Thermal Energy Field Guide

01Solid, liquid, gas: the particle picture

In a solid the particles vibrate about fixed positions. In a liquid they are still close together but slide past each other. In a gas they are far apart and move fast in straight lines between collisions. From solid to gas the forces between the particles get weaker and their potential energy gets bigger.

02Temperature is the average kinetic energy: use kelvin

The two scales have the same step size, so a change of 1 °C is a change of 1 K. But only the kelvin scale starts at absolute zero, where the particles have their lowest possible kinetic energy. The average kinetic energy is proportional to T in kelvin: at 20 °C = 293 K it is 1.5 × 1.38 × 10⁻²³ × 293 = 6.07 × 10⁻²¹ J.

03Internal energy: kinetic energy plus potential energy

0.10 kg of ice starts at −20 °C. The first 4200 J raise its temperature: the random kinetic energy of the molecules grows. Then the temperature stops at 0 °C while the next 33 400 J break bonds and melt the ice: the potential energy grows. Supply 20 kJ in total: the ice is at 0 °C and 47 g has melted. The internal energy is the sum of both bars.

04Specific heat capacity: same energy, different rises

Each block has a mass of 1.0 kg and gets the same energy. ΔT = Q/(mc), so a small c gives a big temperature rise. 10 kJ warms water by only 2.38 K but copper by 26.0 K: water needs 10.9 times as much energy per kelvin.

05Mixing: the energy lost by the hot = the energy gained by the cold

Hot water is poured into 0.30 kg of water at 20 °C. Nothing escapes, so the two curves meet at one final temperature and the two bars are equal. 0.20 kg at 80 °C gives (0.20 × 80 + 0.30 × 20)/0.50 = 44 °C: the hot water loses 30 240 J and the cold water gains 30 240 J.

06Reading a heating curve, one part at a time

This is the 150 g sample of worked example 5. Move the time and watch which equation fits each part. Every part uses Q = Pt. The first sloping part gives the power: 0.15 × 2000 × 40/100 = 120 W. On the flat part the sample melts at 55 °C: 120 × 400 = 48 000 J melts 0.15 kg, so L = 3.2 × 10⁵ J kg⁻¹.

07From ice to warm water: one step for each change

0.20 kg of ice at −10 °C becomes water at 40 °C. Do it in steps and add them: warm the ice (mcΔT, 4200 J), melt it (mL, 66 800 J), warm the water (mcΔT, 33 600 J). Total: 104 600 J. Use the right c for each phase, and never use mcΔT across 0 °C.

08Conduction through a wall

A wall of area 10 m² has 20 °C inside and 5 °C outside. The temperature falls steadily through the wall. A 20 cm brick wall (k = 0.60 W m⁻¹ K⁻¹) lets through 0.60 × 10 × 15/0.20 = 450 W: about 3.9 × 10⁷ J every day. Make it thicker and the rate falls; change the material and k changes the rate.

09Convection: the fluid carries the energy

Water at the bottom of the pan warms, expands and becomes less dense, so it rises above the flame. Cooler, denser water sinks at the sides to take its place. The loop is a convection current. Unlike conduction, the fluid itself moves; that is why convection cannot happen in a solid or in a vacuum.

10Radiated power grows as T⁴

A black sphere of radius 0.10 m (A = 4πr² = 0.126 m²) at 600 K radiates 5.67 × 10⁻⁸ × 0.126 × 600⁴ = 923 W. The curve rises steeply: double T and P becomes 2⁴ = 16 times bigger. Always use kelvin: heating from 50 °C to 250 °C is 323 K → 523 K, a factor of 6.9, not 5⁴.

11Wien’s law: the colour tells the temperature

The curve shows how much radiation a black body gives out at each wavelength. As T rises the peak moves to shorter wavelengths: λ_maxT = 2.9 × 10⁻³ m K. The Sun (5800 K) peaks at 500 nm, in the visible. A 3000 K star peaks in the infrared and looks red; a 10 000 K star peaks in the ultraviolet and looks blue-white.

12Luminosity and apparent brightness

A star sends its power L out equally in all directions. At a distance d it is spread over a sphere of area 4πd², so b = L/(4πd²). The Sun gives 3.9 × 10²⁶ W; at 1 AU (1.5 × 10¹¹ m) that is 1.38 × 10³ W m⁻². Double the distance: the same rays cover 4 times the area, so b falls to a quarter.

13Practice and self-check

Before the exam, can you…

• describe the particles in a solid, a liquid and a gas (arrangement, motion, forces)?
• change °C to K, and say why Ē_k = (3/2)k_BT uses kelvin?
• define internal energy, naming the molecules (atoms, particles) and both kinds of energy?
• use Q = mcΔT and Q = Pt, and solve a mixing problem with lost = gained?
• read every part of a heating curve and explain the flat parts with molecules?
• define specific latent heat and use Q = mL, doing one step for each change?
• check first whether all the ice melts?
• use ΔQ/Δt = kAΔT/Δx with Δx in metres, and explain conduction and convection?
• use L = σAT⁴ (T in kelvin, A = 4πR²) and λ_maxT = 2.9 × 10⁻³ m K?
• use b = L/(4πd²) to find a luminosity or a distance?