First Law of Thermodynamics

The law

The change in internal energy of a system equals the heat added to the system minus the work done by the system:

ΔU = Q − W

where:

• U — internal energy (total microscopic energy of all particles).
• Q — heat added TO the system (positive for heat in).
• W — work done BY the system (positive for work out).

This is energy conservation applied to thermodynamic systems. Energy can change form (heat, work, internal) but is never created or destroyed.

State functions vs process variables

Quantity	Type	Notes
Internal energy U	State function	Depends only on T, P, V, composition
Temperature T	State function	—
Pressure P	State function	—
Volume V	State function	—
Heat Q	Process variable	Depends on path taken
Work W	Process variable	Depends on path taken
Entropy S	State function	(But ΔS depends on details — see 2nd law)

Common thermodynamic processes

Process	Constraint	Q, W, ΔU relations
Isothermal (constant T)	ΔT = 0	For ideal gas, ΔU = 0 → Q = W
Adiabatic (no heat)	Q = 0	ΔU = −W (work done at expense of internal energy)
Isobaric (constant P)	ΔP = 0	W = P·ΔV; Q = ΔU + W
Isochoric (constant V)	ΔV = 0	W = 0; Q = ΔU
Cyclic	Returns to same state	ΔU = 0 → Q_net = W_net

Engines and the first law

An engine takes heat from a hot reservoir (Q_h), uses some to do work (W), and dumps waste heat to a cold reservoir (Q_c).

Q_h = W + Q_c
W = Q_h − Q_c

Efficiency:

η = W / Q_h = (Q_h − Q_c) / Q_h = 1 − Q_c/Q_h

System	Typical efficiency
Car gasoline engine	20-30%
Diesel engine	30-40%
Combined-cycle gas turbine	~60%
Best Rankine cycle (steam)	~50%
Solar PV cell	~20-25%
Bicycle (human → mechanical)	~95%
Carnot ideal (T_h = 800 K, T_c = 300 K)	~63%

Worked examples

Adiabatic compression of gas

A gas is compressed quickly so no heat exchange occurs (Q = 0). Work is done ON the gas (W < 0 by convention). By first law:

ΔU = Q − W = 0 − W = −W (positive since W is negative)

Internal energy increases, so temperature rises. This is why bike pumps get hot — adiabatic compression of air.

Isothermal expansion of ideal gas

Gas expands at constant temperature. For ideal gas, U depends only on T, so ΔU = 0. Therefore Q = W. The gas absorbs heat from the surroundings and does equal work on the piston.

JavaScript — first law calculations

// First law: ΔU = Q - W
function deltaU(Q, W) { return Q - W; }

// Engine efficiency
function engineEfficiency(Q_hot, Q_cold) {
  const W = Q_hot - Q_cold;
  return W / Q_hot;
}

console.log(`Engine: Q_h = 1000 J, Q_c = 700 J → η = ${(engineEfficiency(1000, 700) * 100).toFixed(1)}%`);

// Work in isobaric process
function isobaricWork(P, V_initial, V_final) {
  return P * (V_final - V_initial);
}

// Work in isothermal expansion of ideal gas
function isothermalWork(n, T, V_initial, V_final, R = 8.314) {
  return n * R * T * Math.log(V_final / V_initial);
}

// 1 mole of gas at 300 K, doubles volume
console.log(`Isothermal expansion: ${isothermalWork(1, 300, 1, 2).toFixed(0)} J`); // ~1729 J

// Adiabatic process: T·V^(γ-1) = constant for ideal gas
function adiabaticTemperature(T_initial, V_initial, V_final, gamma = 1.4) {
  // (γ - 1) is heat capacity ratio adjustment
  return T_initial * Math.pow(V_initial / V_final, gamma - 1);
}

// Compress air to half volume adiabatically (e.g., diesel)
console.log(`Adiabatic compression to half: T_initial=300K → T_final=${adiabaticTemperature(300, 2, 1).toFixed(1)}K`);
// 300 × 2^0.4 = 396 K (123°C)

// Cycle work (net)
function cycleWork(Q_in, Q_out) { return Q_in - Q_out; }

// Bicycle pump efficiency: human → mechanical work
function pumpEfficiency(useful_work, calories_burned) {
  // 1 kcal = 4184 J
  const energy_in = calories_burned * 4184;
  return useful_work / energy_in;
}

console.log(pumpEfficiency(50, 0.05).toFixed(3));  // 50 J output, ~50 cal of metabolic energy

Where the first law shows up

• Engineering thermodynamics. Heat engines (cars, power plants), refrigerators, heat pumps, HVAC.
• Chemistry. Heats of formation, reaction energies, enthalpy changes.
• Biology. Metabolic energy balance, food calories, exercise physiology.
• Astrophysics. Stellar structure (energy balance in stars), gravitational collapse.
• Atmospheric science. Adiabatic cooling/heating in air masses, weather formation.
• Materials science. Heat capacity, phase transitions, mechanical work on materials.
• Industrial process. Distillation, refining, manufacturing efficiency calculations.

Common mistakes

• Confusing Q and ΔT. Heat Q is energy transferred. Temperature change ΔT depends on Q AND heat capacity. Q = mc·ΔT for sensible heat; phase changes have Q with no ΔT.
• Sign convention errors. Q > 0 if heat enters system. W > 0 if system does work (expansion). Mistakes here flip the formula.
• Treating heat as a substance. Heat is energy in transit. It "flows" but isn't a thing stored in a body. (Although caloric theory historically thought so — disproved by Rumford and Joule.)
• Forgetting it's energy CONSERVATION. The 1st law doesn't say what's possible — it says what's required. Many "possible" processes (e.g., heat flowing from cold to hot spontaneously) violate the 2nd law, not the 1st.
• Misapplying state vs process variables. ΔU between states is well-defined. Q and W aren't — depend on path. Don't write "U = Q + W" as if Q and W were stored quantities.
• Confusing internal energy with mechanical energy. Internal U is microscopic energy of constituents (kinetic + potential of atoms). Mechanical energy is bulk motion + position energy. Different things.