Matter exists in three common physical states: solid, liquid and gas. The state of a substance depends on the balance between the kinetic energy of its particles, which tends to spread them apart, and the intermolecular forces, which tend to pull them together. This chapter examines the gaseous state in detail, develops the ideal gas laws, and then treats real gases, the kinetic molecular theory and the liquid state.
Gases are the easiest state to describe quantitatively because their particles are far apart and weakly interacting. This simplicity made the gas laws the foundation of physical chemistry. Boyle, Charles and Avogadro established the fundamental relationships between pressure, volume, temperature and the number of moles, and the ideal gas equation combines them into a single powerful expression.
Real gases deviate from ideal behaviour, particularly at high pressure and low temperature, because intermolecular attractions become significant and molecules occupy finite volume. Johannes van der Waals corrected the ideal gas equation to account for these effects. This chapter also covers the liquefaction of gases, the vapour pressure of liquids and the liquid crystal state, giving a complete picture of how matter behaves under different conditions.
Gases have no fixed shape or volume; they expand to fill their container completely. They are highly compressible because the particles are far apart with large empty spaces between them. Gases have very low densities compared with solids and liquids, mix readily with one another by diffusion, and exert pressure on the walls of their container through the bombardment of their particles.
The macroscopic properties used to describe a gas are pressure (P), volume (V), temperature (T) and the amount of substance (n). Pressure is defined as force per unit area and is measured in atmosphere (atm), bar, pascal (Pa) or torr, where 1 atm = 1.01325 bar = 1.01325 x 10^5 Pa = 760 mm Hg = 760 torr. Temperature must be expressed on the Kelvin scale in gas law calculations because the Celsius scale can give negative values.
The measured properties of a gas are related to the average kinetic energy of its particles. Increasing temperature raises the average kinetic energy, which increases the frequency and force of particle collisions with the container walls and hence the pressure. This microscopic picture links the observable gas laws to molecular motion.
Boyle's law states that at constant temperature, the volume of a fixed amount of gas is inversely proportional to its pressure:
$$V \propto \frac{1}{P} \quad \text{or} \quad PV = \text{constant}$$
Charles's law states that at constant pressure, the volume of a fixed amount of gas is directly proportional to its absolute temperature:
$$V \propto T$$
Gay Lussac's law states that at constant volume, the pressure of a fixed amount of gas is directly proportional to its absolute temperature. Avogadro's law states that equal volumes of all gases at the same temperature and pressure contain equal numbers of molecules.
Combining all these laws gives the ideal gas equation:
$$PV = nRT$$
where R is the universal gas constant, R = 0.0821 L atm K^-1 mol^-1 = 8.314 J K^-1 mol^-1. For a fixed amount of gas undergoing a change of state, the combined gas law applies:
$$\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$$
Dalton's law of partial pressures states that the total pressure of a mixture of non-reacting gases equals the sum of the partial pressures of the individual gases:
$$P_{\text{total}} = p_1 + p_2 + p_3 + \cdots$$
The kinetic molecular theory explains gas behaviour with a set of assumptions. Gases consist of a large number of identical particles moving randomly in all directions. The volume of the individual particles is negligible compared with the volume of the container. The particles exert no force on each other except during brief collisions, which are perfectly elastic. The average kinetic energy of the particles is directly proportional to the absolute temperature.
From these assumptions, the theory derives the pressure of a gas and the relationship between average kinetic energy and temperature:
$$KE_{\text{avg}} = \frac{3}{2}kT = \frac{3}{2}\frac{R}{N_A}T$$
where k is the Boltzmann constant. At a given temperature, all gases have the same average kinetic energy, but lighter molecules move faster. The root mean square speed of gas molecules is:
$$u_{\text{rms}} = \sqrt{\frac{3RT}{M}}$$
The distribution of molecular speeds is described by the Maxwell-Boltzmann distribution. As temperature increases, the distribution shifts toward higher speeds and broadens. The average speed, most probable speed and rms speed differ, with the rms speed always the largest of the three.
Real gases deviate from ideal behaviour, especially at high pressure and low temperature. The compressibility factor is defined as:
$$Z = \frac{PV}{nRT}$$
For an ideal gas, Z = 1 at all pressures. For real gases, Z deviates from 1, being less than 1 at moderate pressures (because attractive forces dominate) and greater than 1 at very high pressures (because the finite volume of molecules becomes significant). Gases such as H2 and He show Z greater than 1 even at moderate pressures because they have weak attractive forces.
Van der Waals corrected the ideal gas equation for the finite volume of molecules and the intermolecular attractions:
$$\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT$$
The constant a accounts for the attractive forces between molecules, while b accounts for the finite volume of molecules. The van der Waals equation behaves like the ideal gas equation when P is low and T is high.
The conditions under which a real gas approximates ideal behaviour are low pressure, high temperature and small molecular size. The behaviour of a real gas is represented by isotherms known as Andrews curves, which show how pressure varies with volume at constant temperature.
A gas can be liquefied by compression and cooling. For every gas there is a temperature above which it cannot be liquefied by pressure alone, called the critical temperature (Tc). Below the critical temperature, a gas can be liquefied by applying sufficient pressure. Above it, the substance exists only as a gas, however high the pressure.
At the critical temperature, the gas liquefies at the critical pressure (pc) occupying the critical volume (Vc). The critical constants are related to the van der Waals constants:
$$T_c = \frac{8a}{27Rb}, \quad p_c = \frac{a}{27b^2}, \quad V_c = 3b$$
Gases with high critical temperatures, such as ammonia and carbon dioxide, are easily liquefied because their intermolecular forces are strong. Helium and hydrogen, with very low critical temperatures, are the hardest to liquefy. The Joule-Thomson effect, the cooling of a gas when it expands through a throttle, is used in the liquefaction of gases.
Liquids have a definite volume but no definite shape; they take the shape of their container. The particles in a liquid are close together but can move past one another, giving liquids their fluidity. The density of a liquid is much higher than that of its gas but slightly lower than that of its solid (with the exception of water).
Vapour pressure is the pressure exerted by the vapour in equilibrium with its liquid at a given temperature. It increases with temperature because more molecules gain enough kinetic energy to escape the liquid surface. A liquid boils when its vapour pressure equals the atmospheric pressure. Since atmospheric pressure decreases at high altitude, water boils at a lower temperature in the mountains.
Boiling, evaporation and condensation are important liquid-state processes. Evaporation occurs at the surface of a liquid at any temperature and causes cooling because the most energetic molecules escape. Boiling occurs throughout the liquid at its boiling point. The liquid crystal state shows properties intermediate between liquids and solids, such as flowing like a liquid while maintaining some molecular order.
| Law | Relationship | Statement |
|---|---|---|
| Boyle's law | V proportional to 1/P (constant T) | PV = constant |
| Charles's law | V proportional to T (constant P) | V/T = constant |
| Gay Lussac's law | P proportional to T (constant V) | P/T = constant |
| Avogadro's law | V proportional to n (constant P, T) | V/n = constant |
| Dalton's law | P total = sum of partial pressures | Applicable to gas mixtures |
| Property | Solid | Liquid | Gas |
|---|---|---|---|
| Shape | Fixed | Container shaped | Container shaped |
| Volume | Fixed | Fixed | Container volume |
| Compressibility | Negligible | Very low | High |
| Density | High | High | Very low |
| Particle motion | Vibrational | Translational, close | Rapid, far apart |
| Quantity | Value |
|---|---|
| R | 0.0821 L atm K^-1 mol^-1 |
| R | 8.314 J K^-1 mol^-1 |
| Standard temperature | 273.15 K (0 degrees Celsius) |
| Standard pressure | 1 atm = 760 mm Hg = 1.01325 bar |
| Molar volume at STP | 22.4 L mol^-1 |
| Avogadro's number | 6.022 x 10^23 mol^-1 |
| Boltzmann constant | 1.38 x 10^-23 J K^-1 |
The states of matter represent different balances between kinetic energy and intermolecular forces. The gaseous state, with its weak interactions, obeys the elegant ideal gas laws embodied in PV = nRT, and the kinetic molecular theory provides the microscopic explanation for these macroscopic relationships. Real gases deviate from ideal behaviour through intermolecular attractions and molecular volume, corrected by the van der Waals equation, and their liquefaction depends on the critical temperature. The liquid state is understood through vapour pressure, boiling and evaporation. Together, these ideas describe how matter responds to pressure and temperature and provide the physical foundation for phase behaviour, equilibria and thermodynamic studies in the chapters ahead.