Heat and Temperature

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your WAEC exam.

Key Distinction

• Temperature is a measure of the degree of hotness — how hot or cold a body is (scalar quantity)
• Heat is the transfer of thermal energy from a hotter body to a colder one due to a temperature difference

Think of temperature as the “level” and heat as the “amount transferred.”

Essential Formulas

• Heat energy: $Q = mc\Delta \theta$
• Heat energy (latent): $Q = mL$
• Heat capacity: $C = \frac{Q}{\Delta \theta}$
• Specific heat capacity: $c = \frac{Q}{m\Delta \theta}$
• Thermal equilibrium: heat lost = heat gained

Quick Facts

• $c_{\text{water}} = 4200 , \text{J kg}^{-1}\text{K}^{-1}$ (or $4180$ in older textbooks)
• $L_{\text{ice}} = 3.34 \times 10^5 , \text{J kg}^{-1}$ (latent heat of fusion)
• $L_{\text{steam}} = 2.26 \times 10^6 , \text{J kg}^{-1}$ (latent heat of vaporisation)
• $1 , \text{calorie} = 4.2 , \text{J}$ (calorimetry connection)
• Temperature is measured in °C or K; they have the same size, so $\Delta \theta$ in °C = $\Delta T$ in K

⚡ WAEC Exam Tip: “Specific heat capacity” questions are very common in WAEC. A typical question: “A heater supplies 2000 J of energy to 0.5 kg of a liquid whose $c = 4000 , \text{J kg}^{-1}\text{K}^{-1}$. Find the temperature rise.” Use $Q = mc\Delta \theta \implies \Delta \theta = \frac{Q}{mc}$.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for building a solid understanding of heat and temperature concepts.

Temperature and Its Measurement

Temperature is a scalar quantity that measures the average kinetic energy of particles in a substance. When we heat a substance, we increase the average kinetic energy of its molecules. When we cool it, we decrease it.

Thermometers work on the principle that a suitable thermometric property (such as the length of a mercury column, electrical resistance, or emf of a thermocouple) changes in a predictable way with temperature.

Thermometric Scales

Scale	Lower fixed point	Upper fixed point	Number of divisions
Celsius (°C)	Ice point (0°C)	Steam point (100°C)	100
Kelvin (K)	Absolute zero (−273.15°C)	Triple point of water (273.16 K)	—
Fahrenheit (°F)	32°F	212°F	180

To convert: $T(\text{K}) = \theta(°C) + 273$. The Kelvin is the SI unit of temperature.

Specific Heat Capacity ($c$)

The specific heat capacity of a substance is the amount of heat energy required to raise the temperature of 1 kg of the substance by 1 K (or 1°C). Units: $\text{J kg}^{-1}\text{K}^{-1}$.

Different substances have different $c$ values — water has a very high $c$ (absorbs a lot of heat for a small temperature rise), which is why coastal areas have more moderate climates than inland areas.

Heat Capacity ($C$)

Heat capacity is the total heat energy required to raise the temperature of a body (of any mass) by 1 K: $C = mc$. Units: $\text{J K}^{-1}$. It depends on both the mass and the material.

Calorimetry and the Principle of Mixtures

In a closed system, heat lost by the hotter body equals heat gained by the cooler body:

$$m_1 c_1 (\theta_1 - \theta_f) = m_2 c_2 (\theta_f - \theta_2)$$

where $\theta_f$ is the final equilibrium temperature.

Worked Example (Principle of Mixtures)

A 0.2 kg iron bar at 100°C is dropped into 0.5 kg of water at 20°C. Find the final temperature. ($c_{\text{iron}} = 460 , \text{J kg}^{-1}\text{K}^{-1}$, $c_{\text{water}} = 4200 , \text{J kg}^{-1}\text{K}^{-1}$)

Solution: Heat lost by iron = Heat gained by water

$m_{\text{iron}} c_{\text{iron}} (100 - \theta_f) = m_{\text{water}} c_{\text{water}} (\theta_f - 20)$

$0.2 \times 460 \times (100 - \theta_f) = 0.5 \times 4200 \times (\theta_f - 20)$

$92(100 - \theta_f) = 2100(\theta_f - 20)$

$9200 - 92\theta_f = 2100\theta_f - 42000$

$51200 = 2192\theta_f$

$\theta_f \approx 23.4°C$

Change of State: Latent Heat

When a substance changes state (e.g., ice → water, water → steam), the temperature remains constant while latent heat is absorbed or released.

• Latent heat of fusion ($L_f$): heat needed to melt 1 kg of solid at its melting point
• Latent heat of vaporisation ($L_v$): heat needed to vaporise 1 kg of liquid at its boiling point

The total heat required: $Q = mc\Delta \theta + mL$ (heat to reach melting/boiling point + latent heat).

⚡ WAEC Exam Tip: In calorimetry questions, always state which substance is losing heat and which is gaining heat before setting up the equation. Watch out for questions where one substance undergoes a change of state — in these, the final temperature is the melting/boiling point, not something in between.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage including derivations, heat transfer mechanisms, and WAEC past question patterns.

The Kinetic Theory Interpretation of Temperature

From the kinetic theory of gases: $\frac{1}{2}m\overline{c^2} = \frac{3}{2}kT$. Temperature is directly proportional to the mean kinetic energy per molecule. This is the fundamental statistical basis for temperature — it arises from the random motion of particles. For solids and liquids, where molecules are closely packed and interact strongly, the relationship is more complex, but the principle of thermal energy being related to molecular motion remains.

Methods of Heat Transfer

Heat can be transferred in three ways:

1. Conduction: Heat transfer through a solid (or stationary fluid) without bulk movement of the material. Mechanism: fast-moving molecules transfer kinetic energy to neighbouring slow-moving molecules through collisions. Metals are good conductors because their free electrons carry energy rapidly. Insulators (e.g., wool, fibreglass, air) have few free electrons. Good conductors: copper, aluminium, silver. Poor conductors: wood, plastic, glass.

   Rate of heat flow: $\frac{Q}{t} = \frac{kA(\theta_1 - \theta_2)}{d}$ where $k$ is the thermal conductivity, $A$ is cross-sectional area, $d$ is thickness.

2. Convection: Heat transfer by movement of the fluid itself (bulk motion). Hot fluid expands, becomes less dense, rises; cold fluid is denser, sinks — creating a convection current. This is why refrigerators are placed near the top of a kitchen (cold air sinks) and cookers heat from below (hot air rises). Convection cannot occur in solids.

3. Radiation: Heat transfer by electromagnetic waves (infrared). All objects above absolute zero emit thermal radiation. The rate of radiation is given by Stefan’s law: $P = e\sigma A T^4$, where $\sigma = 5.67 \times 10^{-8} , \text{W m}^{-2}\text{K}^{-4}$ (Stefan-Boltzmann constant) and $e$ is the emissivity (0 ≤ e ≤ 1, where e = 1 for a perfect black body). Unlike conduction and convection, radiation can travel through a vacuum — this is how the Sun’s energy reaches the Earth.

The Vacuum Flask (Dewar Vessel)

A vacuum flask minimises all three heat transfer mechanisms:

• Double-walled glass/metal with a vacuum between walls → eliminates conduction and convection
• Silvered walls → reduces radiation (high reflectivity, low emissivity)
• Supported on insulating mounts → minimises conducted heat loss through the base

Result: liquids stay hot (or cold) for hours.

Dewar’s original design used a double-walled vessel with vacuum insulation — the same principle in modern flasks. Heat gain/loss is extremely slow.

Specific Heat Capacity by Cooling

An alternative experimental method: a body at temperature $\theta_1$ cools to $\theta_2$ in time $t$ in a calorimeter with water equivalent $w$. By Newton’s law of cooling (an approximation), rate of cooling is proportional to temperature difference:

$$\frac{d\theta}{dt} = -k(\theta - \theta_0)$$

where $\theta_0$ is ambient temperature. Integrating gives a logarithmic cooling curve. This method is used when direct electrical heating is impractical.

Latent Heat: Molecular Interpretation

During a change of state at constant temperature:

• At the melting point, the solid lattice is breaking — work must be done against intermolecular forces
• At the boiling point, molecules overcome atmospheric pressure and leave the liquid surface — again, work against forces

The latent heat represents the energy needed to overcome these intermolecular forces without increasing temperature (all the energy goes into potential energy change, not kinetic).

WAEC Past Question Patterns

From WAEC Physics papers (2020–2024), the most common question types are:

1. Specific heat capacity calculation — “X g of substance at T1 is mixed with Y g of water at T2. Find the final temperature.” — Apply heat balance.
2. Latent heat — “Find the heat required to convert 0.5 kg of ice at −10°C to steam at 100°C” — Break into stages: ice warm-up + melt + water warm-up + vaporise.
3. Thermal conductivity — “Two rods of different materials have the same dimensions. Which is the better insulator?” — Compare $k$ values.
4. Convection vs conduction — Conceptual questions about why sea breezes form, or why refrigerators are placed near the top of a kitchen.

⚡ Common Pitfalls to Avoid

• Confusing heat capacity with specific heat capacity — heat capacity is for the whole object
• Forgetting to convert mass from g to kg (divide by 1000)
• Adding the latent heat for the wrong change of state (fusion vs vaporisation)
• Using $c_{\text{water}} = 4200$ when the question might use $c_{\text{water}} = 4180$ — use the given value
• Not including the water equivalent of the calorimeter when required (some questions add this explicitly)
• Mixing up “heat lost = heat gained” sign convention — just set up the equation with appropriate $\Delta \theta$

Key Formulas Quick Reference

Quantity	Formula	Units
Heat energy (sensible)	$Q = mc\Delta \theta$	J
Specific heat capacity	$c = Q/(m\Delta \theta)$	$\text{J kg}^{-1}\text{K}^{-1}$
Heat capacity	$C = mc$	$\text{J K}^{-1}$
Latent heat	$Q = mL$	J
Thermal equilibrium	$\Sigma Q_{\text{lost}} = \Sigma Q_{\text{gained}}$	—
Stefan radiation	$P = e\sigma A T^4$	W

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