Gases and the absolute scale of temperature

1. Overview

This topic explores how gas particles behave when we change their environment. By understanding the relationship between pressure, volume, and temperature at a molecular level, we can predict how gases will respond in everything from bicycle pumps to weather balloons.

Key Definitions

• Pressure: The force exerted by gas particles colliding with the walls of a container per unit area.
• Absolute Zero: The lowest possible temperature (0 Kelvin or -273°C), where particles have minimum kinetic energy and effectively stop moving.
• Kelvin Scale: An absolute temperature scale that starts at absolute zero.
• Kinetic Energy: The energy an object has due to its motion. In gases, temperature is a direct measure of the average kinetic energy of the particles.

Core Content

(a) Effect of Temperature on Pressure (Constant Volume)

When the temperature of a gas increases in a fixed-volume container:

• The particles gain more kinetic energy and move faster.
• They collide with the container walls more frequently.
• They hit the walls with greater force.
• Result: The pressure increases.

(b) Effect of Volume on Pressure (Constant Temperature)

When the volume of a gas is decreased (compressed) at a constant temperature:

• The particles are pushed closer together.
• The particles collide with the walls more frequently because there is less space to travel between impacts.
• Result: The pressure increases. (Conversely, increasing volume decreases pressure).

Temperature Conversions

To work with the absolute scale, you must be able to convert between Celsius (°C) and Kelvin (K).

• Equation: $T (\text{in K}) = \theta (\text{in °C}) + 273$

Worked Example:

• Question: A gas is at $25°C$. What is this temperature in Kelvin?
• Answer: $T = 25 + 273 = 298\text{ K}$

---

Extended Content (Extended Only)

The $PV = \text{constant}$ Relationship (Boyle’s Law)

For a fixed mass of gas at a constant temperature, pressure and volume are inversely proportional. If you double the pressure, the volume halves.

• Equation: $P_1 V_1 = P_2 V_2$ (where 1 is the initial state and 2 is the final state)
• Graphical Representation:
  • A graph of Pressure vs. Volume shows a curve (hyperbola). As volume decreases, pressure rises exponentially.
  • A graph of Pressure vs. 1/Volume shows a straight line through the origin, proving the inverse relationship.

Worked Example:

• Question: A gas occupies $2.0\text{ m}^3$ at a pressure of $100,000\text{ Pa}$. If the gas is compressed to $0.5\text{ m}^3$ at a constant temperature, what is the new pressure?
• Answer:
  1. $P_1 V_1 = P_2 V_2$
  2. $100,000 \times 2.0 = P_2 \times 0.5$
  3. $200,000 = P_2 \times 0.5$
  4. $P_2 = 200,000 / 0.5 = 400,000\text{ Pa}$

---

Key Equations

1. Temperature Conversion: $T\text{ (K)} = \theta\text{ (°C)} + 273$
   • $T$: Temperature in Kelvin (K)
   • $\theta$: Temperature in Celsius (°C)
2. Boyle's Law: $PV = \text{constant}$ or $P_1 V_1 = P_2 V_2$
   • $P$: Pressure (Pascals, Pa)
   • $V$: Volume ($\text{m}^3$ or $\text{cm}^3$)

---

Common Mistakes to Avoid

• ❌ Wrong: Thinking that increasing the volume of a container makes the pressure go up.
• ✓ Right: Recognize that an increase in volume reduces the rate of molecular collisions, leading to a drop in pressure (unless temperature is increased significantly to compensate).
• ❌ Wrong: Suggesting that faster particle speed leads to lower pressure.
• ✓ Right: Always remember that higher speed means harder and more frequent collisions, which increases pressure.
• ❌ Wrong: Forgetting that a "standard" container usually already has air in it.
• ✓ Right: Unless a container is a vacuum, it starts with an initial atmospheric pressure of approximately $10^5\text{ Pa}$.
• ❌ Wrong: Drawing a straight line for a Pressure vs. Volume graph.
• ✓ Right: Use a curve to show the dynamic, non-linear relationship that occurs when gas particles are compressed.

---

Exam Tips

1. Use Key Words: When explaining pressure changes, always use the terms "frequency of collisions" and "force of impact." Examiners look specifically for these.
2. Check Units: Ensure volume units are the same on both sides of the $P_1 V_1 = P_2 V_2$ equation. You don't always have to convert to $\text{m}^3$, but they must match!
3. Temperature Link: If a question mentions "Average Kinetic Energy," they are asking about Temperature. If they mention "Temperature," they are asking about Average Kinetic Energy.