Work Done by a Thermodynamic System: MDCAT Physics notes
Work Done by a Thermodynamic System notes for MDCAT: W = PΔV, isothermal, isobaric, isochoric and adiabatic processes, gas-law examples and traps.
Work done by a gas
A gas in a cylinder with a movable piston of area $A$ pushes with force $F = PA$. If the piston moves out by $\Delta y$ at constant pressure, the gas does work
$$W = F\Delta y = PA\Delta y = P\Delta V = P(V_2 - V_1)$$
- Expansion ($\Delta V > 0$): work is done by the gas (positive).
- Compression ($\Delta V < 0$): work is done on the gas.
- No volume change: no work.
- On a $P$–$V$ graph, work is the area under the curve.
Example: a gas expands from $1\ \text{m}^3$ to $3\ \text{m}^3$ at 2 Pa: $W = 2 \times 2 = 4\ \text{J}$.
First law link
$$Q = \Delta U + W$$
Heat supplied to a system increases its internal energy and/or lets it do work.
Thermodynamic processes
| Process | Constant | Law | First law becomes |
|---|---|---|---|
| Isothermal | Temperature (throughout) | Boyle's: $PV = \text{constant}$ | $\Delta U = 0$, $Q = W$ |
| Isobaric | Pressure | Charles's: $V/T = \text{constant}$ | $Q = \Delta U + P\Delta V$ |
| Isochoric | Volume | $P/T = \text{constant}$ | $W = 0$, $Q = \Delta U$ |
| Adiabatic | No heat exchange ($Q = 0$) | $PV^\gamma = \text{constant}$ | $W = -\Delta U$ |
Isothermal
Temperature stays constant throughout, so the internal energy of an ideal gas does not change. During isothermal expansion the pressure falls, the gas does work, and it absorbs an equal amount of heat. For isothermal compression, $PV$ remains constant.
Isobaric
$V \propto T$ in kelvin. Heating from 150°C (423 K) to 300°C (573 K) gives $V_2/V_1 = 573/423 \approx 1.35$: the volume becomes less than double, even though the Celsius temperature doubled.
Isochoric
Volume is fixed, so the work done is zero; all heat supplied raises the internal energy (and temperature).
Adiabatic
The process is so rapid, or so well insulated, that no heat enters or leaves.
- Adiabatic compression: work done on the gas raises its internal energy, so temperature increases (e.g. a bicycle pump warms up; diesel engine ignition).
- Adiabatic expansion: the gas does work at the expense of internal energy, so it cools. The rapid escape of air from a burst tyre is adiabatic, and the escaping air feels cool. Cloud formation in rising air is another example.
Key formulas
- $W = P\Delta V = P(V_2 - V_1)$
- $Q = \Delta U + W$
- Isothermal: $P_1V_1 = P_2V_2$
- Isobaric: $V_1/T_1 = V_2/T_2$
- Adiabatic: $PV^\gamma = \text{constant}$
Common MDCAT traps
- Use kelvin in Charles's law: 150°C to 300°C is less than double the volume.
- Isochoric: work is zero, not heat.
- Boyle's law ($PV$ constant) applies to isothermal processes.
- Adiabatic compression raises temperature; adiabatic expansion lowers it.
- Isothermal: temperature constant throughout, not just at the start.
Quick revision
- Burst tyre: adiabatic.
- Isobaric: pressure constant.
- Isochoric: volume constant.
- Isothermal: $\Delta U = 0$ for an ideal gas.
- Work = area under the $P$–$V$ curve.