States of Matter: Gases and Liquids

Matter exists in three primary classical states—solid, liquid, and gas—as a consequence of the dynamic competition between two fundamental microscopic factors:

• Intermolecular Forces (Attractive Potential Energy): Forces that tend to hold molecules together, favoring condensed states (liquids and solids). These include:
  • Dispersion (London) Forces: Weak attractive forces between non-polar molecules or noble gases arising from temporary, instantaneous dipole moments that induce complementary dipoles in neighboring atoms. Potential energy varies as $V \propto -rac{1}{r^6}$.
  • Dipole-Dipole Forces: Attractive electrostatic interactions between molecules possessing permanent dipole moments (e.g., $\text{HCl}, \text{SO}_2$). For stationary polar molecules, $V \propto -rac{1}{r^3}$; for rotating polar molecules in gases or liquids, $V \propto -rac{1}{r^6}$.
  • Dipole-Induced Dipole Forces: Interactions where a permanent dipole induces a dipole in a non-polar neighbor (e.g., $\text{HCl}$ interacting with $\text{Ar}$ or $\text{Cl}_2$), also exhibiting $V \propto -rac{1}{r^6}$.
  • Hydrogen Bonding: A particularly strong, directional dipole-dipole interaction occurring between hydrogen bonded to small, highly electronegative atoms ($\text{F}, \text{O}, \text{N}$) and an electronegative lone pair on another molecule.
• Thermal Energy (Kinetic Energy): The energy arising from molecular motion, directly proportional to absolute temperature ($T$). Thermal energy promotes randomness, separation, and chaos, favoring the gaseous state.

A gas is characterized macroscopically by four measurable state variables: Pressure ($P$), Volume ($V$), Temperature ($T$), and Amount of substance ($n$). The empirical gas laws govern these variables when two are held constant:

Combining the four proportionalities yields the unified Ideal Gas Equation:

where $R$ is the Universal Gas Constant, representing the work done per degree Kelvin per mole of an ideal gas ($R = \frac{PV}{nT} = \frac{\text{Work}}{\text{mol} \cdot \text{K}}$).

Values of Universal Gas Constant ($R$):

• $R = 0.082057 \approx 0.0821 \; \text{L} \cdot \text{atm} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}$ (Use when $P$ is in atm and $V$ is in liters).
• $R = 0.08314 \; \text{L} \cdot \text{bar} \cdot \text{K}^{-1} \cdot \text{mol}^{-1} = 0.08314 \; \text{dm}^3 \cdot \text{bar} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}$.
• $R = 8.314 \; \text{J} \cdot \text{K}^{-1} \cdot \text{mol}^{-1} = 8.314 \; \text{N} \cdot \text{m} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}$ (Standard SI unit).
• $R = 8.314 \times 10^7 \; \text{erg} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}$ (CGS unit).
• $R = 1.987 \approx 2.0 \; \text{cal} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}$ (Thermochemical unit).

For a fixed mass of gas ($n = \text{constant}$), the Combined Gas Law relates two arbitrary thermodynamic states:

Since the number of moles is $n = \frac{m}{M}$ (where $m$ is mass and $M$ is molar mass), substituting into $PV = nRT$ yields:

This fundamental relation reveals that at constant temperature and pressure, the density of an ideal gas is directly proportional to its molar mass ($d \propto M$).

Formulated by John Dalton in 1801, this law applies to mixtures of non-reacting gases confined in a container of fixed volume $V$ at constant temperature $T$:

For an ideal mixture of $k$ gases with mole numbers $n_1, n_2, \dots, n_k$, each gas follows $p_i = \frac{n_i RT}{V}$. Therefore:

Dividing the partial pressure of component $i$ by the total pressure yields:

where $x_i$ is the mole fraction of component $i$ ($\sum x_i = 1$). The partial pressure of any component in a gas mixture is the product of its mole fraction and the total pressure.

When an insoluble or sparingly soluble gas (e.g., $\text{H}_2, \text{O}_2, \text{N}_2$) is prepared in the laboratory, it is typically collected over water by downward displacement. The collected gas inside the gas jar is saturated with water vapour:

The partial pressure exerted by water vapour in dynamic equilibrium with liquid water at a specific temperature is known as the Aqueous Tension ($p_{\text{aq}}$):

Thomas Graham (1833) experimentally investigated the rates at which gases disperse through one another (diffusion) or escape through a pinhole into a vacuum (effusion):

For two gases 1 and 2 at identical temperature and pressure:

Operational Expressions for Rate ($r$):

• In terms of volume ($V$) diffused per time ($t$): $r = \frac{V}{t} \implies \frac{r_1}{r_2} = \frac{V_1 / t_1}{V_2 / t_2} = \sqrt{\frac{M_2}{M_1}}$. If equal volumes diffuse ($V_1 = V_2$), then $\frac{t_2}{t_1} = \sqrt{\frac{M_2}{M_1}}$.
• In terms of distance ($l$) travelled in a tube of uniform cross-section: $r = \frac{l}{t} \implies \frac{l_1}{l_2} = \sqrt{\frac{M_2}{M_1}}$ (for identical time $t$).
• In terms of moles ($n$) effused per time ($t$): $\frac{n_1 / t_1}{n_2 / t_2} = \sqrt{\frac{M_2}{M_1}}$.
• When pressures are unequal ($P_1 \neq P_2$): The rate of effusion is directly proportional to pressure: $$\frac{r_1}{r_2} = \frac{P_1}{P_2} \sqrt{\frac{M_2}{M_1}}$$

Practical Applications: Graham law is used in separation of isotopes (such as gaseous diffusion of $^{235}\text{UF}_6$ and $^{238}\text{UF}_6$), determination of unknown molar masses, and detecting explosive methane leaks in coal mines using atmolysis.

Developed by Bernoulli, Clausius, Maxwell, and Boltzmann, the Kinetic Molecular Theory of Gases (KMTG) provides a microscopic mechanical explanation for macroscopic gas laws based on the following fundamental postulates:

1. Particulate Nature & Vast Separation: A gas consists of an enormous number of extremely minute, identical spherical particles (atoms or molecules). The actual volume occupied by the individual gas molecules is negligible compared to the total volume of the container ($V_{\text{molecules}} \ll V_{\text{container}}$).
2. Continuous Random Motion: Gas molecules are in ceaseless, chaotic, rapid straight-line motion in all possible directions. During this motion, they collide with one another and with the walls of the container.
3. Negligible Intermolecular Forces: Under ordinary conditions, attractive and repulsive forces between gas molecules are negligible. The molecules behave completely independently of one another.
4. Perfect Elasticity of Collisions: All collisions (molecule-molecule and molecule-wall) are perfectly elastic. Although individual molecules may exchange momentum and kinetic energy during a collision, the total translational kinetic energy of the entire system remains conserved.
5. Origin of Gas Pressure: The pressure exerted by a gas is the cumulative macroscopic force resulting from the continuous bombardment of gas molecules per unit area of the container walls ($P = \frac{F}{A} = \frac{\Delta p / \Delta t}{A}$).
6. Absolute Temperature and Kinetic Energy: The average kinetic energy of gas molecules is directly proportional to the absolute temperature of the gas ($E_k \propto T$). At absolute zero ($0 \; \text{K}$), molecular motion theoretically ceases.

By analyzing the rate of change of momentum of $N$ molecules, each of mass $m$, colliding with the mutually perpendicular planar walls of a cubic container of edge length $L$ and volume $V = L^3$, Clausius derived the Kinetic Gas Equation:

where $u_{\text{rms}}$ is the root-mean-square speed of the gas molecules. For 1 mole of gas ($N = N_A$, so $m N_A = M$, the molar mass):

Total Kinetic Energy Expressions:

• Total Kinetic Energy for $n$ moles: $E_k = \frac{1}{2} M_{\text{total}} u_{\text{rms}}^2 = \frac{1}{2} (n M) \left(\frac{3RT}{M}\right) = \frac{3}{2} nRT$.
• Translational Kinetic Energy per mole ($n=1$): $E_{k, \text{molar}} = \frac{3}{2} RT$. (Notice this depends strictly on temperature $T$ and is completely independent of the nature or molar mass of the gas!).
• Translational Kinetic Energy per molecule: Dividing by Avogadro's number $N_A$: $$E_{k, \text{molecule}} = \frac{3}{2} \frac{R}{N_A} T = \frac{3}{2} k_B T$$ where $k_B = \frac{R}{N_A} = 1.3806 \times 10^{-23} \; \text{J} \cdot \text{K}^{-1}$ is the Boltzmann Constant.

Because gas molecules constantly undergo millions of random collisions per second, their speeds are continuously changing. In 1859, James Clerk Maxwell and Ludwig Boltzmann applied statistical mechanics to derive the probability distribution function of molecular speeds in a three-dimensional gas:

where $\frac{dN_u}{N}$ represents the fraction of total gas molecules possessing speeds between $u$ and $u + du$.

Salient Features of the Maxwell-Boltzmann Distribution Curve:

• The fraction of molecules with zero speed is zero (curve starts at origin, $(0,0)$).
• The curve rises sharply with increasing speed, reaches a distinctive maximum corresponding to the most probable speed ($u_{\text{mp}}$), and then descends asymptotically toward zero at very high speeds, forming an asymmetrical bell curve with a long tail.
• The total area under the curve represents the total fraction of molecules and equals unity: $\int_0^\infty \frac{1}{N} \frac{dN_u}{du} du = 1$.
• Effect of Temperature: As temperature increases ($T_2 > T_1$), the peak shifts to the right (higher speed) and simultaneously broadens and flattens (lower peak height). This proves that at higher temperatures, a significantly greater fraction of molecules possess high speeds.
• Effect of Molar Mass: At a given temperature, lighter gases (e.g., $\text{H}_2, \text{He}$) have flatter, broader distributions shifted toward higher speeds, whereas heavier gases (e.g., $\text{O}_2, \text{Cl}_2$) have narrower, taller peaks shifted toward lower speeds.

To characterize the speed distribution of billions of gas molecules moving in all directions, three mathematically distinct statistical speeds are employed:

1. Most Probable Speed ($u_{\text{mp}}$ or $\alpha$):

   The speed possessed by the maximum fraction of gas molecules at a given temperature. It corresponds to the mathematical peak (maximum) of the Maxwell-Boltzmann distribution curve (found by setting $\frac{d}{du}\left[\frac{dN_u}{du}\right] = 0$):

   $$u_{\text{mp}} = \sqrt{\frac{2RT}{M}} = \sqrt{\frac{2k_B T}{m}} = \sqrt{\frac{2PV}{M}} = \sqrt{\frac{2P}{d}}$$
2. Average Speed ($u_{\text{av}}$ or $\bar{u}$):

   The arithmetic mean of the speeds of all gas molecules present in the sample ($u_{\text{av}} = \frac{u_1 + u_2 + \dots + u_N}{N}$):

   $$u_{\text{av}} = \sqrt{\frac{8RT}{\pi M}} = \sqrt{\frac{8k_B T}{\pi m}} = \sqrt{\frac{8PV}{\pi M}} = \sqrt{\frac{8P}{\pi d}} \approx \sqrt{\frac{2.546 RT}{M}}$$
3. Root-Mean-Square Speed ($u_{\text{rms}}$ or $c$):

   The square root of the arithmetic mean of the squares of the speeds of all molecules ($u_{\text{rms}} = \sqrt{\frac{u_1^2 + u_2^2 + \dots + u_N^2}{N}}$). This speed represents the direct kinetic energy equivalence of the gas:

   $$u_{\text{rms}} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3k_B T}{m}} = \sqrt{\frac{3PV}{M}} = \sqrt{\frac{3P}{d}}$$

Comparing the mathematical expressions under identical conditions of temperature ($T$) and molar mass ($M$):

All three speeds obey identical functional proportionalities:

• For the same gas at different temperatures $T_1$ and $T_2$: $$\frac{u_1}{u_2} = \sqrt{\frac{T_1}{T_2}}$$
• For two different gases 1 and 2 at the same temperature $T$: $$\frac{u_1}{u_2} = \sqrt{\frac{M_2}{M_1}}$$
• Equal Speed Condition: If the root-mean-square speed of gas 1 at temperature $T_1$ equals the root-mean-square speed of gas 2 at temperature $T_2$: $$\sqrt{\frac{3RT_1}{M_1}} = \sqrt{\frac{3RT_2}{M_2}} \implies \frac{T_1}{M_1} = \frac{T_2}{M_2}$$

Under experimental conditions of high pressure and low temperature, real gases show marked deviations from ideal gas behavior. The Dutch physicist Johannes Diderik van der Waals (1873) identified two faulty assumptions in the classical Kinetic Molecular Theory:

1. Assumption 1 (Negligible Volume of Molecules): KMT assumes that gas molecules occupy zero volume. In reality, molecules are hard, incompressible spheres possessing finite volume. At high pressure, the volume of the container becomes very small, making the volume occupied by the molecules themselves significant. Hence, the actual free space available for molecular movement is $(V - nb)$, where $b$ is the excluded volume or co-volume per mole.
2. Assumption 2 (Zero Intermolecular Attraction): KMT assumes no forces of attraction exist between molecules. In reality, intermolecular attractive forces pull striking molecules inward, reducing the net momentum transferred per collision to the wall. The measured pressure $P_{\text{real}}$ is less than the ideal pressure $P_{\text{ideal}}$ by a pressure correction factor $p' = \frac{a n^2}{V^2}$.

Introducing the pressure correction $P_{\text{ideal}} = P + \frac{a n^2}{V^2}$ and volume correction $V_{\text{ideal}} = V - nb$ into the ideal gas equation yields the celebrated van der Waals Equation:

Physical Significance and Units of van der Waals Constants:

• van der Waals Constant $a$: Measures the magnitude of intermolecular attractive forces.
  • Units: $\text{atm} \cdot \text{L}^2 \cdot \text{mol}^{-2}$ or $\text{bar} \cdot \text{dm}^6 \cdot \text{mol}^{-2}$ or $\text{N} \cdot \text{m}^4 \cdot \text{mol}^{-2}$ (SI).
  • Higher $a$ values indicate stronger intermolecular attractions, implying easier liquefaction (e.g., $a_{\text{SO}_2} > a_{\text{NH}_3} > a_{\text{CO}_2} > a_{\text{CH}_4} > a_{\text{O}_2} > a_{\text{N}_2} > a_{\text{H}_2} > a_{\text{He}}$).
• van der Waals Constant $b$: Measures the effective incompressible volume (co-volume) of the molecules.
  • Units: $\text{L} \cdot \text{mol}^{-1}$ or $\text{dm}^3 \cdot \text{mol}^{-1}$ or $\text{m}^3 \cdot \text{mol}^{-1}$ (SI).
  • $b = 4 N_A v = 4 N_A \left(\frac{4}{3}\pi r^3\right)$, proving that excluded volume is exactly four times the actual spherical volume of the molecules.

The deviation of a real gas from ideality is quantitatively evaluated by the Compressibility Factor ($Z$):

• Ideal Gas: $Z = 1$ under all conditions of temperature and pressure.
• Case I: Low to Moderate Pressure Regime ($P < 200 \; \text{atm}$): $V_m$ is relatively large, making $b$ negligible compared to $V_m$ ($V_m - b \approx V_m$): $$\left(P + \frac{a}{V_m^2}\right)V_m = RT \implies PV_m + \frac{a}{V_m} = RT \implies Z = \frac{PV_m}{RT} = 1 - \frac{a}{V_m RT}$$ Since $\frac{a}{V_m RT} > 0$, $Z < 1$. Attractive forces dominate; the gas is more compressible than an ideal gas ($V_{\text{real}} < V_{\text{ideal}}$).
• Case II: High Pressure Regime ($P > 500 \; \text{atm}$): $P$ is very large, so the attractive correction $\frac{a}{V_m^2}$ is negligible compared to $P$ ($P + \frac{a}{V_m^2} \approx P$): $$P(V_m - b) = RT \implies PV_m - Pb = RT \implies Z = \frac{PV_m}{RT} = 1 + \frac{Pb}{RT}$$ Since $\frac{Pb}{RT} > 0$, $Z > 1$. Molecular volume repulsions dominate; the gas is less compressible than an ideal gas ($V_{\text{real}} > V_{\text{ideal}}$).
• Case III: Hydrogen ($\text{H}_2$) and Helium ($\text{He}$): Because of extremely small molecular masses and minimal polarizability, London dispersion forces are virtually non-existent ($a \approx 0$) at ordinary temperatures. Thus: $$Z = 1 + \frac{Pb}{RT} > 1 \quad \text{at all pressures!}$$ The curves for $\text{H}_2$ and $\text{He}$ lie entirely above $Z = 1$ without showing an initial dip.
• Boyle Temperature ($T_b$): The specific temperature at which the attractive term $\frac{a}{V_m}$ and repulsive term $\frac{Pb}{RT}$ cancel out over an appreciable pressure range, making $Z \approx 1$: $$T_b = \frac{a}{Rb}$$

Thomas Andrews (1869) studied the $P-V$ isotherms of carbon dioxide ($\text{CO}_2$), demonstrating that a gas can only be liquefied below a specific threshold temperature known as the Critical Temperature ($T_c$).

Expanding the 1-mole van der Waals equation as a cubic equation in $V_m$:

At the critical point $(P_c, T_c)$, the three roots of this cubic equation coalesce into a single real root: $(V_m - V_c)^3 = 0 \implies V_m^3 - 3V_c V_m^2 + 3V_c^2 V_m - V_c^3 = 0$. Equating coefficients yields the fundamental Critical Constants:

In a closed evacuated container partially filled with liquid, molecules with kinetic energy exceeding the average escape into the vapour phase (evaporation). Simultaneously, vapour molecules collide with the liquid surface and are recaptured (condensation). Eventually, a dynamic equilibrium is reached:

Vapour Pressure: The equilibrium pressure exerted by the vapour phase above the liquid at a constant temperature.

• Factors Influencing Vapour Pressure:
  • Nature of Liquid: Weaker intermolecular forces result in higher volatility and higher vapour pressure (e.g., $P_{\text{diethyl ether}} > P_{\text{ethanol}} > P_{\text{water}}$).
  • Temperature: Vapour pressure increases exponentially with temperature according to the Clausius-Clapeyron equation: $\ln\left(\frac{P_2}{P_1}\right) = \frac{\Delta H_{\text{vap}}}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)$.
• Boiling Point: The temperature at which the equilibrium vapour pressure of the liquid equals the surrounding external atmospheric pressure.
  • Normal Boiling Point: Boiling temperature when external pressure is exactly $1 \; \text{atm} = 1.01325 \; \text{bar}$ (for pure water, $100.0^\circ\text{C}$).
  • Standard Boiling Point: Boiling temperature when external pressure is exactly $1 \; \text{bar} = 0.987 \; \text{atm}$ (for pure water, $99.6^\circ\text{C}$).

Molecules within the bulk of a liquid experience isotropic (equal) cohesive forces in all directions from surrounding neighbors. However, molecules at the liquid-gas interface lack liquid neighbors above them and experience a net inward unbalanced cohesive force pulling them toward the bulk.

• Units: SI unit is $\text{N} \cdot \text{m}^{-1}$ or $\text{J} \cdot \text{m}^{-2}$; CGS unit is $\text{dyn} \cdot \text{cm}^{-1}$ or $\text{erg} \cdot \text{cm}^{-2}$ ($1 \; \text{N} \cdot \text{m}^{-1} = 10^3 \; \text{dyn} \cdot \text{cm}^{-1}$).
• Physical Consequences:
  • Spherical Shape of Liquid Drops: To minimize surface free energy, liquids naturally minimize surface area. For a given volume, a sphere possesses the minimum surface area, causing raindrops, soap bubbles, and mercury droplets to assume spherical shapes.
  • Capillary Action: The rise or fall of liquids inside narrow capillary tubes: $$h = \frac{2\gamma \cos\theta}{r \rho g}$$ where $\theta$ is the contact angle. When adhesive forces (liquid-glass) exceed cohesive forces ($\theta < 90^\circ$, e.g., water), capillary rise occurs and the meniscus is concave. When cohesive forces exceed adhesive forces ($\theta > 90^\circ$, e.g., mercury), capillary depression occurs and the meniscus is convex.
  • Effect of Temperature: As temperature increases, kinetic energy weakens cohesive forces; surface tension decreases linearly and becomes zero at the critical temperature $T_c$.

When a real liquid flows smoothly through a pipe or across a flat surface, it moves in parallel layers or laminae with varying velocities (laminar flow). The layer in direct contact with the solid boundary is stationary ($u = 0$), while successive layers move with progressively increasing velocities up to a maximum velocity $u_{\text{max}}$ at the center.

The velocity gradient perpendicular to the direction of flow is $\frac{du}{dz}$. Isaac Newton established that the internal frictional shearing force ($F$) acting between adjacent liquid layers of contact area $A$ is proportional to the area and the velocity gradient:

where $\eta$ is the Coefficient of Viscosity, defined as the tangential shearing force per unit area required to maintain a unit velocity gradient between two parallel liquid layers unit distance apart.

• Units of Viscosity:
  • SI Unit: $\text{Pa} \cdot \text{s} = \text{N} \cdot \text{s} \cdot \text{m}^{-2} = \text{kg} \cdot \text{m}^{-1} \cdot \text{s}^{-1}$.
  • CGS Unit: Poise (P) $= \text{dyn} \cdot \text{s} \cdot \text{cm}^{-2} = \text{g} \cdot \text{cm}^{-1} \cdot \text{s}^{-1}$.
  • Conversion: $1 \; \text{Pa} \cdot \text{s} = 10 \; \text{Poise} = 1000 \; \text{centipoise (cP)}$. (Water at $20^\circ\text{C}$ has $\eta \approx 1.002 \; \text{cP} = 1.002 \times 10^{-3} \; \text{Pa} \cdot \text{s}$).
• Effect of Temperature on Viscosity:
  • Liquids: Viscosity decreases rapidly with increasing temperature because elevated thermal energy overcomes intermolecular hydrogen bonding and van der Waals attractions (empirically $\eta = A e^{E_a / RT}$).
  • Gases: In stark contrast to liquids, gas viscosity increases with temperature ($\eta \propto \sqrt{T}$) due to increased momentum transfer between rapidly colliding gas molecules.