Introduction

This chapter introduces the kinetic theory of gases, which explains the behavior of gases based on the concept that gases consist of rapidly moving atoms or molecules. This theory was developed in the 19th century by scientists like Maxwell and Boltzmann. It successfully explains the molecular interpretation of pressure and temperature, aligns with gas laws and Avogadro's hypothesis, and accurately predicts the specific heat capacities of many gases. Furthermore, it relates observable properties like viscosity, conduction, and diffusion to molecular parameters, allowing for estimations of molecular sizes and masses.

Molecular Nature Of Matter

A fundamental idea in physics, emphasized by Richard Feynman, is that matter is made up of atoms. This "Atomic Hypothesis" posits that all substances are composed of tiny particles that are in perpetual motion, attracting each other at a distance but repelling when compressed. Ancient Indian (Kanada) and Greek (Democritus) scholars also speculated about the existence of atoms. John Dalton is credited with modern atomic theory, which explains the laws of definite and multiple proportions in chemical combinations. Dalton's theory, combined with Gay-Lussac's and Avogadro's laws (which state that equal volumes of all gases at the same temperature and pressure contain the same number of molecules), explains the behavior of gases and the formation of compounds.

Modern science has confirmed that atoms, composed of nuclei and electrons (which are further made of protons, neutrons, and quarks), constitute matter. The size of an atom is approximately 1 angstrom (10^-10 m). In solids and liquids, atoms are closely packed, with separations of a few angstroms. In gases, interatomic distances are much larger, on the order of tens of angstroms, and molecules are relatively free to move. The mean free path is the average distance a molecule travels between collisions. In gases, this path is thousands of angstroms long. The concept of dynamic equilibrium describes gases, where molecules are in constant motion, colliding and changing velocities, but average properties remain constant.

Behaviour Of Gases

The properties of gases are simpler to understand than those of solids and liquids due to the large separation between gas molecules and the negligible interaction forces, except during collisions. Gases, especially at low pressures and high temperatures (well above their liquefaction points), approximate the ideal gas law:

$$ PV = KT $$

where T is the absolute temperature, and K is a constant. Introducing the idea of molecules, K is proportional to the number of molecules (N), with K = Nk_B, where k_B is the Boltzmann constant (1.38 × 10^-23 J K^-1). This leads to Avogadro's hypothesis: equal volumes of all gases at the same temperature and pressure have the same number of molecules (Avogadro number, N_A ≈ 6.02 × 10²³).

The ideal gas law can be written as:

$$ PV = \mu RT $$

where $\mu$ is the number of moles and R is the universal gas constant (R = N_Ak_B ≈ 8.314 J mol^-1 K^-1). Other forms include $PV = k_B NT$ and $P = k_B nT$ (where n is number density), and $P = \rho RT/M$ (where $\rho$ is mass density and M is molar mass).

A gas that perfectly obeys this law under all conditions is called an ideal gas. Real gases deviate from ideal behavior, particularly at high pressures and low temperatures, but approach ideal behavior at low pressures and high temperatures where molecular interactions are minimal.

The ideal gas law leads to:

• Boyle's Law: At constant temperature, PV = constant.
• Charles' Law: At constant pressure, V ∝ T.

For a mixture of ideal gases, the total pressure is the sum of the partial pressures of each gas (Dalton's Law of Partial Pressures), $P = P_1 + P_2 + ...$, where $P_i = \mu_i RT/V$ is the partial pressure of gas i.

Kinetic Theory Of An Ideal Gas

Pressure Of An Ideal Gas

The kinetic theory models a gas as a large number of molecules in continuous random motion, colliding elastically with each other and the container walls. The pressure exerted by a gas is due to the force exerted by these molecules colliding with the walls.

Consider a gas in a cube of side length 'l' and area 'A' (A = l²). A molecule with velocity component v_x in the x-direction hitting a wall perpendicular to the x-axis will rebound with velocity -v_x due to elastic collision. The change in momentum of the molecule is $\Delta p_x = -mv_x - (mv_x) = -2mv_x$. By the conservation of momentum, the momentum imparted to the wall is $2mv_x$.

In a time $\Delta t$, the molecules within a volume Av_xΔt can hit the wall. Assuming half of these are moving towards the wall, the number of collisions with the wall in time $\Delta t$ is $\frac{1}{2} n A v_x \Delta t$, where n is the number density of molecules.

The total momentum transferred to the wall in time $\Delta t$ is $Q = (2mv_x) \times (\frac{1}{2} n A v_x \Delta t) = n m A v_x^2 \Delta t$. The force on the wall is $F = Q/\Delta t = n m A v_x^2$. The pressure P is $F/A = n m v_x^2$. Since gas molecules have random motion in all directions, the average of the square of velocity components are equal: $\overline{v_x^2} = \overline{v_y^2} = \overline{v_z^2}$. Therefore, the mean square velocity $\overline{v^2} = \overline{v_x^2} + \overline{v_y^2} + \overline{v_z^2} = 3\overline{v_x^2}$.

Thus, the pressure is given by:

$$ P = n m \overline{v_x^2} = n m \frac{\overline{v^2}}{3} $$

This can be written as $P = \frac{1}{3} \frac{N}{V} m \overline{v^2}$, where N is the total number of molecules and V is the volume. This equation connects the macroscopic property (pressure) with the microscopic property (mean square speed of molecules).

Kinetic Interpretation Of Temperature

Relating the pressure equation to the ideal gas law ($PV = \frac{2}{3} N (\frac{1}{2} m \overline{v^2})$ and $PV = Nk_B T$), we get:

$$ \frac{2}{3} N \left(\frac{1}{2} m \overline{v^2}\right) = Nk_B T $$

This implies:

$$ \frac{1}{2} m \overline{v^2} = \frac{3}{2} k_B T $$

This is a crucial result: the average translational kinetic energy of a molecule in an ideal gas is directly proportional to the absolute temperature of the gas and is independent of the gas's nature, pressure, or volume. Temperature is a measure of the average kinetic energy of the molecules. The root-mean-square (rms) speed of a molecule is given by:

$$ v_{rms} = \sqrt{\overline{v^2}} = \sqrt{\frac{3k_B T}{m}} $$

For a mixture of gases at the same temperature, the average kinetic energy per molecule is the same for all gases. However, molecules of heavier gases move slower than lighter gases at the same temperature because $v_{rms} \propto 1/\sqrt{m}$.

Law Of Equipartition Of Energy

The Law of Equipartition of Energy, a principle from classical statistical mechanics, states that in thermal equilibrium, the total energy is distributed equally among all possible modes of energy absorption, with each mode having an average energy of $\frac{1}{2}k_B T$.

Degrees of freedom represent the independent ways a molecule can store energy:

• Translational degrees of freedom: A molecule moving in 3D space has 3 translational degrees of freedom (corresponding to motion along x, y, and z axes). Each contributes $\frac{1}{2}k_B T$ to the energy.
• Rotational degrees of freedom: Diatomic molecules can rotate about axes perpendicular to the bond axis. They have 2 rotational degrees of freedom, each contributing $\frac{1}{2}k_B T$. Rotation along the bond axis is generally negligible due to small moment of inertia and quantum effects.
• Vibrational degrees of freedom: Molecules can vibrate along the bond axis. Each vibrational mode contributes both kinetic and potential energy, each contributing $\frac{1}{2}k_B T$, for a total of $k_B T$ per vibrational mode.

The total internal energy of a gas molecule is the sum of energies from all its degrees of freedom.

Specific Heat Capacity

The specific heat capacities of gases can be predicted using the Law of Equipartition of Energy:

Monatomic Gases

Monatomic gases (like Helium, Neon, Argon) have only 3 translational degrees of freedom. The internal energy per mole is $U = \frac{3}{2}RT$. Thus, the molar specific heat at constant volume is:

$$ C_v = \frac{dU}{dT} = \frac{3}{2}R $$

The molar specific heat at constant pressure is $C_p = C_v + R = \frac{5}{2}R$. The ratio $\gamma = C_p/C_v = 5/3 \approx 1.67$.

Diatomic Gases

Diatomic gases (like O₂, N₂), treated as rigid rotators, have 3 translational and 2 rotational degrees of freedom, totaling 5 degrees of freedom. The internal energy per mole is $U = \frac{5}{2}RT$. Thus:

$$ C_v = \frac{5}{2}R $$

And $C_p = \frac{7}{2}R$, with $\gamma = C_p/C_v = 7/5 = 1.40$. If vibrational modes are considered, $C_v$ increases.

Polyatomic Gases

Polyatomic gases have 3 translational, 3 rotational degrees of freedom, and a certain number of vibrational modes (f). The internal energy per mole is $U = (\frac{3}{2} + \frac{3}{2} + f)RT = (\frac{6+f}{2})RT$. Therefore:

$$ C_v = \frac{6+f}{2}R $$

And $C_p = C_v + R = (\frac{8+f}{2})R$. The ratio $\gamma = C_p/C_v = \frac{8+f}{6+f}$. Note that $C_p - C_v = R$ holds for all ideal gases.

Specific Heat Capacity Of Solids

For solids, assuming atoms vibrate in 3 dimensions, each atom has 3 translational degrees of freedom. The internal energy per mole is $U = 3RT$. At constant pressure, the specific heat capacity is $C_v \approx C_p \approx 3R$. This prediction generally agrees with experimental values for many solids at room temperature (Dulong-Petit law), though exceptions exist and the agreement breaks down at low temperatures.

Mean Free Path

The mean free path (l) is the average distance a molecule travels between successive collisions. Even though gas molecules move at high speeds, their paths are constantly interrupted by collisions. The mean free path depends on the density of the gas and the size of the molecules.

Assuming molecules are spheres of diameter 'd', and focusing on one molecule moving with average speed $\overline{v}$, it collides with any molecule within a distance 'd' of its path. In time $\Delta t$, it sweeps a volume $\pi d^2 \overline{v} \Delta t$. If 'n' is the number density of molecules, the number of collisions in $\Delta t$ is $n \pi d^2 \overline{v} \Delta t$. The average time between collisions ($\tau$) is $\tau = 1/(n \pi d^2 \overline{v})$.

The mean free path is then $l = \overline{v} \tau = 1/(n \pi d^2)$. A more accurate treatment considering the relative motion of molecules gives $l = 1/(\sqrt{2} n \pi d^2)$.

For air at STP, with $d \approx 2 \times 10^{-10}$ m and $n \approx 2.7 \times 10^{25}$ m^-3, the mean free path is about $2.9 \times 10^{-7}$ m (or 1500 times the molecular diameter), and the collision time is about $6.1 \times 10^{-10}$ s. This large mean free path is characteristic of gaseous behavior, allowing gases to diffuse and fill containers.

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