Difficulty: Intermediate | Prerequisites: Ch. 11 (first law), Ch. 12.1–12.3 (ideal gas basics)
Materials expand when heated (thermal expansion), absorb specific amounts of heat per degree of temperature change (specific heat), and can change phase at fixed temperatures (heat of transformation). Calorimetry uses energy conservation to find final temperatures when hot and cold objects mix. Heat moves by conduction, convection, and radiation. Fluids exert pressure that increases with depth, and buoyancy is the upward force on submerged objects equal to the weight of displaced fluid.
Specific heat (c)
The amount of heat needed to raise the temperature of 1 kg of a substance by 1 K (or 1°C). Units: J/(kg·K). Different materials have different specific heats; water's is unusually high at 4190 J/(kg·K).
Heat of transformation (L)
The energy per kilogram needed to change a substance's phase (solid to liquid, or liquid to gas) without changing its temperature. Also called latent heat.
Heat of vaporisation
The energy per kilogram to convert liquid to gas (or vice versa). Much larger than the heat of fusion for the same substance.
Thermal expansion
The tendency of materials to increase in size when heated. Applies to length (linear expansion) and volume (volume expansion).
Linear thermal expansion coefficient (α)
ΔL = αLΔT. The fractional change in length per degree of temperature change.
Volume thermal expansion coefficient (β)
ΔV = βVΔT. For isotropic solids, β ≈ 3α.
Bimetallic strip
Two metals bonded together that bend when heated, because they have different expansion coefficients. Used in thermostats.
Calorimetry
The measurement of heat exchange. Based on conservation of energy: all heat lost by hot objects equals the heat gained by cold objects in an isolated system.
Conduction
Heat flow directly through a physical material, from hot regions to cold regions. Rate depends on thermal conductivity, cross-sectional area, temperature difference, and thickness.
Convection
Heat transfer by the bulk movement of a fluid. Occurs when a heated fluid rises and cooler fluid sinks to replace it.
Radiation
Energy emitted by all objects in the form of electromagnetic waves. Does not require a medium. Rate depends on temperature and surface properties.
Diffusion
Net movement of molecules from a region of higher concentration to a region of lower concentration.
Density (ρ)
Mass per unit volume: ρ = m/V. Units: kg/m³.
Buoyant force (F_b)
The upward force exerted by a fluid on a submerged or partially submerged object. F_b = ρ_fluid × V_displaced × g. Equal to the weight of the displaced fluid (Archimedes' principle).
Hydrostatic pressure
Pressure at a depth d in a fluid of uniform density: P = P_surface + ρgd. Pressure increases linearly with depth.
Volume thermal expansion
ΔV = βVΔT, where β is the coefficient of volume expansion
Applies to solids, liquids, and gases
Linear thermal expansion
ΔL = αLΔT, where α is the coefficient of linear expansion
For isotropic solids: β ≈ 3α
Bimetallic strips
Two metals with different α values bonded together
When heated, the metal with the higher α expands more, causing the strip to bend
If comparing two metals in a thermal expansion diagram, the metal that expanded more has the larger α. This is the basis for simple thermostats.
Example pattern
Two cubes of different materials (e.g. aluminium and concrete) with different initial volumes and initial temperatures
Find the ratio of their volume changes: ΔV_A / ΔV_B using ΔV = βVΔT for each
Pay attention to whether ΔT is the same for both, or different
Specific heat
Q = McΔT, where M is mass, c is specific heat, ΔT is temperature change
This gives the heat needed to produce a temperature change ΔT for a mass M with specific heat c
Molar form: Q = nCΔT, where C is molar specific heat (capital C)
Heat of transformation (phase changes)
During a phase change, temperature does not change. All the energy goes into breaking or forming molecular bonds.
Q = ML, where L is the latent heat (heat of fusion or heat of vaporisation)
Heat of fusion: energy to go from solid to liquid
Heat of vaporisation: energy to go from liquid to gas (much larger than heat of fusion)
If the process is reversed (gas to liquid, liquid to solid), the same amount of energy is released
Two basic ideas for solving calorimetry problems:
Energy in the system is conserved: Q_net = 0, which means Q_lost + Q_gained = 0
The system reaches thermal equilibrium at some final temperature T_f where all objects have the same temperature
Standard setup (no phase change)
M_w c_w (T_f - T_i,w) + M_s c_s (T_f - T_i,s) = 0
Solve for T_f
The hot object loses heat (its Q is negative), the cold object gains heat (its Q is positive), and they sum to zero
With phase changes
If a phase change occurs, you must account for the heat of transformation separately: Q_total = Q_temperature change + Q_phase change
Check whether the available energy is sufficient to complete the phase change before assuming the final state
Example from the source notes (Lecture 8, p. 20)
0.120 kg steel pellets at some initial temperature mixed with 0.080 kg of water
Boiling water at T = 100°C, water at T = 20°C
Heat is transferred between pellets and water; no phase changes occur
Set up: M_w c_w (T_f - T_i,w) + M_s c_s (T_f - T_i,s) = 0
The source notes flag this concept as important and say to start from the derivations on Lecture 9, page 7
Finding T_f
T_f = (M_w c_w T_i,w + M_s c_s T_i,s) / (M_w c_w + M_s c_s)
Make sure T is in kelvin or Celsius (not Fahrenheit) and be consistent
Three mechanisms by which heat moves from one place to another.
Conduction
Heat flows directly through a physical material, from hot to cold
Rate: Q/Δt = kA(ΔT/L), where k is thermal conductivity, A is cross-sectional area, ΔT is temperature difference, and L is thickness
Metals are good conductors; insulators (wood, air, foam) are poor conductors
Convection
Heat transfer by the bulk movement of a circulating fluid
Occurs when a heated fluid becomes less dense, rises, and is replaced by cooler fluid sinking
Natural convection is driven by gravity and density differences; forced convection uses a fan or pump
Radiation
Energy emitted by all objects in the form of electromagnetic waves (infrared, visible light, etc.)
Does not require a medium; this is how the Sun heats the Earth through the vacuum of space
Rate: P = eσAT⁴ (Stefan-Boltzmann law), where e is emissivity (0 to 1), σ = 5.67 × 10⁻⁸ W/(m²·K⁴), A is surface area, T is absolute temperature
Net radiation between an object and its surroundings: P_net = eσA(T⁴ - T_surroundings⁴)
What it is
Net movement of molecules from a region of higher concentration to a region of lower concentration
Driven by random molecular motion, not by any external force
Root-mean-square distance
r_rms = √(6Dt), where D is the diffusion coefficient and t is time
This gives the typical distance a molecule has diffused after time t
Diffusion is slow over large distances because it scales with √t, not t
Fick's law
Describes the rate of diffusion across a boundary
For situations converting between distance, time, and concentration, use the relationship between r_rms and the diffusion coefficient
Time to diffuse some distance: t = r² / (6D)
Density
ρ = m/V
Liquids of uniform density: the density is the same everywhere in the fluid
Typical values: ρ_water = 1000 kg/m³, ρ_oil ≈ 900 kg/m³
Pressure in fluids
Liquids exert force on the walls of their containers
Pressure at all points on a horizontal depth line through a connected fluid of a single kind is the same
Pressure increases with depth: P₂ = P₁ + ρgd, where d is the depth below the surface
Atmospheric pressure at the surface: P_atm ≈ 1 atm = 1.013 × 10⁵ Pa
Buoyancy (Archimedes' principle)
F_b = ρ_fluid × g × V_displaced
The buoyant force equals the weight of the displaced fluid
An object floats when F_b = W (its weight), which happens when ρ_object < ρ_fluid
Maximum displaced volume = V_object (fully submerged). If F_b at full submersion is still less than W, the object sinks.
Two-liquid manometer problems
Common exam setup: a U-tube with two liquids that do not mix (e.g. water and oil)
One end may be sealed or open to the atmosphere
Strategy: find a horizontal level where pressures must be equal, then write P expressions for each side at that level
Example pattern from the source notes:
P₁ = P_atm + ρ_oil × g × h_oil (open side)
P₂ = P_atm + ρ_water × g × h_water (other side)
Set P₁ = P₂ at the same horizontal level and solve
Biology application (from Lecture 11, p. 23)
Pressure differences in biological systems (blood pressure, pressure in lungs) follow the same hydrostatic pressure formula
The source notes flag a biology question from this lecture as exam-relevant
Q = Mc\Delta T \quad \text{(specific heat)}Q = ML \quad \text{(phase change)}\Delta L = \alpha L \Delta T \quad ; \quad \Delta V = \beta V \Delta T\frac{Q}{\Delta t} = \frac{kA\Delta T}{L} \quad \text{(conduction)}P = e\sigma A T^4 \quad \text{(radiation, Stefan-Boltzmann)}r_{rms} = \sqrt{6Dt} \quad \text{(diffusion)}P_2 = P_1 + \rho g d \quad \text{(hydrostatic pressure)}F_b = \rho_{fluid} \cdot g \cdot V_{displaced} \quad \text{(buoyancy)}T_f = \frac{M_1 c_1 T_1 + M_2 c_2 T_2}{M_1 c_1 + M_2 c_2} \quad \text{(calorimetry, no phase change)}Bimetallic strips are the working principle behind older thermostats and some circuit breakers. When the strip bends past a threshold, it makes or breaks an electrical contact.
Car radiators rely on all three heat transfer mechanisms: conduction through the metal fins, convection as air flows past them, and radiation from the hot surfaces. The fins increase surface area to maximise conduction and convection.
Diffusion governs how oxygen moves from your lungs into your blood and how ink spreads in still water. The √t scaling explains why diffusion works well over short distances (cells) but is far too slow for transporting substances across an entire body.
Students often assume that during a phase change, temperature rises. It does not. All the energy goes into breaking molecular bonds until the phase change is complete.
Students sometimes think buoyant force depends on the object's mass. It does not. It depends on the volume of fluid displaced and the density of the fluid.
Students confuse conduction and convection. Conduction is heat moving through a stationary material. Convection requires the material itself to move.
Students occasionally think radiation requires air or some medium. It does not. Radiation travels through vacuum, which is how the Sun heats the Earth.
⚠️ Calorimetry problems (mixing hot and cold objects) are a staple of this exam. Know how to set up Q_lost + Q_gained = 0 and solve for T_f.
⚠️ The two-liquid U-tube manometer problem is highlighted in the source notes. Know how to equate pressures at the same horizontal level.
⚠️ Thermal expansion comparison problems (which material expands more) appear frequently. Remember: ΔV = βVΔT, so you need to compare β, V, and ΔT for each material.
⚠️ Know the three heat transfer mechanisms and be able to identify which is operating in a given scenario.
⚠️ The biology question from Lecture 11, p. 23, is flagged as exam-relevant. Review pressure in biological systems.
True or False: During a phase change, the temperature of the substance increases. (False. Temperature stays constant during a phase change.)
Fill in the blank: The buoyant force on an object equals the weight of the ______ ______. (displaced fluid)
True or False: Convection can occur in a solid. (False. Convection requires fluid flow.)
Fill in the blank: Pressure in a fluid increases with ______. (depth)
True or False: The root-mean-square diffusion distance scales linearly with time. (False. It scales with √t.)
Q: A 0.5 kg aluminium block (c = 900 J/(kg·K)) at 200°C is dropped into 2 kg of water (c = 4190 J/(kg·K)) at 20°C. What is the final equilibrium temperature?
A: M_Al × c_Al × (T_f - 200) + M_w × c_w × (T_f - 20) = 0. Solving: 0.5(900)(T_f - 200) + 2(4190)(T_f - 20) = 0. So 450T_f - 90000 + 8380T_f - 167600 = 0. Therefore 8830T_f = 257600, giving T_f ≈ 29.2°C.
Q: A wooden block (ρ = 600 kg/m³) with volume 0.01 m³ is placed in water (ρ = 1000 kg/m³). Does it float? If so, what fraction is submerged?
A: ρ_block < ρ_water, so it floats. At equilibrium, F_b = W: ρ_water × g × V_submerged = ρ_block × g × V_block. So V_submerged / V_block = ρ_block / ρ_water = 600/1000 = 0.6. 60% is submerged.
Q: A copper rod (k = 400 W/(m·K)) is 0.5 m long with a cross-section of 0.001 m². One end is at 100°C, the other at 20°C. What is the rate of heat conduction?
A: Q/Δt = kAΔT/L = 400 × 0.001 × (100 - 20) / 0.5 = 400 × 0.001 × 80 / 0.5 = 64 W.
Q: If a molecule takes 10 seconds to diffuse 1 mm, roughly how long will it take to diffuse 1 cm?
A: Distance scales as √t, so if you multiply distance by 10, time multiplies by 100. About 1000 seconds.
Q: In a U-tube with water on one side and oil (ρ = 900 kg/m³) on the other, both open to the atmosphere, the oil column is 10 cm taller than the water column. What is the height of the water column?
A: At the interface level: P_atm + ρ_oil × g × h_oil = P_atm + ρ_water × g × h_water. The atmospheric terms cancel. ρ_oil × h_oil = ρ_water × h_water. If h_oil = h_water + 0.10, then 900(h_water + 0.10) = 1000 × h_water. Solving: 900h + 90 = 1000h, so 100h = 90, h_water = 0.90 m = 90 cm.
Calorimetry is a direct application of the first law of thermodynamics (Ch. 11.4) to systems where heat is the only energy transfer.
Heat transfer mechanisms (conduction, convection, radiation) explain why the second law of thermodynamics always holds in practice: there are always pathways for energy to spread out.
Fluid pressure and buoyancy connect to forces and Newton's laws from earlier in the course. The buoyant force is just the net upward pressure force on a submerged object.
Specific heat, heat of transformation, latent heat, heat of fusion, heat of vaporisation, calorimetry, thermal equilibrium, thermal expansion, linear expansion, volume expansion, bimetallic strip, conduction, convection, radiation, Stefan-Boltzmann law, emissivity, diffusion, Fick's law, root-mean-square distance, density, buoyancy, Archimedes' principle, buoyant force, hydrostatic pressure, gauge pressure, U-tube manometer, Pascal's principle, PHYS 115, Physics 115 Exam 1