First Law of Thermodynamics

During the mid-nineteenth century, James Joule demonstrated experimentally that mechanical work and the heat produced by that work are equivalent quantities — related by $W = J \times H$, where $J$ is called the mechanical equivalent of heat. This equivalence is the experimental seed from which the First Law of Thermodynamics grows: it turns the earlier, qualitative observation (heating and doing work both change internal energy) into a precise, quantitative relationship.

Deriving the work-done formula. Consider the gas-in-cylinder system from Section 4.4.4, with cross-sectional piston area $A$ and gas pressure $p$. The gas exerts a total outward force $F = pA$ on the piston. If the piston moves through an infinitesimal distance $dx$, the infinitesimal work done by this force is $dW = F\,dx = pA\,dx$. But $A\,dx$ is exactly the infinitesimal change in the cylinder's volume, $dV$ — so

For a finite change from an initial volume $V_i$ to a final volume $V_f$, the total work done by the gas is the sum (integral) of all these infinitesimal contributions:

During expansion, the gas molecules striking the outward-moving piston lose momentum to it, exerting an outward pressure that pushes the piston through a finite distance — the gas does positive work on its surroundings. During compression, the reverse happens: molecules striking the inward-moving piston gain momentum from it, so the gas does negative work (equivalently, the surroundings do positive work on the gas).

Combining heat and work: the First Law. Suppose heat $Q$ is added to the system and the system does no work during the process — its internal energy simply increases by the full amount of heat added, $\Delta U = Q$. Now suppose instead the system does some work $W$ to expand its volume, with no heat added at all — the system must draw the energy for this work from its own internal energy, so its internal energy decreases: $\Delta U = -W$. In general, both effects act together, and combining them gives the mathematical statement of the First Law of Thermodynamics:

Rearranged, this is equally often written as

In words: the change in a system's internal energy equals the heat supplied to it minus the work done by it on its surroundings — or, equivalently, the heat supplied to a system is partly used to raise its internal energy and partly used to do work on its surroundings.

Both $Q$ and $W$ can independently be positive, negative, or zero, so $\Delta U$ can be positive (heat added exceeds work done, internal energy rises), negative (work done exceeds heat added, internal energy falls), or zero (heat added exactly balances work done — this special case, as we will see, is exactly what characterises an isothermal process for an ideal gas, in Section 4.7.3.2). …