, SO_2$).
$Z > 1$ (Positive Deviation): Incompressible molecular volume dominates; observed in $H_2$ and $He$ even at moderate pressures due to tiny mass and negligible attractions.

Johannes Diderik van der Waals (1873) resolved non-ideality by applying two corrective terms to the ideal gas law:

Pressure Correction ($P_{\text{corrected}} = P + \frac{an^2}{V^2}$): Corrects for attractive forces pulling inward on boundary molecules. Constant $a$ measures the strength of intermolecular attraction.
Volume Correction ($V_{\text{corrected}} = V - nb$): Corrects for actual molecular volume. Constant $b$ represents the excluded volume or effective co-volume:

$$b = 4 V_m$$

Excluded volume is four times the actual physical volume of the spherical molecules.

The Van der Waals Equation

$$\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT$$
Units of constant $a$: $\text{atm}\cdot\text{dm}^6\cdot\text{mol}^{-2}$ (or $\text{N}\cdot\text{m}^4\cdot\text{mol}^{-2}$).
Units of constant $b$: $\text{dm}^3\cdot\text{mol}^{-1}$ (or $\text{m}^3\cdot\text{mol}^{-1}$).

Crucial Physiological Boundary

Is excluded volume equal to actual molecular volume?

No! Excluded volume b equals four times (4x) the actual molecular volume because each molecule excludes a spherical collision space around itself.

Provincial Board Variance

PTB: PTB derives that excluded volume b is 4 times the volume of individual gas molecules.

KTB: KTB notes gases with strong dipoles (SO2, NH3) have much higher a constants than non-polar noble gases.

States of Matter & Gaseous Bulk Properties

Macroscopic Characteristics of Gases

Atmospheric Pressure & Unit Conversions

Standard Atmospheric Pressure Equivalences at Sea Level

Kinetic Molecular Theory & Speed Distributions

Molecular Speed Formulations

Boyle's Law & Isothermal Compression

Boyle's Law Graphical Archetypes

Charles's Law & Absolute Zero Extrapolation

Physical Nature of Absolute Zero

Avogadro's Law & Molar Volume at STP

Standard Molar Constants at STP

Ideal Gas Equation & Universal Gas Constant R

Molar Mass & Gas Density Expressions

Universal Gas Constant R Across Physical Units

Real Gas Non-Ideality & Van der Waals Mechanics

The Compressibility Factor (Z)

The Van der Waals Equation

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