Gases and the absolute scale of temperature Revision Notes

A gas consists of a large number of tiny particles (atoms or molecules) moving randomly in all directions. These particles collide with each other and with the walls of their container.

Pressure is the force exerted per unit area by gas particles colliding with the container walls. It depends on the frequency and force of these collisions.

The kinetic energy of gas particles is directly related to temperature: as temperature increases, particles move faster and collide more forcefully, increasing pressure if volume is constant.

• Gas particles move randomly and collide with container walls.
• Pressure arises from these collisions.
• Higher particle speed means more forceful collisions and higher pressure.

When the volume of a gas is fixed, increasing the temperature increases the average kinetic energy of the particles.

Faster particles collide more frequently and with greater force against the container walls, causing the pressure to rise.

This relationship is directly proportional: pressure (p) is proportional to absolute temperature (T) when volume (V) and amount of gas are constant, expressed as p ∝ T (at constant V).

• Pressure increases as temperature increases at constant volume.
• Particle kinetic energy increases with temperature.
• More energetic collisions cause higher pressure.

At constant temperature, the average kinetic energy of gas particles remains unchanged.

If the volume of the gas decreases, particles have less space to move, so they collide more frequently with the container walls, increasing pressure.

Conversely, increasing volume decreases collision frequency and pressure decreases accordingly, showing an inverse relationship: p ∝ 1/V (at constant T).

• Pressure is inversely proportional to volume at constant temperature.
• Smaller volume means more frequent collisions and higher pressure.
• Larger volume means fewer collisions and lower pressure.

Gay-Lussac's Law: For a fixed mass of gas at constant volume, pressure is directly proportional to absolute temperature: p/T = constant.

Boyle's Law: For a fixed mass of gas at constant temperature, pressure multiplied by volume is constant: pV = constant.

These laws assume the amount of gas and the temperature or volume are held constant as specified.

• p/T = constant (constant volume)
• pV = constant (constant temperature)
• Temperature must be in kelvin for these laws to apply.

A graph of pressure (y-axis, in pascals) against temperature (x-axis, in kelvin) at constant volume is a straight line passing through the origin, showing direct proportionality.

A graph of pressure (y-axis) against volume (x-axis, in cubic metres) at constant temperature is a hyperbola, showing inverse proportionality.

The gradient of the pressure-temperature graph indicates how pressure changes per unit temperature increase, while the area under the curve is not physically meaningful in this context.

• Pressure vs Temperature graph: straight line through origin (constant volume).
• Pressure vs Volume graph: hyperbola (constant temperature).
• Gradient shows rate of pressure change with temperature.

Temperature is a measure of the average kinetic energy of the particles in a substance. The Celsius (°C) scale is commonly used in everyday life and is based on the freezing point of water at 0 °C and boiling point at 100 °C under standard atmospheric pressure.

The Kelvin (K) scale is the absolute temperature scale used in physics. It starts at absolute zero, the theoretical temperature at which particles have minimum kinetic energy and molecular motion nearly stops. Absolute zero is 0 K, which corresponds to −273.15 °C.

The Kelvin scale is important because many gas laws and thermodynamic equations require temperature to be measured from absolute zero to avoid negative values and to maintain proportionality with kinetic energy.

• Celsius scale: 0 °C = freezing point of water, 100 °C = boiling point of water.
• Kelvin scale: 0 K = absolute zero (−273.15 °C).
• Kelvin is an absolute scale; Celsius is relative to water's phase changes.

The relationship between temperature in degrees Celsius (T(°C)) and temperature in Kelvin (T(K)) is linear and given by the formula:

\[ T(K) = T(°C) + 273 \] This means to convert a temperature from Celsius to Kelvin, add 273.

To convert from Kelvin to Celsius, rearrange the formula: \[ T(°C) = T(K) - 273 \] This means subtract 273 from the Kelvin temperature to get degrees Celsius.

• Formula: T(K) = T(°C) + 273
• Rearranged: T(°C) = T(K) - 273
• Always use Kelvin in gas law calculations to avoid negative temperatures.

Convert 25 °C to Kelvin using the formula. Step 1: Write down the formula: \[ T(K) = T(°C) + 273 \] Step 2: Substitute the known value: \[ T(K) = 25 + 273 \] Step 3: Calculate the sum: \[ T(K) = 298 \] Step 4: Write the answer with units: 25 °C = 298 K.

Common error: Using 273.15 instead of 273 or forgetting to add 273, which leads to incorrect temperature values. For IGCSE level, adding 273 is accepted and sufficient.

• Formula: T(K) = T(°C) + 273
• Substitution: T(K) = 25 + 273
• Answer: T(K) = 298 K
• Common error: Forgetting to add 273 or subtracting incorrectly.

The Kelvin scale is absolute because it starts at absolute zero, the lowest possible temperature where particles have minimum kinetic energy. This makes it essential for scientific calculations involving gases and thermodynamics, where temperature must be proportional to particle energy.

Unlike Celsius, Kelvin temperatures cannot be negative. This avoids confusion in calculations and ensures all temperatures are positive values, which is important for laws such as the ideal gas law.

Absolute zero is approximately −273.15 °C, but for IGCSE calculations, 273 is used for simplicity.

• 0 K = absolute zero, no molecular motion.
• Kelvin scale is absolute and non-negative.
• Used in gas laws and thermodynamics for proportionality with kinetic energy.

The Celsius scale is based on water's freezing and boiling points; the Kelvin scale is absolute, starting at absolute zero.

To convert Celsius to Kelvin, add 273: \( T(K) = T(°C) + 273 \).

To convert Kelvin to Celsius, subtract 273: \( T(°C) = T(K) - 273 \). Kelvin is essential in physics because it relates directly to particle kinetic energy and avoids negative temperatures in calculations.

• Kelvin and Celsius scales differ by 273 units.
• Use Kelvin in physics calculations involving temperature.
• Remember the formula and its rearrangement for conversions.

For a fixed mass of gas at constant temperature (an isothermal process), the pressure (p) and volume (V) of the gas are related such that their product remains constant. This means that if the volume increases, the pressure decreases proportionally, and vice versa.

Mathematically, this is expressed as: p × V = constant

This relationship is known as Boyle's Law. It applies only when the temperature and the amount of gas remain unchanged. The pressure is measured in pascals (Pa) or newtons per square metre (N/m²), and volume is measured in cubic metres (m³) or litres (L).

• pV = constant for a fixed mass of gas at constant temperature.
• Pressure and volume are inversely proportional under these conditions.
• Units: pressure in pascals (Pa), volume in cubic metres (m³) or litres (L).

Gas pressure arises from gas particles colliding with the walls of their container. When the volume decreases, the gas particles have less space to move, so they collide more frequently with the container walls, increasing the pressure.

Conversely, increasing the volume gives particles more space, reducing collision frequency and pressure.

Since temperature is constant, the average kinetic energy of the particles remains unchanged, so the change in pressure is solely due to the change in collision frequency caused by volume change.

• Pressure results from particle collisions with container walls.
• Decreasing volume increases collision frequency and pressure.
• Increasing volume decreases collision frequency and pressure.
• Temperature constant means average kinetic energy of particles is constant.

If a gas changes from an initial state (p₁, V₁) to a final state (p₂, V₂) at constant temperature, the equation can be written as: p₁V₁ = p₂V₂

This allows calculation of an unknown pressure or volume when the other three quantities are known.

Example calculation: A gas at 200 kPa occupies 0.5 m³. If the volume is compressed to 0.2 m³ at constant temperature, find the new pressure p₂.

• Use p₁V₁ = p₂V₂ for changes at constant temperature.
• Rearranged to find unknown: p₂ = (p₁V₁) / V₂ or V₂ = (p₁V₁) / p₂.
• Example: p₂ = (200000 Pa × 0.5 m³) / 0.2 m³ = 500000 Pa = 500 kPa.

Plotting pressure (p) on the vertical axis (y-axis) and volume (V) on the horizontal axis (x-axis) for a fixed mass of gas at constant temperature produces a curve called an isotherm.

The graph is a rectangular hyperbola, showing that pressure decreases as volume increases.

The area under the curve between two volumes does not have a direct physical meaning in this context, but the shape clearly shows the inverse relationship: doubling the volume halves the pressure, and so on.

• Graph axes: pressure (Pa) on y-axis, volume (m³) on x-axis.
• Graph shape: rectangular hyperbola (inverse proportionality).
• As volume increases, pressure decreases.
• Curve is called an isotherm (constant temperature).

In experiments to verify pV = constant, a sealed syringe or a gas-filled container with a movable piston can be used to vary volume and measure pressure.

Variables: independent variable is volume, dependent variable is pressure, temperature is kept constant (e.g., by performing the experiment at room temperature and allowing time for thermal equilibrium).

Measurements should be repeated to improve accuracy and identify anomalies. Use precise pressure gauges and volume markings for better results.

• Use apparatus like a syringe with pressure sensor to vary volume and measure pressure.
• Keep temperature constant during measurements.
• Repeat measurements for reliability.
• Use accurate instruments to reduce uncertainty.