Ideal gases

Temperature – Cambridge IGCSE/A‑Level (9702)

Learning Objective

Develop a solid understanding of how temperature is defined, measured and related to molecular energy. By the end of this section you should be able to:

• Explain thermal equilibrium and describe a simple experimental demonstration.
• Convert between the Celsius and Kelvin scales (Fahrenheit is optional).
• Apply the concepts of specific heat capacity and specific latent heat in quantitative problems.
• State why the Kelvin scale is required for any gas‑law calculation (A‑Level extension).

1. Thermal Equilibrium

Definition: Two or more bodies are in thermal equilibrium when they are at the same temperature and no net heat flows between them over a measurable period of time.

• Heat always flows from the hotter body (higher temperature) to the colder body (lower temperature) until equilibrium is reached.
• Temperature determines the direction of heat flow; it is not the amount of heat itself.
• Experimental demonstration (calorimeter): Place two metal blocks of different initial temperatures in an insulated container and insert a thermometer. After a few minutes the thermometer reads a single temperature – the system has reached thermal equilibrium.
• Zero °C = freezing point of water at 1 atm; 0 K = absolute zero, the theoretical point at which molecular motion ceases.

2. Temperature Scales

• Celsius (°C) – based on the freezing (0 °C) and boiling (100 °C) points of water at 1 atm.
• Kelvin (K) – the absolute (thermodynamic) scale; 0 K = absolute zero. One kelvin has exactly the same magnitude as one degree Celsius (ΔK = Δ°C).
• Fahrenheit (°F) – optional, mainly used in the United States. It is not required for the IGCSE syllabus and is therefore placed in a side box.

Optional – Fahrenheit (extra)

Water freezes at 32 °F and boils at 212 °F. Conversion formulas are given in the next table for completeness.

3. Conversion Between Scales

Quantity	°C	K	°F (optional)
Freezing point of water	0	273.15	32
Boiling point of water	100	373.15	212
Absolute zero	-273.15	0	-459.67

General conversion formulas (required)

• From Celsius to Kelvin:
  $$T(\mathrm{K}) = T(^{\circ}\mathrm{C}) + 273.15$$
• From Kelvin to Celsius:
  $$T(^{\circ}\mathrm{C}) = T(\mathrm{K}) - 273.15$$

Optional Fahrenheit formulas

• °C → °F: $$T(^{\circ}\mathrm{F}) = \frac{9}{5}\,T(^{\circ}\mathrm{C}) + 32$$
• °F → °C: $$T(^{\circ}\mathrm{C}) = \frac{5}{9}\,\bigl[T(^{\circ}\mathrm{F}) - 32\bigr]$$

4. Thermodynamic Temperature & Kinetic Energy

Kelvin is defined so that temperature is directly proportional to the average translational kinetic energy of the particles in an ideal gas:

$$\langle E_{\text{kin}}\rangle = \frac{3}{2}\,k_{\!B}\,T$$

where \(k_{\!B}=1.38\times10^{-23}\ \mathrm{J\,K^{-1}}\) is Boltzmann’s constant. Because this relationship holds only when the zero point is absolute zero, **Kelvin must be used in any equation that links temperature to molecular energy**, such as the ideal‑gas law.

5. Specific Heat Capacity

Definition: The amount of heat required to raise the temperature of 1 kg of a substance by 1 K (or 1 °C). Symbol: \(c\). Units: J kg⁻¹ K⁻¹.

Heat‑energy equation:

$$Q = mc\Delta T$$

• \(Q\) – heat added or removed (J)
• \(m\) – mass of the substance (kg)
• \(c\) – specific heat capacity (J kg⁻¹ K⁻¹)
• \(\Delta T = T_{\text{final}}-T_{\text{initial}}\) (K or °C)

Reference Table – Specific Heat Capacities (selected)

Substance	c (J kg⁻¹ K⁻¹)
Water	4180
Ice	2100
Aluminium	900
Copper	385

Worked Example – Heating Water

How much heat is needed to raise 50 g of water from 20 °C to 80 °C?

1. Convert mass: \(m = 50\ \text{g} = 0.050\ \text{kg}\).
2. Temperature change: \(\Delta T = 80 - 20 = 60\ \text{K}\).
3. Apply \(Q = mc\Delta T\):
   $$Q = (0.050)(4180)(60) = 1.25\times10^{4}\ \text{J}$$
4. Result: \(Q \approx 12.5\ \text{kJ}\) of heat must be supplied.

6. Specific Latent Heat

Latent heat is the heat required to change the phase of a substance without changing its temperature.

• Fusion (melting) – \(L_{\!f}\): heat needed to melt 1 kg of solid.
• Vapourisation – \(L_{\!v}\): heat needed to vaporise 1 kg of liquid.

Heat‑energy equation for a phase change:

$$Q = mL$$

Units: J kg⁻¹.

Reference Table – Specific Latent Heats (water)

Process	Symbol	Value (J kg⁻¹)
Fusion (melting)	\(L_{\!f}\)	3.34 × 10⁵
Vapourisation (boiling)	\(L_{\!v}\)	2.26 × 10⁶

Worked Example – Melting Ice

How much heat is required to melt 200 g of ice at 0 °C?

1. Convert mass: \(m = 0.200\ \text{kg}\).
2. Apply \(Q = mL_{\!f}\):
   $$Q = (0.200)(3.34\times10^{5}) = 6.68\times10^{4}\ \text{J}$$
3. Result: \(Q \approx 66.8\ \text{kJ}\).

Worked Example – Boiling Water (Vapourisation)

How much heat is needed to convert 150 g of water at 100 °C into steam at the same temperature?

1. Convert mass: \(m = 0.150\ \text{kg}\).
2. Use the vapourisation value \(L_{\!v}=2.26\times10^{6}\ \text{J kg}^{-1}\).
   $$Q = mL_{\!v}= (0.150)(2.26\times10^{6}) = 3.39\times10^{5}\ \text{J}$$
3. Result: \(Q \approx 339\ \text{kJ}\).

7. Link to the Ideal‑Gas Law (A‑Level Extension)

Because the ideal‑gas equation uses absolute temperature, Kelvin must be used – this is the direct link to Syllabus 15 (Ideal gases).

The ideal‑gas equation is

$$pV = nRT$$

• \(p\) – pressure (Pa or atm)
• \(V\) – volume (m³ or L)
• \(n\) – amount of substance (mol)
• \(R\) – universal gas constant
• \(T\) – absolute temperature (K)

Two convenient values of \(R\):

• \(R = 8.314\ \mathrm{J\,mol^{-1}\,K^{-1}}\) (Pa·m³ units)
• \(R = 0.0821\ \mathrm{L\,atm\,mol^{-1}\,K^{-1}}\) (L·atm units)

Worked Example – A‑Level

Calculate the pressure exerted by 2.00 mol of an ideal gas confined in a 5.00 L container at 25 °C.

1. Convert temperature: \(T = 25 + 273.15 = 298.15\ \text{K}\).
2. Use \(R = 0.0821\ \mathrm{L\,atm\,mol^{-1}\,K^{-1}}\) (because \(V\) is in litres).
3. Apply the equation (solve for \(p\)):
   $$p = \frac{nRT}{V} = \frac{(2.00)(0.0821)(298.15)}{5.00} = 9.8\ \text{atm}$$
4. Convert to pascals if required: \(1\ \text{atm}=1.013\times10^{5}\ \text{Pa}\) → \(p \approx 9.9\times10^{5}\ \text{Pa}\).

8. Common Mistakes to Avoid

• Using °C or °F directly in equations that require absolute temperature – always convert to kelvin first.
• Mixing units (e.g., litres with the SI value of \(R\)). Choose the form of \(R\) that matches the volume units you are using.
• Confusing heat (energy) with temperature change – remember \(Q = mc\Delta T\) or \(Q = mL\) involve energy, not temperature.
• Assuming the numerical value of the gas constant is the same in all unit systems.
• Omitting the “no net heat flow” condition when describing thermal equilibrium.