Gases

Postulates of the Kinetic Theory of Gases

The kinetic theory of gases explains the behavior of an ideal gas. Its main postulates are as follows:

The forces of attraction and repulsion between gas molecules are negligible.
Gas molecules move randomly and collide with each other and the walls of the container.
Collisions between molecules are perfectly elastic, meaning no energy is lost. Although individual molecules may gain or lose energy during collisions, the total kinetic energy remains constant.
The actual volume of the gas molecules is negligible compared to the volume of the container. Gas molecules are extremely small relative to the space between them.
The temperature of a gas reflects the average kinetic energy of its molecules.

Gas Laws

To study the behavior of gases in relation to volume, temperature, and pressure, the following conditions are investigated:

Variation of volume with pressure at constant temperature (Boyle’s Law): $ PV = \text{constant} $
Variation of volume with temperature at constant pressure (Charles’ Law): $ \frac{V}{T} = \text{constant} $
Variation of pressure with temperature at constant volume (Pressure Law): $ \frac{P}{T} = k $

Boyle’s Law

Boyle’s law states that the pressure of a fixed mass of gas varies inversely with its volume at constant temperature.

\[ P \propto \frac{1}{V} \] \[ P = kV \] \[ P_1 V_1 = P_2 V_2 \]

Charles’ Law

Charles' law states that for a fixed mass of gas at constant pressure, the volume is proportional to its absolute temperature.

\[ V \propto T \] \[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]

Gas Law Example: Applying Charles’s Law

A gas occupies a volume of 20.0 dm³ at 373 K. Its volume at 746 K (with constant pressure) will be determined using Charles’s Law.

Given Data:

$ V_1 = 20.0 \, \text{dm}^3 $
$ T_1 = 373 \, \text{K} $
$ T_2 = 746 \, \text{K} $
$ V_2 = ? $

Applying Charles’s Law:

Charles’s Law states:

\[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]

Rearranging for $ V_2 $:

\[ V_2 = V_1 \times \frac{T_2}{T_1} \]

Calculation:

\[ V_2 = 20.0 \times \frac{746}{373} \] \[ V_2 = 40.0 \, \text{dm}^3 \]

Pressure Law

Pressure law states that the pressure of a fixed mass of gas at constant volume is proportional to its absolute temperature.

\[ P \propto T \] \[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \]

Absolute Zero of Temperature

When graphs of volume–temperature or pressure–temperature are extrapolated backward, they cut the temperature axis at -273°C. This temperature is called absolute zero. It is the temperature at which the volume of the gas theoretically becomes zero as it is cooled. At this temperature, gas molecules stop moving completely.

However, this is a theoretical assumption since gases typically liquefy before reaching this temperature.

General Gas Law

The general gas law combines Boyle’s Law, Charles’ Law, and the Pressure Law.

From Boyle’s Law:

\[ PV = k \]

From Charles’ Law:

\[ \frac{V}{T} = k \]

From Gay-Lussac’s (Pressure) Law:

\[ \frac{P}{T} = k \]

Combining these, we get:

\[ \frac{PV}{T} = k \] \[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \]

This equation is known as the general gas law.

It can also be written as:

\[ PV = nRT \]

Where:

$ n $ = number of moles of gas
$ R $ = universal gas constant = 8.31 J K$^{-1}$ mol$^{-1}$

Dalton’s Law of Partial Pressure

Dalton’s law states that the total pressure exerted by a mixture of gases at constant temperature is equal to the sum of the pressures each gas would exert if it were alone.

Formula:

$$ P_T = P_1 + P_2 + P_3 + \ldots + P_n $$

where:

P_T = total pressure
P₁ to P_n = partial pressures of the gases

Graham’s Law of Diffusion

Graham’s law states that the rate at which a gas diffuses at constant temperature and pressure is inversely proportional to the square root of its density.

Formula:

$$ \frac{R_1}{R_2} = \sqrt{\frac{q_2}{q_1}} $$

where:

R₁ and R₂ = rates of diffusion of the two gases
q₁ and q₂ = densities of the gases

Since density is related to molar mass, Graham’s law can also be expressed as:

$$ \frac{R_1}{R_2} = \sqrt{\frac{M_2}{M_1}} $$

where:

M₁ and M₂ = molar masses of the gases

Recall that density (𝑞) is mass (m) per unit volume (v):

$$ q = \frac{m}{v} $$

Therefore, another version of Graham’s law using volume is:

$$ \frac{R_1}{R_2} = \sqrt{\frac{V_1}{V_2}} $$

Gay-Lussac’s Law of Combining Volumes

Gay-Lussac’s law states that when gases react, they do so in simple whole number ratios by volume, provided temperature and pressure remain constant. The same applies to the volumes of gaseous products formed.

Avogadro’s Law

Avogadro’s law states that equal volumes of all gases, at the same temperature and pressure, contain the same number of molecules, regardless of the gases’ chemical or physical properties.