“Heat in, work out”
The first law sign convention
Heat supplied to the gas is positive, work done by the gas is positive. ΔQ = ΔU + ΔW then reads: what you put in either stays inside or comes out as work.
Class 11 Physics · Gujarat Board, CBSE, GUJCET and JEE
Thermodynamics formulas
Thermodynamics is bookkeeping for energy: heat in, work out, and what is left inside. The sign convention is the whole battle. Get ΔQ = ΔU + ΔW right and the four processes fall out of it almost automatically.
15 formulas, 4 memory hooks, and the 5 mistakes that cost the most marks.
The first law
| What | Formula | Symbols and notes |
|---|---|---|
| First law | ΔQ = ΔU + ΔW | heat supplied to the gas is positive, work done by the gas is positive |
| Work done by a gas | W = ∫p dV | the area under the p-V curve |
| Internal energy of an ideal gas | U = (f/2)nRT | f is the degrees of freedom; U depends on temperature alone |
The four processes
| What | Formula | Symbols and notes |
|---|---|---|
| Isothermal | T constant, ΔU = 0, W = nRT ln(V₂/V₁) | slow process, in contact with a reservoir |
| Adiabatic | Q = 0, ΔU = −W, pVγ = constant | fast process, insulated from the surroundings |
| Adiabatic work | W = (p₁V₁ − p₂V₂)/(γ − 1) | also nR(T₁ − T₂)/(γ − 1) |
| Isobaric | p constant, W = pΔV = nRΔT | Q = nCpΔT |
| Isochoric | V constant, W = 0, ΔQ = ΔU | all the heat raises the temperature |
Heat capacities
| What | Formula | Symbols and notes |
|---|---|---|
| Mayer's relation | Cp − Cv = R | per mole, for an ideal gas |
| Ratio of heat capacities | γ = Cp/Cv | 1.67 monatomic, 1.40 diatomic, 1.33 triatomic |
| Monatomic gas | Cv = 3R/2, Cp = 5R/2 | three translational degrees of freedom |
| Diatomic gas | Cv = 5R/2, Cp = 7R/2 | three translational plus two rotational |
Engines and refrigerators
| What | Formula | Symbols and notes |
|---|---|---|
| Efficiency of a heat engine | η = 1 − Q₂/Q₁ | work out over heat in |
| Carnot efficiency | η = 1 − T₂/T₁ | kelvin only; the highest possible between two temperatures |
| Coefficient of performance | β = Q₂/W = T₂/(T₁ − T₂) | for a refrigerator, and it can exceed 1 |
Taught by Darshit Bhatt, founder and lead faculty at Darshit Sir Institute. Instrumentation and Control engineer, with a master's degree and a chartered degree, teaching in Paldi, Ahmedabad since 2006.
Know the formula but not when to use it?
Darshit Sir has taught this in Paldi since 2006, in the classroom and live online.
Memory hooks
How to remember this chapter.
“Isothermal is slow, adiabatic is sudden”
Telling the two apart
Isothermal needs time for heat to flow, so it is slow. Adiabatic gives heat no time at all, so it is sudden, like a bursting tyre.
“Gamma: 5 over 3, 7 over 5”
Values of γ
Monatomic 5/3 = 1.67, diatomic 7/5 = 1.40. More ways to store energy means a smaller γ.
“Carnot is the ceiling”
Maximum possible efficiency
No engine between the same two temperatures can beat 1 − T₂/T₁, and reaching it would need infinitely slow, reversible operation.
Where marks go
The mistakes that cost the most.
- Using °C in the Carnot efficiency. Both temperatures must be in kelvin.
- Saying ΔU = 0 for an adiabatic process. That is the isothermal one; in an adiabatic process Q = 0 instead.
- Forgetting that internal energy of an ideal gas depends only on temperature, so any process returning to the same T has ΔU = 0.
- Using W = pΔV when the pressure is not constant. Then the work is the area under the curve, not a product.
- Expecting a coefficient of performance below 1. For a refrigerator it is normally greater than 1, and that is not a violation of anything.
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