Thermodynamics

The First Law of Thermodynamics is energy conservation applied to heat and work together. If $\Delta Q$ is the heat supplied to a system, $\Delta W$ the work done by it, and $\Delta U$ the resulting change in internal energy:

Work done by a gas is the area under its P-V curve, $W = \int P\,dV$. Two special cases drive the simulation above. Isothermal (T fixed, so $PV=\text{const}$, Boyle’s law): internal energy of an ideal gas depends only on T, so $\Delta U = 0$ and all the heat supplied becomes work:

Adiabatic (insulated, $Q=0$, so $PV^\gamma=\text{const}$ where $\gamma = C_p/C_v$): the First Law collapses to $\Delta U = -W$, every joule of work done by the gas comes directly out of its own internal energy, which is exactly why compressing it (doing work on it, $W<0$) raises its temperature:

A similar bookkeeping argument at constant pressure, combined with the ideal gas law, gives the standard relation between the two molar specific heats of a gas:

It’s easy to picture a gas as “containing” some heat, the way a tank contains water. NCERT is explicit that this picture is wrong: heat and work are both just modes of energy transfer, not quantities a system holds. What a gas in a given state actually has is internal energy, a state variable with a definite value. Heat supplied and work done can both depend on the path taken between two states; internal energy never does.

The second trap is confusing isothermal with adiabatic, they sound like they could mean similar things, and they describe opposite constraints. Isothermal fixes temperature and lets heat flow freely to maintain it. Adiabatic fixes heat flow at zero and lets temperature swing freely, often dramatically, as the simulation’s colour change shows. A rapidly rising parcel of air cools by this exact adiabatic mechanism, no heat has left it, it has simply expanded and spent its own internal energy doing work against the lower pressure outside, which is a large part of why clouds form at altitude.

A third trap: assuming a clever enough design could one day build a 100%-efficient engine. The Second Law rules this out completely, not as a practical limitation but as a structural one. Even the Carnot engine, the best any engine can ever be between two fixed temperatures, tops out at $\eta = 1 - T_2/T_1$, strictly less than 1 unless the cold reservoir is at absolute zero.