9.2 The Ideal Gas Law

The ideal gas law describes the behavior of gases under various conditions. It relates pressure, volume, temperature, and the number of gas particles, assuming certain simplifications about molecular interactions and size.

This model helps predict how gases respond to changes in their environment. While not perfect for all real-world scenarios, it provides a useful framework for understanding gas behavior in many practical applications.

Properties of an Ideal Gas

The classical model of an ideal gas is built upon several key assumptions that simplify our understanding of gas behavior. 🧪

• Gas particles have random instantaneous velocities and move freely in all directions
• The volume of individual gas particles is negligibly small compared to the total volume the gas occupies
• Collisions between gas particles are perfectly elastic, meaning kinetic energy is conserved
• Gas particles experience no forces except during collisions (no intermolecular attractions)
• Energy is transferred only through work and heat

These assumptions allow us to develop a mathematical relationship between the macroscopic properties of gases. The ideal gas equation connects pressure, volume, temperature, and the amount of gas:

$PV = nRT = Nk_B T$

Where:

• $P$ = pressure (in pascals, Pa)
• $V$ = volume (in cubic meters, m³)
• $n$ = number of moles
• $R$ = ideal gas constant (8.31 J/mol·K)
• $T$ = temperature (in Kelvin, K)
• $N$ = number of gas particles
• $k_B$ = Boltzmann constant (1.38 × 10⁻²³ J/K)

The behavior of ideal gases can be visualized through various graphs that illustrate important relationships. 📈

• Pressure-volume graphs show an inverse relationship (Boyle's Law): as pressure increases, volume decreases when temperature is constant
• Volume-temperature graphs show a direct relationship (Charles' Law): as temperature increases, volume increases when pressure is constant
• Pressure-temperature graphs show a direct relationship (Gay-Lussac's Law): as temperature increases, pressure increases when volume is constant

When examining a pressure-temperature graph and extrapolating to where pressure would equal zero, we arrive at absolute zero (0 K or -273.15°C). 🥶

Practice Problem 1: Ideal Gas Law Application

Solution

To solve this problem, we'll use the ideal gas law and recognize that when volume and number of moles remain constant, pressure and temperature are directly proportional.

Starting with the ideal gas law: $PV = nRT$

Since volume and number of moles are constant, we can write: $\frac{P_1}{T_1} = \frac{P_2}{T_2}$

Rearranging to solve for the new pressure: $P_2 = P_1 \times \frac{T_2}{T_1}$

Substituting the given values: $P_2 = 1.5 \text{ atm} \times \frac{450 \text{ K}}{300 \text{ K}} = 1.5 \text{ atm} \times 1.5 = 2.25 \text{ atm}$

Therefore, the new pressure is 2.25 atm.

Practice Problem 2: Absolute Zero Extrapolation

Solution

This problem requires us to extrapolate to find absolute zero using Charles' Law, which states that volume is directly proportional to temperature when pressure is constant.

According to Charles' Law: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$

First, we need to convert the temperature to Kelvin: $T_1 = 27°C + 273.15 = 300.15 \text{ K}$

We want to find the temperature at which the volume becomes zero: $V_2 = 0 \text{ L}$

Rearranging the equation: $T_2 = T_1 \times \frac{V_2}{V_1} = 300.15 \text{ K} \times \frac{0 \text{ L}}{2.0 \text{ L}} = 0 \text{ K}$

Therefore, the volume would theoretically become zero at 0 K (or -273.15°C), which is absolute zero. This illustrates why absolute zero is the lowest possible temperature - the volume of a gas cannot be negative.