Difficulty: Introductory to Intermediate | Prerequisites: Basic algebra, unit conversions
Energy comes in several forms (kinetic, potential, thermal, chemical) and can be converted between them, but the total energy of an isolated system is always conserved. The first law of thermodynamics says the change in thermal energy of a system equals the work done on it plus the heat transferred into it. The second law adds a direction: entropy always increases in irreversible processes, meaning energy spontaneously spreads out rather than concentrating.
Energy
The capacity to do work or cause change. Measured in joules (J). In simple terms, it is whatever lets something move, heat up, or change state.
Kinetic energy (K)
Energy of motion. K = ½mv². Think of it as the energy a moving object carries because of its speed.
Potential energy (U)
Stored energy due to position or configuration. Gravitational potential energy: U_g = mgy. Think of it as energy waiting to be released when something falls or springs back.
Thermal energy (E_th)
The total kinetic energy of all the atoms and molecules in an object, associated with their random motion. Think of it as the internal "jiggle" energy of matter.
Temperature (T)
A measure of the average kinetic energy per atom or molecule in an object. Not the same as thermal energy: a cup of boiling water and a bathtub of warm water can have different temperatures but similar thermal energies.
Heat (Q)
Energy transferred between objects because of a temperature difference. "Energy transferred thermally." It is not something an object possesses; it is energy in transit.
Work (W)
Energy transferred to or from a system by mechanical means (forces acting over a distance). Positive W means energy enters the system; negative W means energy leaves.
Entropy (S)
A measure of the number of microstates (microscopic arrangements) consistent with a given macrostate. S = k_B ln Ω. In simple terms, entropy counts how many ways the atoms can be rearranged without changing the big-picture properties.
Irreversible process
A process that can only occur in one direction spontaneously. All macroscopic processes involving friction, mixing, or heat flow across a temperature difference are irreversible.
Reversible process
A process at the molecular level that can run equally well in either direction. An idealisation: real macroscopic processes are never perfectly reversible.
Efficiency (e)
The fraction of input energy that is converted to useful output. e = energy out (useful) / energy in. Always less than 1 for real systems.
Chemical energy (E_chem)
Energy stored in the bonds of molecules, released or absorbed during chemical reactions. Food energy is a common example.
Calorie / kilocalorie
1 food Calorie (capital C) = 1 kcal = 1000 cal = 4190 J. The unit used on nutrition labels. 1 cal (lowercase) = 4.19 J.
Multiplicity (Ω)
The number of microstates corresponding to a given macrostate. Higher multiplicity means higher entropy.
The energy principle states that the total energy of a system changes only when energy is transferred in or out:
ΔE = ΔK + ΔU + ΔE_th + ΔE_chem + ...
Work is positive when energy enters the system, negative when energy leaves
The efficiency equation
e = what you get out / what you put in
The efficiency equation diagram (from lectures) is a key visual: energy input on one side, useful energy output on the other, with waste energy (usually thermal) leaving the system
Two types of energy limitations
Process limitations: obstacles that reduce efficiency but could in principle be overcome (e.g. friction, air resistance). A better design can reduce these losses.
Fundamental limitations: limits set by physics itself. No amount of engineering can exceed them (e.g. the second law of thermodynamics caps heat engine efficiency).
Gravitational potential energy
U_g = mgy, where y is measured from a chosen reference point
When solving for ΔU_g: ΔU_g = mgΔy
The human body is an energy-conversion system. Food (chemical energy) is converted into kinetic energy (movement), thermal energy (body heat), and gravitational potential energy (climbing stairs).
Energy accounting in the body
ΔE_chem + ΔU_g + ΔE_th = 0 (for a closed system with no external work)
|ΔE_chem| = ΔU_g + ΔE_th
The body is roughly 25% efficient at converting chemical energy to mechanical work. The remaining 75% becomes thermal energy.
Unit conversions for food energy
1 food Calorie (Cal) = 1 kcal = 4190 J
1 kJ = 1000 J
100 Cal = 100 kcal = 419,000 J ≈ 4.19 × 10⁵ J
Worked example pattern (stair climbing)
Find ΔU_g = mgΔy for climbing one flight of stairs
Use 25% efficiency to find ΔE_chem: |ΔE_chem| = ΔU_g / 0.25
Convert to Calories by dividing by 4190 J/Cal
Compare to food energy (e.g. a banana is roughly 100 Cal, a Big Mac around 600 Cal)
Power
Power = energy / time, measured in watts (W)
1 W = 1 J/s
A human climbing stairs at 15 km/h expends roughly 1150 W
Key distinctions
Thermal energy is the total energy of all atoms' random motion in an object. It depends on mass, temperature, and the type of material.
Temperature is the average kinetic energy per atom. Two objects can have the same temperature but vastly different thermal energies if they differ in size.
Heat is energy transferred between objects because of a temperature difference. It is not a property of an object.
Temperature conversions
T(°C) = (5/9)(T(°F) - 32)
T(°F) = (9/5)T(°C) + 32
5°C change = 9°F change (useful shortcut for temperature differences)
Relating ΔE_th to ΔT
Adding heat to an object raises its thermal energy, which raises its temperature (provided no phase change occurs)
The relationship between ΔE_th and ΔT depends on the material's specific heat (covered in 12.5)
The first law is a specific form of energy conservation for thermal systems:
ΔE_th = W + Q
W = work done on the system (positive when energy enters)
Q = heat transferred into the system (positive when energy enters)
Sign conventions (important)
W > 0: work done on the system (energy in)
W < 0: work done by the system (energy out)
Q > 0: heat transferred into the system
Q < 0: heat transferred out of the system
Special cases
If only thermal energy changes (no other energy forms): ΔE_th = W + Q
If the system is isolated (no heat transfer, no work): W = 0, Q = 0, so ΔE_th = 0. The thermal energy does not change.
If no work is done: W = 0, so ΔE_th = Q. All heat goes to changing thermal energy.
If no heat is transferred: Q = 0, so ΔE_th = W. Only work changes the thermal energy.
Helpful tip
Always think about the PV graph. Work done by a gas equals the area under the curve on a pressure-volume diagram. This connects the first law to the ideal gas processes in Chapter 12.
Entropy basics
Entropy S = k_B ln Ω, where k_B = 1.38 × 10⁻²³ J/K (Boltzmann's constant) and Ω = multiplicity (number of microstates)
More microstates = higher entropy
Entropy is maximised when a system reaches equilibrium
The second law
Energy spontaneously spreads out. It flows from hot to cold, never the reverse, without external work.
The total entropy of an isolated system never decreases.
Irreversible processes (friction, free expansion, mixing) always increase entropy
Interactions at the molecular level are reversible, but macroscopic processes are not, because there are overwhelmingly more disordered microstates than ordered ones
Connecting to probability
Think of the 1D random walk from Lecture 4. A particle is equally likely to step left or right, but the number of paths that end near the centre vastly outnumbers the paths that end at an extreme. This is why systems evolve toward high-multiplicity macrostates: not because any law forbids the reverse, but because it is astronomically unlikely.
Cheat codes for counting microstates
For N items in 2 bins: total microstates = 2^N
To find microstates on one side: use combinations. How many ways to have n items on one side out of N total? C(N, n) = N! / (n!(N-n)!)
Example: 4 particles, 2 on each side: C(4,2) = 4! / (2!·2!) = 6 microstates
\Delta E = \Delta K + \Delta U + \Delta E_{th} + \Delta E_{chem}e = \frac{\text{useful energy out}}{\text{total energy in}}U_g = mgy \quad ; \quad \Delta U_g = mg\Delta y|\Delta E_{chem}| = \Delta U_g + \Delta E_{th}T(^{\circ}C) = \frac{5}{9}(T(^{\circ}F) - 32)\Delta E_{th} = W + Q \quad \text{(First Law)}S = k_B \ln \Omega \quad \text{(Entropy)}1 \text{ Cal} = 1 \text{ kcal} = 4190 \text{ J}The efficiency concept explains why car engines waste so much energy as heat: a petrol engine is roughly 20-25% efficient, similar to the human body. The rest is thermal energy that must be removed by the cooling system.
Calorie counting on food labels is a direct application of E_chem and the body's energy budget. When you eat 2000 Cal, your body converts that chemical energy into motion, heat, and (if climbing) gravitational potential energy.
Students often confuse temperature and thermal energy. Temperature is the average energy per particle. Thermal energy is the total. A large cold lake has far more thermal energy than a small cup of boiling water, even though the cup is hotter.
Students often think heat is something an object "has." It is not. Heat is energy in transit between objects at different temperatures. Once transferred, it becomes thermal energy.
Students frequently mix up the sign conventions for work. In this course, W > 0 means work done on the system (energy in), not by the system. Check your sign carefully every time.
Students sometimes believe entropy means "disorder" in a vague, qualitative sense. It has a precise definition: S = k_B ln Ω. It counts microstates, not mess.
⚠️ The energy bar chart (from the efficiency diagram) is highlighted as "super useful" in the source notes. Expect a question asking you to trace energy through a system and account for all forms.
⚠️ Unit conversions between Calories, kilocalories, and joules appear frequently. Memorise: 1 Cal = 1 kcal = 4190 J.
⚠️ Sign conventions for W and Q in the first law are a common source of lost marks. State your sign convention explicitly in your working.
⚠️ Entropy problems often ask you to count microstates using combinations. Know C(N,n) = N! / (n!(N-n)!) and be comfortable with small examples (N = 4 or 6).
True or False: Heat is a form of energy stored inside an object. (False. Heat is energy in transit between objects at different temperatures.)
Fill in the blank: The first law of thermodynamics states ΔE_th = ______ + ______. (W + Q)
True or False: If you double the mass of an object at the same temperature, you double its thermal energy. (True.)
Fill in the blank: Entropy is maximised when a system reaches ______. (Equilibrium.)
True or False: A process that increases entropy can run spontaneously in reverse. (False. Irreversible processes do not spontaneously reverse.)
Q: A 70 kg person climbs 3 flights of stairs (each 2.7 m). If the body is 25% efficient, how much chemical energy is used? Express your answer in joules and in Calories.
A: ΔU_g = mg(3 × 2.7) = 70 × 9.8 × 8.1 ≈ 5557 J. At 25% efficiency: E_chem = 5557 / 0.25 ≈ 22,228 J ≈ 22.2 kJ. In Calories: 22,228 / 4190 ≈ 5.3 Cal.
Q: A closed system has no heat transfer and no work done on it. What happens to its thermal energy?
A: Nothing. ΔE_th = W + Q = 0 + 0 = 0. The thermal energy is unchanged.
Q: You have 4 coins. How many microstates correspond to the macrostate "2 heads, 2 tails"?
A: C(4,2) = 4! / (2! × 2!) = 6 microstates.
Q: Explain why a hot cup of coffee left on a table always cools down, never heats up further.
A: The second law of thermodynamics. Heat flows spontaneously from hot to cold. The macrostate where energy is spread between the coffee and the room has overwhelmingly more microstates (higher entropy) than the macrostate where all the energy is concentrated in the coffee. The system evolves toward the higher-entropy state.
Q: Convert 98.6°F (normal body temperature) to Celsius.
A: T(°C) = (5/9)(98.6 - 32) = (5/9)(66.6) = 37°C.
The first law of thermodynamics connects directly to ideal gas processes (Ch. 12.3), where you apply ΔE_th = W + Q to specific types of PV changes (isobaric, isochoric, isothermal, adiabatic).
Entropy and the second law set the stage for understanding why heat engines have maximum theoretical efficiencies, a topic that extends into later thermodynamics courses.
The energy-in-the-body calculations connect to calorimetry (Ch. 12.6), where you track heat transfer between objects using conservation of energy.
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