Behaviour of Gases — Boyle, Charles & Avogadro

Why gases are the simple case

Properties of gases are easier to describe than those of solids and liquids. In a gas the molecules are far apart and their mutual interactions are negligible except during the brief instant of a collision. With the forces between molecules effectively switched off, the bulk behaviour collapses to one experimental relation. NCERT writes it for a fixed sample as $PV = KT$, where $T$ is the absolute (kelvin) temperature and $K$ is constant for that sample but grows with the amount of gas.

This simplicity holds only in a regime — low pressure and high temperature, well above the temperature at which the gas would liquefy or solidify. There the molecules are widely spaced, interactions vanish, and the gas approaches what we will define below as an ideal gas. The four classical gas laws are simply this one relation read with one variable held fixed at a time.

The four empirical gas laws

The gas laws share an identical logical shape: hold some quantities constant, and two of the remaining variables are tied by a simple proportionality. Reading them as a set, rather than as four unrelated facts, is the fastest route through this topic.

Each is a special case of $PV = KT$ with the labelled variables frozen. Charles's, Gay-Lussac's and Avogadro's laws are direct proportionalities, so all three graph as straight lines through the origin; Boyle's law is an inverse proportionality, so it graphs as a hyperbola. Three of the four demand that $T$ be measured on the absolute (kelvin) scale, which is the most exploited trap in the chapter.

Boyle's law in detail

Fix the amount of gas and the temperature in $PV = KT$ and the right-hand side is constant, so

$$PV = \text{constant} \qquad (T,\, n \text{ fixed}).$$

At constant temperature, the pressure of a given mass of gas varies inversely with its volume. Halve the volume and the pressure doubles. On a $P$–$V$ graph each fixed temperature gives one curve, a rectangular hyperbola called an isotherm; a higher temperature shifts the whole hyperbola outward. NCERT notes that experimental $P$–$V$ curves agree with Boyle's law best at high temperatures and low pressures — the same ideal regime.

Charles's law in detail

Now fix the pressure. NCERT reads off from $PV = KT$ that $V \propto T$: at fixed pressure, the volume of a given mass of gas is proportional to its absolute temperature. The graph of $V$ against absolute $T$ is a straight line through the origin. Plotted against Celsius temperature instead, the same data is still a straight line, but it intercepts the temperature axis at $-273.15~^\circ\text{C}$ — the experimental signpost to absolute zero.

The companion pressure law (Gay-Lussac) fixes the volume instead: $P \propto T$. Heating a gas in a sealed rigid container raises its pressure in direct proportion to absolute temperature. Both are the same straight-line proportionality with a different variable held constant.

Avogadro's law and the mole

Avogadro's law (originally a hypothesis) states that equal volumes of all gases at the same temperature and pressure contain equal numbers of molecules. NCERT reaches it by writing $K = N k_B$ in $PV = KT$, where $N$ is the number of molecules and $k_B$ is the same constant for every gas:

$$\frac{PV}{NT} = k_B = \text{constant for all gases}.$$

If $P$, $V$ and $T$ are the same, then $N$ is the same — irrespective of the chemical identity of the gas. Equivalently, the number density (molecules per unit volume) is identical for all gases at a fixed temperature and pressure.

The number of molecules in 22.4 litres of any gas at STP is the Avogadro number $N_A = 6.02 \times 10^{23}$, and that amount of substance is one mole. Avogadro inferred the equality of numbers in equal volumes from chemical reactions; kinetic theory later justified it from first principles.

Combining into the ideal gas equation

Putting $K = N k_B$ back into $PV = KT$ gives the molecular form $PV = N k_B T$. Grouping molecules into moles via $N = \mu N_A$ and defining $R = N_A k_B$ produces the form most used in NEET:

$$PV = \mu R T = N k_B T,$$

where $\mu$ is the number of moles, $N$ the number of molecules, $R$ the universal gas constant and $k_B$ the Boltzmann constant. The number of moles itself can be read two ways NCERT gives explicitly,

$$\mu = \frac{N}{N_A} = \frac{M}{M_0},$$

where $M$ is the mass of the sample and $M_0$ its molar mass. Writing $P = k_B n T$ with $n = N/V$ the number density, or $P = \rho R T / M_0$ with $\rho$ the mass density, are two further rearrangements of the same equation. A gas that satisfies $PV = \mu R T$ exactly at all pressures and temperatures is, by definition, an ideal gas — a theoretical model that no real gas matches perfectly.

The constants R and k

$R$ and $k_B$ are the same physics seen at two scales, and confusing them is a standard NEET error. $R$ is the gas constant per mole; $k_B$ is the gas constant per molecule. They are linked through Avogadro's number.

Dalton's law of partial pressures

For a mixture of non-interacting ideal gases — $\mu_1$ moles of gas 1, $\mu_2$ of gas 2, and so on — sharing a vessel of volume $V$ at temperature $T$, the equation of state is

$$PV = (\mu_1 + \mu_2 + \dots)RT = \frac{\mu_1 RT}{V} + \frac{\mu_2 RT}{V} + \dots = P_1 + P_2 + \dots$$

The term $P_i = \mu_i RT/V$ is the partial pressure of component $i$ — the pressure that gas alone would exert in the same vessel at the same temperature. The total pressure of a mixture of ideal gases is the sum of the partial pressures. This is Dalton's law of partial pressures, and it follows immediately from applying $PV = \mu RT$ to the whole mixture.

The ideal gas as a limit; deviations of real gases

No real gas is truly ideal. The ideal model assumes point molecules with no intermolecular forces — true only when molecules are far apart, which means low pressure or high temperature. As NCERT puts it, at low pressures or high temperatures the molecules are far apart and molecular interactions are negligible, so the gas behaves like an ideal one. The clean diagnostic is the dimensionless ratio $PV/\mu RT$, which equals exactly 1 for an ideal gas at every state point.

NCERT's Fig. 12.1 shows the same gas at three temperatures: all curves approach ideal behaviour as pressure falls and as temperature rises, and the higher-temperature curve hugs the ideal line over a wider pressure range. This is the qualitative content NEET expects — the direction of approach to ideality, not a quantitative equation of state for real gases.