Gas laws and the Kelvin scale

Introduction to Gas Laws and the Kelvin Scale

In this chapter, we explore how gases behave when we change their environment. Whether you are pumping up a bicycle tyre, watching a hot air balloon rise, or wondering why a bag of crisps puffs up on an aeroplane, you are seeing the Gas Laws in action. We will look at how pressure, volume, and temperature are all connected, and why scientists use a special temperature scale called the Kelvin scale to make sense of it all.

Note: To understand how these gas molecules move and cause pressure, you can refer to the chapter on "Ideal gas molecules and kinetic theory".

The Kelvin Scale and Absolute Zero

In everyday life, we use the Celsius scale (\(^\circ\text{C}\)). However, in Physics, we need a scale that starts at the "true" beginning of temperature.

What is Absolute Zero?

Imagine cooling a gas down. The particles move slower and slower. Eventually, you reach a temperature where the particles have no kinetic energy and stop moving entirely. This temperature is called Absolute Zero.

• Absolute Zero is approximately \(-273 ^\circ\text{C}\).
• It is the coldest possible temperature in the universe.

The Kelvin Scale

The Kelvin scale starts at Absolute Zero (\(0\text{ K}\)). It does not use degrees, just the letter K. Because it starts at \(-273 ^\circ\text{C}\), the conversion is very simple:

\(\text{Temperature in K} = \text{Temperature in } ^\circ\text{C} + 273\)

\(\text{Temperature in } ^\circ\text{C} = \text{Temperature in K} - 273\)

Quick Review Examples:
- If water boils at \(100 ^\circ\text{C}\), its temperature in Kelvin is \(100 + 273 = 373\text{ K}\).
- If a gas is at \(300\text{ K}\), its temperature in Celsius is \(300 - 273 = 27 = 27 ^\circ\text{C}\).

Key Takeaway:

The Kelvin temperature of a gas is proportional to the average kinetic energy of its molecules. If you double the Kelvin temperature, you double the average kinetic energy of the particles.

The Gas Laws: Qualitative Behavior

Before we look at the math, let's understand how gases behave "qualitatively" (using words and descriptions).

1. Pressure and Volume (at constant temperature)

Imagine you have a gas in a sealed syringe. If you push the plunger in (decreasing the volume), the gas particles are squashed into a smaller space. They will hit the walls of the syringe more often, which increases the pressure.

• If volume decreases, pressure increases.
• If volume increases, pressure decreases.

2. Pressure and Temperature (at constant volume)

Imagine a rigid, sealed container of gas. If you heat the gas (increasing the temperature), the particles gain more kinetic energy and move faster. They hit the walls harder and more frequently.

• If temperature increases, pressure increases.
• If temperature decreases, pressure decreases.

The Gas Laws: Calculations

For your exam, you will need to use two main formulas to calculate changes in a gas. Don't worry—these formulas are usually provided on your exam equation sheet!

Boyle’s Law: Pressure and Volume

For a fixed mass of gas at a constant temperature:

\(p_1V_1 = p_2V_2\)

Where:
- \(p_1\) and \(V_1\) are the starting pressure and volume.
- \(p_2\) and \(V_2\) are the final pressure and volume.

Example: A gas has a volume of \(2\text{ m}^3\) at a pressure of \(100,000\text{ Pa}\). If the volume is squashed to \(1\text{ m}^3\), what is the new pressure?
\(100,000 \times 2 = p_2 \times 1\)
\(200,000 = p_2\)
The new pressure is \(200,000\text{ Pa}\).

The Pressure Law: Pressure and Temperature

For a fixed mass of gas at a constant volume:

\(\frac{p_1}{T_1} = \frac{p_2}{T_2}\)

CRITICAL RULE: In this formula, the temperature MUST be in Kelvin. If the question gives you Celsius, convert it to Kelvin first by adding \(273\)!

Example: A sealed tank of gas has a pressure of \(100\text{ kPa}\) at \(27 ^\circ\text{C}\). It is heated to \(127 ^\circ\text{C}\). What is the new pressure?
1. Convert temperatures to Kelvin:
\(T_1 = 27 + 273 = 300\text{ K}\)
\(T_2 = 127 + 273 = 400\text{ K}\)
2. Use the formula:
\(\frac{100}{300} = \frac{p_2}{400}\)
\(0.333 = \frac{p_2}{400}\)
\(p_2 = 0.333 \times 400 = 133.3\text{ kPa}\)

Summary and Common Mistakes

Quick Summary:
- Absolute Zero is \(-273 ^\circ\text{C}\) or \(0\text{ K}\).
- Kelvin temperature is directly related to kinetic energy.
- Pressure and Volume are inversely related (as one goes up, the other goes down).
- Pressure and Temperature are directly related (as one goes up, the other goes up).

Common Mistakes to Avoid:
- Forgetting to convert to Kelvin: This is the most common error in Physics exams. Always check the units of temperature before calculating.
- Mixing up \(p_1\) and \(p_2\): Be careful when reading the question to identify which values belong together at the start and which belong at the end.
- Units: Ensure your units for pressure (e.g., \(\text{Pa}\) or \(\text{kPa}\)) and volume (e.g., \(\text{m}^3\) or \(\text{cm}^3\)) are consistent on both sides of the equation.

Did you know?
The reason you shouldn't leave an aerosol can in the sun is the Pressure Law. As the sun heats the gas inside the constant-volume metal can, the pressure increases so much that the can might eventually burst!