First Law of Thermodynamics

1. Definitions (Must Know)

A. First law of thermodynamics (syllabus sign convention)

The first law of thermodynamics is conservation of energy for a thermodynamic system:

Δ U = Q + W

where (for the system):

• Δ U is the change in internal energy (J).
• Q is the energy transferred to the system by heating (J).
• W is the work done on the system (J).

B. Work done by vs work done on

• Work done by the gas, W_by: energy transferred from the gas to the surroundings.
• Work done on the gas, Wₒₙ: energy transferred to the gas by the surroundings.

They are related by: Wₒₙ = -W_by

C. Internal energy is a state function

Internal energy U is a state function: it depends only on the system’s state (e.g. p, V, T), not on the path taken.

So for the same initial and final states, Δ U is the same even if Q and W are different.

2. Key Ideas (What Earns Marks)

• In this syllabus, use:
  • Δ U = Q + Wₒₙ
• Many textbooks use the alternative form:
  • Δ U = Q - W_by
  • which is the same equation, since Wₒₙ = -W_by.
• At constant volume, Δ V = 0 ⇒ W_by = 0 so:
  • Δ U = Q
• For expansion/compression against constant external pressure p:
  • W_by = pΔ V
  • Wₒₙ = -pΔ V
• Unit check: Pa·m³ = J.

3. Detailed Explanations

A. A safe workflow for first-law questions

1. Define the system (usually “the gas”).
2. Decide the sign convention you will use (this site uses Wₒₙ).
3. Identify what the process tells you:
   • isochoric ( Δ V = 0) → W_by = 0,
   • no heating (no net energy transfer by heating) → Q = 0,
   • constant temperature ( Δ T = 0) → for an ideal gas, Δ U = 0.
4. Compute W (if needed) using the external pressure given.
5. Substitute into Δ U = Q + Wₒₙ.

B. Work term for constant external pressure: W_by = pΔ V

If a gas expands against a constant external pressure p: W_by = pΔ V

So in the first law (syllabus convention): Wₒₙ = -pΔ V

Sign check:

• expansion: Δ V > 0 ⇒ W_by > 0 and Wₒₙ < 0
• compression: Δ V < 0 ⇒ W_by < 0 and Wₒₙ > 0

C. What Δ U means in words (for explanation marks)

• Δ U > 0: internal energy increases.
• Δ U < 0: internal energy decreases.

For an ideal gas, internal energy is tied mainly to particle kinetic energy, so Δ U tracks temperature change.

4. Common Mistakes

• Mixing sign conventions (using Δ U = Q + W but treating W as work done by the system).
• Using gas pressure instead of the constant external pressure specified for pΔ V.
• Forgetting V must be in m³ (convert from L or cm³).
• Writing “heat” as a stored quantity (it is a transfer; internal energy U is stored).

5. Exam Tips

• State the sign convention explicitly if the question looks ambiguous (“W is work done on the gas”).
• Always write a final energy-balance sentence, e.g. “Q adds energy, expansion does work out, net internal energy increases.”
• Quick special cases:
  • constant volume → Δ U = Q
  • no heating ( Q = 0) → Δ U = Wₒₙ

6. Worked Examples

7. Mind Stretchers

8. Practice (Quiz)