16.4 Gas Laws, Ideal Gas Equation, Kinetic Molecular Theory & Real Gases

Gas Laws, Ideal Gas Equation, Kinetic Molecular Theory & Real Gases

In the Intermediate Chemistry Part I curriculum and the ECAT examination, the behavior of gases provides the primary testing ground for kinetic theory, thermodynamic equations of state, and non-ideal fluid dynamics. Examination problems require candidates to manipulate Boyle's, Charles's, and Dalton's laws under rapid calculation constraints, evaluate Graham's effusion ratios without square-root calculators, and interpret the physical significance of van der Waals constants ($a$ and $b$). Understanding these kinetic models ensures mastery over quantitative chemical physics questions.

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1. Parameters of the Gaseous State & Pressure Units

A gas is macroscopically defined by four mutually interdependent state variables: Pressure ($P$), Volume ($V$), Absolute Temperature ($T$), and Amount of Substance ($n$).

Pressure Unit Equivalences

Pressure is force per unit area ($P = \frac{F}{A}$). On the ECAT, proficiency with international and customary pressure units is critical:

$1\text{ atm} = 760\text{ mmHg} = 760\text{ torr} = 101,325\text{ Pa} = 101.325\text{ kPa} = 1.01325\text{ bar} = 14.7\text{ psi}$

$1\text{ torr} = 1\text{ mmHg} = 133.322\text{ Pa}$

$1\text{ bar} = 10^5\text{ Pa} = 100\text{ kPa} \approx 0.987\text{ atm}$

Temperature Scales and Absolute Zero

Gas law calculations strictly require thermodynamic absolute temperature in Kelvin:

$T\text{ (K)} = t\text{ (}^\circ\text{C)} + 273.15 \approx t + 273$

$\text{Rankine Scale: } T\text{ (R)} = t\text{ (}^\circ\text{F)} + 459.67$

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2. The Empirical Gas Laws

1. Boyle's Law (Robert Boyle, 1662)

"At constant temperature, the volume of a fixed mass of gas is inversely proportional to the applied pressure."

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

• Graphical Representations (Isotherms):
  1. $P$ versus $V$: An equilateral rectangular hyperbola asymptotic to both axes.
  2. $P$ versus $\frac{1}{V}$: A straight line passing through the origin with slope equal to $k = nRT$.
  3. $PV$ versus $P$: A horizontal line parallel to the pressure axis.
  • Temperature Effect: Increasing temperature shifts the $P-V$ isotherm outward away from both axes, because higher kinetic energy expands the gas at any given pressure.

2. Charles's Law (Jacques Charles, 1787)

"At constant pressure, the volume of a fixed mass of gas is directly proportional to its absolute thermodynamic temperature."

$V \propto T \implies \frac{V}{T} = k' \implies \frac{V_1}{T_1} = \frac{V_2}{T_2} \quad (P, n \text{ constant})$

• Quantitative Celsius Formulation:

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

For every $1^\circ\text{C}$ rise or fall in temperature, the volume of a gas expands or contracts by $\frac{1}{273.15}$ of its original volume at $0^\circ\text{C}$.

• Absolute Zero ($-273.15^\circ\text{C} = 0\text{ K}$): Extrapolating the $V$ vs $t\text{ (}^\circ\text{C)}$ isobar downward intersects the temperature axis at exactly $-273.15^\circ\text{C}$, where theoretical gas volume reaches zero. In reality, all real gases condense into liquids and freeze into solids long before absolute zero is attained.

3. Avogadro's Law (Amedeo Avogadro, 1811)

"Equal volumes of all gases at the same temperature and pressure contain equal numbers of molecules."

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

At STP ($0^\circ\text{C}, 1\text{ atm}$), $1\text{ mol}$ of any ideal gas occupies $V_m = 22.414\text{ dm}^3$.

4. Gay-Lussac's / Amontons's Law

"At constant volume, the pressure of a fixed mass of gas is directly proportional to its absolute temperature."

$P \propto T \implies \frac{P}{T} = k'' \implies \frac{P_1}{T_1} = \frac{P_2}{T_2} \quad (V, n \text{ constant})$

• Graphical Representation (Isochores): A plot of $P$ versus $T\text{ (K)}$ yields a straight line radiating from absolute zero.

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3. The Ideal Gas Equation & Universal Gas Constant ($R$)

Combining Boyle's, Charles's, and Avogadro's laws yields the Ideal Gas Equation of State:

$PV = nRT$

Substituting $n = \frac{m}{M}$:

$PV = \frac{m}{M}RT$

Density and Molar Mass Derivations

Rearranging for gas density ($d = \frac{m}{V}$ in $\text{g}\cdot\text{dm}^{-3}$ or $\text{g}\cdot\text{L}^{-1}$):

$P = \frac{m}{V}\frac{RT}{M} = \frac{d RT}{M} \implies d = \frac{PM}{RT}$

$\text{Molar Mass from Experimental Density: } M = \frac{d RT}{P}$

ECAT Speed Insight: Gas density is directly proportional to pressure and molar mass, and inversely proportional to absolute temperature. Heavy gases at high pressure and low temperature are maximally dense.

Numerical Values of the Universal Gas Constant ($R$)

The numerical value and physical dimensions of $R$ depend strictly on the units chosen for pressure and volume:

1. In Atmospheric Units ($\text{dm}^3\cdot\text{atm}\cdot\text{K}^{-1}\cdot\text{mol}^{-1}$):

$R = \frac{(1\text{ atm})(22.414\text{ dm}^3)}{(1\text{ mol})(273.15\text{ K})} = 0.08206 \approx 0.0821\text{ dm}^3\cdot\text{atm}\cdot\text{K}^{-1}\cdot\text{mol}^{-1}$

1. In SI Energy Units (Joules / $\text{N}\cdot\text{m}$):

$R = \frac{(101,325\text{ N}\cdot\text{m}^{-2})(0.022414\text{ m}^3)}{(1\text{ mol})(273.15\text{ K})} = 8.3143 \approx 8.314\text{ J}\cdot\text{K}^{-1}\cdot\text{mol}^{-1}$

Because $1\text{ J} = 10^7\text{ ergs}$, $R = 8.314 \times 10^7\text{ erg}\cdot\text{K}^{-1}\cdot\text{mol}^{-1}$.

1. In Thermal Energy (Calories):

$R = \frac{8.314\text{ J}\cdot\text{K}^{-1}\cdot\text{mol}^{-1}}{4.184\text{ J}\cdot\text{cal}^{-1}} \approx 1.987\text{ cal}\cdot\text{K}^{-1}\cdot\text{mol}^{-1} \approx 2.0\text{ cal}\cdot\text{K}^{-1}\cdot\text{mol}^{-1}$

1. In Customary Pressure Units (Torr / mmHg):

$R = 0.0821 \times 760 = 62.36 \approx 62.4\text{ dm}^3\cdot\text{torr}\cdot\text{K}^{-1}\cdot\text{mol}^{-1} = 62,400\text{ cm}^3\cdot\text{torr}\cdot\text{K}^{-1}\cdot\text{mol}^{-1}$

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4. Dalton's Law of Partial Pressures & Aqueous Tension

Dalton's Law Formulation (John Dalton, 1801)

"The total pressure exerted by a mixture of non-reacting gases confined in a container of fixed volume is equal to the sum of the partial pressures that each constituent gas would exert if it alone occupied the entire volume at the same temperature."

$P_{\text{total}} = P_A + P_B + P_C + \dots = \sum_{i=1}^k P_i$

Assuming ideal gas behavior for each component:

$P_i = n_i \left(\frac{RT}{V}\right), \quad P_{\text{total}} = n_{\text{total}} \left(\frac{RT}{V}\right)$

$\frac{P_i}{P_{\text{total}}} = \frac{n_i}{n_{\text{total}}} = X_i \implies P_i = X_i P_{\text{total}}$

where $X_i$ is the mole fraction of gas $i$ ($\sum X_i = 1$).

Collection of Gas Over Water & Aqueous Tension

When a gas (such as $\text{H}_2, \text{O}_2, \text{N}_2$) is collected in the laboratory by the downward displacement of water, it becomes saturated with water vapor. The partial pressure exerted by saturated water vapor at a given temperature is termed aqueous tension ($P_{\text{water vapor}}$):

$P_{\text{moist gas}} = P_{\text{dry gas}} + P_{\text{water vapor}} = P_{\text{dry gas}} + \text{Aqueous Tension}$

$P_{\text{dry gas}} = P_{\text{total (observed)}} - \text{Aqueous Tension}$

To apply ideal gas calculations to the collected gas, one must strictly use $P_{\text{dry gas}}$.

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5. Graham's Law of Diffusion & Effusion

• Diffusion: The spontaneous intermingling of gas molecules from a region of higher concentration to lower concentration driven by random thermal motion until a homogeneous mixture is attained.
• Effusion: The escape of gas molecules through a microscopic pinhole aperture into an evacuated chamber without molecular collisions occurring within the orifice.

Mathematical Statement of Graham's Law (Thomas Graham, 1833)

"Under identical conditions of temperature and pressure, the rate of diffusion or effusion ($r$) of a gas is inversely proportional to the square root of its molar mass or density:"

$r \propto \frac{1}{\sqrt{d}} \propto \frac{1}{\sqrt{M}}$

$\frac{r_1}{r_2} = \sqrt{\frac{d_2}{d_1}} = \sqrt{\frac{M_2}{M_1}}$

Alternative Operational Expressions for Rate ($r$)

Because rate of diffusion can be quantified in terms of volume diffused per unit time ($r = \frac{V}{t}$), distance traversed in a tube ($r = \frac{l}{t}$), or moles effused ($r = \frac{n}{t}$):

1. For equal volumes of gases effusing ($V_1 = V_2$):

$\frac{r_1}{r_2} = \frac{V/t_1}{V/t_2} = \frac{t_2}{t_1} = \sqrt{\frac{M_2}{M_1}}$

1. For equal times of effusion ($t_1 = t_2$):

$\frac{r_1}{r_2} = \frac{V_1/t}{V_2/t} = \frac{V_1}{V_2} = \sqrt{\frac{M_2}{M_1}}$

ECAT Benchmark Example: Compare the effusion rates of $\text{H}_2$ ($M=2$) and $\text{CH}_4$ ($M=16$): $\frac{r_{\text{H}_2}}{r_{\text{CH}_4}} = \sqrt{\frac{16}{2}} = \sqrt{8} \approx 2.83$ Hydrogen effuses nearly $2.83$ times faster than methane.

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6. Kinetic Molecular Theory (KMT) of Gases

Developed by Bernoulli, Clausius, Maxwell, and Boltzmann, the Kinetic Molecular Theory provides a microscopic mechanical explanation for macroscopic gas laws.

Core Postulates of KMT

1. Particulate Nature: All gases consist of a vast ensemble of identical, tiny, discrete particles (atoms or molecules) separated by immense empty distances.
2. Negligible Molecular Volume: The actual physical volume occupied by the gas molecules themselves is entirely negligible compared to the total volume of the container ($V_{\text{molecules}} \ll V_{\text{container}}$).
3. Constant Chaotic Motion: Molecules are in a state of continuous, rapid, chaotic, rectilinear motion, colliding frequently with one another and with the container walls.
4. Origin of Pressure: The pressure exerted by a gas results from continuous momentum transfer during collisions between molecules and the container walls.
5. Perfect Elasticity: All intermolecular and wall collisions are perfectly elastic; no net kinetic energy is dissipated as heat during collisions ($\sum KE = \text{constant}$ at fixed $T$).
6. Absence of Intermolecular Forces: There are no attractive or repulsive forces between ideal gas molecules; they move completely independently of one another.
7. Negligible Gravitational Influence: The kinetic energy of molecular motion is so high that gravitational forces on individual molecules are negligible.
8. Kinetic Interpretation of Temperature: The average translational kinetic energy of gas molecules is directly proportional to absolute thermodynamic temperature:

$\overline{KE} = \frac{1}{2}m\overline{c^2} = \frac{3}{2}kT = \frac{3}{2}\left(\frac{R}{N_A}\right)T$

where $k = 1.3806 \times 10^{-23}\text{ J}\cdot\text{K}^{-1}$ is the Boltzmann constant. At absolute zero ($T = 0\text{ K}$), all translational molecular motion ceases entirely.

Molecular Speeds: Root-Mean-Square Velocity ($c_{\text{rms}}$)

From the fundamental kinetic pressure equation derived by Clausius ($P = \frac{1}{3}\frac{m N}{V}\overline{c^2}$):

$c_{\text{rms}} = \sqrt{\overline{c^2}} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3kT}{m}} = \sqrt{\frac{3P}{\rho}}$

Comparing three characteristic molecular speeds:

• Most Probable Speed ($c_{\text{mp}}$): $c_{\text{mp}} = \sqrt{\frac{2RT}{M}} \approx 1.414\sqrt{\frac{RT}{M}}$
• Average Speed ($\bar{c}$): $\bar{c} = \sqrt{\frac{8RT}{\pi M}} \approx 1.596\sqrt{\frac{RT}{M}}$
• Root-Mean-Square Speed ($c_{\text{rms}}$): $c_{\text{rms}} = \sqrt{\frac{3RT}{M}} \approx 1.732\sqrt{\frac{RT}{M}}$

$c_{\text{mp}} : \bar{c} : c_{\text{rms}} = 1.000 : 1.128 : 1.224$

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7. Non-Ideal (Real) Gases & The Van der Waals Equation

Deviations from Ideality & Compressibility Factor ($Z$)

Real gases deviate markedly from ideal gas behavior under two specific physical conditions: high pressure and low temperature.

The deviation is quantitatively measured by the Compressibility Factor ($Z$):

$Z = \frac{PV}{nRT} = \frac{V_{\text{real}}}{V_{\text{ideal}}}$

• Ideal Gas: $Z = 1$ under all temperatures and pressures.
• Low to Moderate Pressure ($Z < 1$): Intermolecular attractive forces pull molecules toward one another, reducing wall collision frequency and pressure. The gas is more compressible than an ideal gas ($V_{\text{real}} < V_{\text{ideal}}$).
• High Pressure ($Z > 1$): Molecules are compressed so tightly that their finite molecular volume resists further compression. Repulsive forces dominate, making the gas less compressible than an ideal gas ($V_{\text{real}} > V_{\text{ideal}}$).

Flawed Postulates of KMT

Van der Waals pointed out that two foundational postulates of KMT fail for real gases:

1. KMT Postulate 2 is invalid at high pressure: When gas is compressed into a small volume, the actual volume of the molecules constitutes a substantial fraction of the total container space.
2. KMT Postulate 6 is invalid at low temperature: As molecules lose kinetic energy at low temperatures, attractive intermolecular van der Waals forces pull molecules together, leading eventually to liquefaction.

The Van der Waals Equation of State (1873)

Johannes D. van der Waals modified the ideal gas equation by introducing two corrective terms:

$\left[P + \frac{a n^2}{V^2}\right](V - n b) = nRT$

For $1\text{ mole}$ of gas ($n=1$):

$\left[P + \frac{a}{V_m^2}\right](V_m - b) = RT$

Physical Significance of Van der Waals Constants

1. Constant $a$ (Pressure Correction / Attraction Constant):

   • Corrects for intermolecular attractive forces pulling striking molecules backward into the bulk gas.
   • $P_{\text{ideal}} = P_{\text{real}} + \frac{an^2}{V^2}$.
   • Units of $a$: $\text{atm}\cdot\text{dm}^6\cdot\text{mol}^{-2}$ or $\text{N}\cdot\text{m}^4\cdot\text{mol}^{-2}$.
   • Physical Meaning: Measures the strength of attractive forces between gas molecules. Highly polar and easily liquefiable gases ($\text{NH}_3, \text{SO}_2, \text{H}_2\text{O}$) possess high $a$ values; small nonpolar gases ($\text{He}, \text{H}_2$) possess very low $a$ values.

2. Constant $b$ (Volume Correction / Co-Volume):

   • Corrects for the finite physical volume occupied by the gas molecules themselves.
   • $V_{\text{ideal}} = V_{\text{real}} - nb$.
   • Units of $b$: $\text{dm}^3\cdot\text{mol}^{-1}$ or $\text{m}^3\cdot\text{mol}^{-1}$.
   • Physical Meaning: Designated as the effective excluded volume or co-volume. For spherical molecules of radius $r$, the excluded volume per mole is exactly four times the actual molecular volume:

$b = 4 N_A V_{\text{molecule}} = 4 N_A \left(\frac{4}{3}\pi r^3\right)$

When Do Real Gases Approach Ideal Behavior?

Real gases behave most ideally at extremely low pressure (where molecules are far apart, rendering molecular volume and attractions negligible) and extremely high temperature (where rapid thermal kinetic energy overcomes intermolecular attractions).

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8. Liquefaction of Gases and the Fourth State of Matter

A gas can be liquefied only below its critical temperature $T_c$, the temperature above which no amount of pressure produces a liquid. The critical pressure $P_c$ is the pressure needed to liquefy the gas at $T_c$.

Gases with strong intermolecular attractions (large van der Waals $a$) have high critical temperatures. Permanent gases such as oxygen and nitrogen are liquefied by the Linde method: the gas is compressed, cooled, and allowed to expand through a fine nozzle. This Joule–Thomson effect cools the gas because work is done against intermolecular attractions; repeating the cycle lowers the temperature until the gas liquefies.

Plasma is often called the fourth state of matter. It is a gas in which a significant fraction of atoms is ionised into positive ions and free electrons, for example in lightning, neon signs, fluorescent tubes and the Sun. A plasma conducts electricity and responds strongly to electric and magnetic fields, although overall it is electrically neutral.