4.1 Thermal Equilibrium
Thermal Contact and Thermal Equilibrium
When two objects are in thermal contact, heat energy is transferred between them:
- Heat flows at a higher net rate from the hotter object to the cooler object.
- As the temperature difference decreases, the net rate of heat transfer decreases.
- Thermal equilibrium is reached when the net rate of heat transfer between the two objects is zero.
- At thermal equilibrium, both objects reach the same temperature.
Liquid-in-Glass Thermometer Calibration
A thermometer measures temperature based on the thermometric property of a liquid (e.g., volume of mercury). To calibrate a thermometer using two fixed points (ice point $0^\circ\text{C}$ and steam point $100^\circ\text{C}$):
$$\theta = \frac{L_\theta - L_0}{L_{100} - L_0} \times 100^\circ\text{C}$$
- $L_\theta$: Length of mercury column at unknown temperature $\theta$
- $L_0$: Length of mercury column at melting ice point ($0^\circ\text{C}$)
- $L_{100}$: Length of mercury column at boiling steam point ($100^\circ\text{C}$)
4.2 Specific Heat Capacity
Heat Capacity and Specific Heat Capacity
Heat Capacity ($C$): The quantity of heat energy required to raise the temperature of an object by $1^\circ\text{C}$ (or $1\text{ K}$). $C = \frac{Q}{\Delta \theta}$ in $\text{J}^\circ\text{C}^{-1}$ or $\text{J K}^{-1}$.
Specific Heat Capacity ($c$): The quantity of heat energy required to raise the temperature of $1\text{ kg}$ of a substance by $1^\circ\text{C}$ (or $1\text{ K}$):
$$Q = m c \Delta \theta$$
- $Q$: Heat energy supplied or released ($\text{J}$)
- $m$: Mass of the substance ($\text{kg}$)
- $c$: Specific heat capacity ($\text{J kg}^{-1}{^\circ\text{C}}^{-1}$)
- $\Delta \theta$: Temperature change ($\theta_{\text{final}} - \theta_{\text{initial}}$ in $^\circ\text{C}$)
Applications of Specific Heat Capacity
- High $c$ (e.g., Water $c \approx 4200\text{ J kg}^{-1}{^\circ\text{C}}^{-1}$): Heats up and cools down slowly. Excellent coolant in car radiators and regulates coastal climates (sea and land breezes).
- Low $c$ (e.g., Cooking metals): Heats up and cools down rapidly. Ideal for frying pans, cooking pots, and thermometer bulbs.
4.3 Specific Latent Heat
Latent Heat and Phase Changes
Latent Heat: Heat absorbed or released during a change of phase at a constant temperature.
- Specific Latent Heat of Fusion ($l_f$): Quantity of heat required to change $1\text{ kg}$ of a substance from solid to liquid without temperature change.
- Specific Latent Heat of Vaporisation ($l_v$): Quantity of heat required to change $1\text{ kg}$ of a substance from liquid to gas without temperature change.
$$Q = m l$$
- $Q$: Heat energy supplied or released ($\text{J}$)
- $m$: Mass ($\text{kg}$)
- $l$: Specific latent heat ($\text{J kg}^{-1}$)
4.4 Gas Laws
Boyle's Law (Pressure-Volume Relationship)
For a fixed mass of gas at constant temperature, pressure ($P$) is inversely proportional to volume ($V$):
$$P \propto \frac{1}{V} \quad \implies \quad P_1 V_1 = P_2 V_2$$
Charles's Law (Volume-Temperature Relationship)
For a fixed mass of gas at constant pressure, volume ($V$) is directly proportional to absolute temperature ($T$ in Kelvin):
$$V \propto T \quad \implies \quad \frac{V_1}{T_1} = \frac{V_2}{T_2}$$
Note: Temperature must always be converted to Kelvin ($T/\text{K} = \theta/^\circ\text{C} + 273.15$).
Gay-Lussac's Law (Pressure-Temperature Relationship)
For a fixed mass of gas at constant volume, pressure ($P$) is directly proportional to absolute temperature ($T$ in Kelvin):
$$P \propto T \quad \implies \quad \frac{P_1}{T_1} = \frac{P_2}{T_2}$$
Absolute Zero ($0\text{ K}$ or $-273^\circ\text{C}$): The lowest possible temperature where gas pressure and kinetic energy of gas molecules theoretically become zero.