Solids and liquids expand on heating in proportion to the temperature rise; in a mixture, heat lost by hot bodies equals heat gained by cold ones, including latent heat for any change of state.
How it's asked: Relation α : β : γ = 1 : 2 : 3, thermal stress in a clamped rod, temperature scale conversion, anomalous expansion of water, and mixing ice with water to find the final temperature.
Why this is true
Heating increases the amplitude of atomic vibrations, and because the interatomic potential is asymmetric the average spacing grows, so length increases by ΔL = αLΔT. Area and volume involve two and three lengths, which is why β ≈ 2α and γ ≈ 3α for an isotropic solid. If the rod is clamped so it cannot expand, the walls must compress it by exactly the strain αΔT it would have had, producing a thermal stress YαΔT. During melting or boiling, the heat supplied breaks intermolecular bonds instead of raising the kinetic energy, so the temperature stays constant and Q = mL. Calorimetry is just energy conservation in an insulated system: always check first whether the available heat is enough to complete a change of state.
Key points
Key formulae
Linear expansion
ΔL = αLΔT
Expansion coefficients
β = 2α, γ = 3α
Thermal stress
σ = YαΔT
rod clamped at both ends
Sensible heat
Q = mcΔT
Latent heat
Q = mL
Temperature scales
C/100 = (F − 32)/180 = (K − 273.15)/100
Fixed points of temperature scales
| Scale | Ice point | Steam point | Divisions |
|---|---|---|---|
| Celsius | 0 °C | 100 °C | 100 |
| Fahrenheit | 32 °F | 212 °F | 180 |
| Kelvin | 273.15 K | 373.15 K | 100 |
Common traps
- Do not assume all the ice melts — compare heat available with heat needed to melt it first.
- A temperature difference of 1 °C equals 1 K, but 1.8 °F — convert differences and readings differently.
Worked example
20 g of ice at 0 °C is dropped into 100 g of water at 40 °C in an insulated vessel of negligible heat capacity. Take c(water) = 1 cal g⁻¹ °C⁻¹ and L(fusion) = 80 cal g⁻¹. Find the final temperature.
High-yield
α : β : γ = 1 : 2 : 3; thermal stress YαΔT; water densest at 4 °C; temperature is constant during a change of state.
Quick check
Q1.The temperature at which the Celsius and Fahrenheit readings are equal is:
Q2.A metal rod is fixed between two rigid walls and heated by ΔT. The thermal stress developed depends on: