- Apply the first law of thermodynamics to different processes
- Calculate the work done by a gas and the change in internal energy
- Find the efficiency of a heat engine and the Carnot limit
A car engine burns petrol, but only about a quarter of that energy turns the wheels; the rest goes into the air as heat. Why can't engineers build an engine that is 100% efficient? The answer comes from thermodynamics, the science of how heat and work are related.
Internal energy and work done by a gas
The internal energy of a body is the sum of the kinetic and interaction energies of all its molecules. In an ideal gas the molecules do not interact, so the internal energy depends only on temperature. Internal energy can be changed in two ways: by doing work and by heat transfer.
- ΔUchange in internal energy of a monatomic ideal gas, in J
- namount of substance, in mol
- ΔTchange in temperature, in K
- Wwork done by the gas at constant pressure, in J
- ΔVchange in volume, in m³
The work is positive when the gas expands and negative when it is compressed. In any process it equals the area under the p–V graph.
The first law of thermodynamics
- Qheat supplied to the gas, in J (Q < 0 if the gas gives off heat)
- ΔUchange in internal energy, in J
- Wwork done by the gas, in J
Heat given to a gas goes into raising its internal energy and into the work the gas does — this is conservation of energy.
| Process | What happens? | First law |
|---|---|---|
| Isochoric (V = const) | the gas does no work | Q = ΔU |
| Isothermal (T = const) | internal energy is unchanged | Q = W |
| Isobaric (p = const) | it both warms up and does work | Q = ΔU + p · ΔV |
| Adiabatic (Q = 0) | no heat exchange | W = −ΔU |
2 mol of a monatomic ideal gas is heated by 50 K at constant pressure. Find the work done by the gas, the change in internal energy and the heat supplied.
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ΔU = (3/2) · 2 · 8.31 · 50 ≈ 1247 J.
Q = ΔU + W ≈ 1247 + 831 ≈ 2078 J ≈ 2.1 kJ.
60% of the heat warms the gas and 40% becomes work.
In a diesel engine cylinder air is compressed so quickly that there is no time for heat exchange. The piston does 400 J of work on the gas. How does the internal energy of the gas change?
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The work done by the gas itself is W = −400 J (it is compressed).
ΔU = Q − W = 0 − (−400) = +400 J.
The internal energy and the temperature rise sharply — that is why the fuel ignites without a spark.
Heat engines and efficiency
Every heat engine has three parts: a hot reservoir at temperature T₁, a working substance (usually a gas) and a cold reservoir at T₂ (often the surrounding air). In each cycle the working substance takes heat Q₁ from the hot reservoir, does work W and gives heat Q₂ to the cold reservoir.
- ηefficiency
- Q₁, Q₂heat taken from the hot reservoir and given to the cold one, in J
- T₁, T₂absolute temperatures of the hot and cold reservoirs, in K
ηmax is the efficiency of an ideal (Carnot) engine; no real engine working between the same temperatures can do better.
a) In each cycle an engine takes 2000 J from the hot reservoir and gives 1400 J to the cold one. Find the work done and the efficiency. b) The hot reservoir is at 527 °C and the cold one at 27 °C. What is the maximum possible efficiency?
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b) T₁ = 800 K, T₂ = 300 K.
ηmax = (800 − 300) / 800 = 0.625 = 62.5%.
A real engine's efficiency is well below this limit.
The second law of thermodynamics
Heat flows by itself only from a hotter body to a colder one, never the other way. Another form: an engine working in cycles cannot turn all the heat it takes in into work — some must be given to a cold reservoir. That is why a fridge or an air conditioner uses electricity to “pump” heat from cold to hot. The law sets the direction of natural processes and is expressed through entropy: in an isolated system entropy never decreases.
Key points
- Internal energy changes through work and heat transfer; for an ideal gas U depends only on T.
- First law: Q = ΔU + W; at constant pressure W = p · ΔV = n · R · ΔT.
- A heat engine's efficiency η = (Q₁ − Q₂) / Q₁ is always below the Carnot limit ηmax = (T₁ − T₂) / T₁.
- Second law: heat never flows by itself from cold to hot, and heat cannot be turned completely into work.
Check yourself
10 questions. Every correct answer earns XP.