Short on time? This page is the recap. For explanations, Figure 1 and the worked examples, use the full study guide.
Recap
- Internal energy U = kinetic energy of the particles + potential energy of their arrangement. An ideal gas has no internal potential energy.
- For an ideal monatomic gas, U is the total kinetic energy of the atoms and depends only on T.
- Energy crosses the boundary in two ways: heating (Q) and work (W). Both are transfers, not contents.
- The first law is conservation of energy for a closed system. For an isolated system, total energy is constant.
- On a PV diagram, area under the path = |W|; the direction of the path gives the sign.
Key relationships
| Relationship | Symbols and units | Use it to |
|---|---|---|
| ΔU = Q + W | all in J; Q > 0 into gas; W > 0 when work is done on the gas | link heating, work and internal energy |
| W = −PΔV | P in Pa, V in m³ | find work on the gas at constant (or average) pressure |
| U = (3/2)nRT = (3/2)PV | R = 8.31 J/(mol·K), T in K | find U or ΔU for an ideal monatomic gas |
| ΔU_cycle = 0, Q_net = −W_net | W_net |
| Process | Condition | First law becomes |
|---|---|---|
| Isochoric | W = 0 | ΔU = Q |
| Isobaric | P constant | Q = (5/2)nRΔT |
| Isothermal | ΔU = 0 | Q = −W |
| Adiabatic | Q = 0 | ΔU = W |
Assumptions behind the numbers
- The gas is ideal and monatomic unless the question says otherwise.
- The system is closed: no gas leaks in or out.
- Processes are slow enough that P and T are well defined at every point on the PV path.
- Temperatures in U = (3/2)nRT are in kelvin.
Mistakes to avoid
- Wrong sign of W. An expanding gas has W < 0 in this course.
- Confusing isothermal and adiabatic. Isothermal: ΔU = 0. Adiabatic: Q = 0.
- Area always positive. Size from the area, sign from the direction.
- Calling heat a property. A gas has U, not “heat”.
- Using (5/2) for a constant-volume process. At constant V, Q = ΔU = (3/2)nRΔT.
- Forgetting ΔU = 0 around a cycle.
Quick self-check
- A gas is heated with 300 J and does 120 J of work on its surroundings. What is ΔU? (+180 J, because W = −120 J)
- A gas is compressed at a constant 2.0 × 10⁵ Pa from 5.0 × 10⁻³ m³ to 3.0 × 10⁻³ m³. What is the work done on it? (+400 J)
- In which process is all the work done on a gas stored as internal energy? (Adiabatic, Q = 0)
Next: practice questions.