First Law of Thermodynamics: MDCAT Physics notes
First Law of Thermodynamics MDCAT notes: Q = ΔU + W, sign conventions, isothermal, adiabatic, isochoric and isobaric processes, with solved-style tips.
Statement of the first law
The first law of thermodynamics is the law of conservation of energy applied to heat and work. When heat $Q$ is given to a system, part of it raises the internal energy $U$ and the rest is used by the system to do external work $W$:
$$Q = \Delta U + W$$
It is not a law of conservation of heat, work, mass, charge or momentum. Heat and work are both ways of transferring energy; the law says energy is neither created nor destroyed in the transfer.
Sign convention (FSc textbook)
- $Q$ is positive when heat is added to the system, negative when heat is removed.
- $W$ is positive when work is done by the gas (expansion), negative when work is done on the gas (compression).
- $\Delta U$ is positive when the temperature of the gas rises.
Example: 150 J of heat is removed ($Q=-150$ J) while 500 J of work is done on the gas ($W=-500$ J). Then $\Delta U = Q - W = -150-(-500) = +350$ J. The internal energy rises.
Internal energy
For an ideal gas the internal energy is the kinetic energy of its molecules, so it depends only on temperature. Whatever the path, if the temperature does not change then $\Delta U = 0$. Internal energy is a state function; $Q$ and $W$ depend on the path taken.
Work done by a gas at constant pressure is $W = P\Delta V$. If the volume does not change, no work is done.
The four special processes
| Process | Held constant | Condition | First law becomes |
|---|---|---|---|
| Isothermal | Temperature | $\Delta U = 0$, $PV$ = constant (Boyle's law) | $Q = W$ |
| Adiabatic | No heat exchange | $Q = 0$ | $W = -\Delta U$ |
| Isochoric | Volume | $W = 0$ | $Q = \Delta U$ |
| Isobaric | Pressure | $W = P\Delta V$ | $Q = \Delta U + P\Delta V$ |
Isothermal process
The temperature stays constant throughout the process, not just at the start. The gas must be in good thermal contact with a reservoir and the change must be slow. All heat supplied is converted into work.
Adiabatic process
No heat enters or leaves, so $Q = 0$. Work done by the gas comes out of its internal energy and it cools (adiabatic expansion). Work done on the gas all goes into internal energy and it warms (adiabatic compression). Adiabatic changes are either very fast or happen in an insulated container. Examples: air rushing out of a burst tyre, compression stroke in a diesel engine, rapid expansion of rising air in the atmosphere forming clouds.
Isochoric process
No volume change means no work, so all the heat supplied goes into internal energy. This is the process asked when a question says "entire heat is used to increase internal energy".
Link to heat engines
A heat engine takes heat from a hot source, does work, and rejects the rest to a cold sink. The maximum (Carnot) efficiency depends only on the absolute temperatures:
$$\eta = 1 - \frac{T_2}{T_1}$$
Always convert to kelvin first. For a source at 327 °C (600 K) and sink at 27 °C (300 K), $\eta = 1 - 300/600 = 50\%$.
Key formulas
- $Q = \Delta U + W$
- $W = P\Delta V$ (constant pressure)
- Isothermal: $PV = \text{constant}$; adiabatic: $PV^{\gamma} = \text{constant}$
- $\eta_{\text{Carnot}} = 1 - T_2/T_1$ (kelvin)
Common MDCAT traps
- Work done on the gas is negative $W$; in adiabatic compression $\Delta U = +W_{\text{on}}$, so 200 J of work on the gas gives $\Delta U = +200$ J.
- "Heat taken out" means $Q$ is negative; keep both signs before substituting.
- In an adiabatic process $Q = 0$ exactly, not +1 or −1.
- Confusing isochoric (all heat to $\Delta U$) with isothermal (all heat to work).
- Using Celsius instead of kelvin in the efficiency formula.
Quick revision
- First law = conservation of energy.
- Isothermal: $\Delta U = 0$ and $Q = W$.
- Adiabatic: $Q = 0$; expansion cools, compression heats.
- Isochoric: $W = 0$ and $Q = \Delta U$.
- Internal energy of an ideal gas depends only on temperature.