The First Law
Heat in equals internal energy gained plus work done.
Lesson at a glanceShow the key points
- Heating (hotter walls) and work (a moving piston) both change a gas's internal energy.
- First law: ΔQ = ΔU + ΔW (26.1): conservation of energy for any system.
- Signs: ΔQ > 0 heat supplied; ΔW > 0 work done by the system.
- ΔQ and ΔW depend on the path; ΔQ − ΔW = ΔU = U_f − U_i does not: internal energy is a state function.
There are two ways to give a gas energy: heat it, or push on it. The first law of thermodynamics is the energy account that keeps track of both.
Two ways to change internal energy
By heating. Keep a piston fixed and make the cylinder walls hotter than the gas. Gas molecules bouncing off the faster-jiggling wall molecules gain energy on average, and share it: the internal energy rises.
By work. Keep the walls at the gas temperature but push the piston in. A molecule bouncing off an approaching piston rebounds faster (like a ball off a moving bat), so again the internal energy rises.
Usually both happen together: a gas in a can on a hot stove takes in heat and, by pushing its piston out, does work and loses energy.
The first law
If heat ΔQ is given to the gas and it does work ΔW, its energy rises by ΔQ − ΔW. If none of this goes into bulk motion of the gas, it all becomes internal energy ΔU:
| Quantity | Positive when | Negative when |
|---|---|---|
| ΔQ | heat is given to the system | heat leaves the system |
| ΔW | the system does work (expands) | work is done on the system |
| ΔU | internal energy rises | internal energy falls |
For an ideal monatomic gas, internal energy is just the translational KE of its molecules; in general it also includes rotational and vibrational energy and the potential energy of molecular forces.
A gas in a cylinder with a movable piston takes in 150 cal of heat and does 200 J of work as it expands. Find the change in its internal energy.
Internal energy is a state function
Take a system from state i to state f by different routes: ΔQ and ΔW will differ from route to route, but ΔQ − ΔW is always the same, ΔU = U_f − U_i. So there is an internal energy U with a definite value in each state. We write ΔU = U_f − U_i, but never “Q_f − Q_i” or “W_f − W_i”: heat and work belong to processes, not states.
1 kg of water at 100°C becomes steam at 1 atm (≈ 100 kPa). Densities: water 1000, steam 0.60 kg/m³; L = 2.25 × 10⁶ J/kg. Find ΔU.
- 1Internal energy can be changed by heat transfer and by work.
- 2First law: ΔQ = ΔU + ΔW (heat in positive, work by the system positive).
- 3U is a function of state; Q and W depend on the process.
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