The first law as an energy balance

At the end of the previous chapter we wrote the energy balance of a gas. In full form:

$E_{\text{int},B} - E_{\text{int},A} = Q - L_\text{gas} + L_\text{atm}$

where each term has a precise meaning:

• $E_\text{int} = \frac{f}{2}nRT$ is the internal energy of the ideal gas ($f$ = degrees of freedom: $3$ monatomic, $5$ diatomic);
• $Q$ is the heat received by the gas (positive if it enters the gas);
• $L_\text{gas}$ is the work done by the gas on the piston (or on other mechanical objects);
• $L_\text{atm} = P_\text{atm}\,\Delta V_\text{atm}$ is the work done by the atmosphere on the gas.

Key formula — First law

$\ev{\Delta E_\text{int} = Q - L_\text{gas} + L_\text{atm}}$ Valid for any transformation, reversible or irreversible.

The form with $L_\text{gas}$ and $L_\text{atm}$ separated is convenient in piston problems: it accounts for the fact that, while the gas expands, the atmosphere is compressed and therefore does work on the gas. In the compact form often found in textbooks, $\Delta U = Q - L$, the symbol $L$ is the net work done by the gas on the surroundings (atmosphere included).

What reversibility is NOT required for

The first law and conservation of energy hold always. It does not matter whether the transformation is slow or abrupt, balanced or explosive: total energy is conserved regardless. All that is needed is precise knowledge of the initial state and the final state. This is exactly what makes the energy-balance method so powerful for irreversible transformations, where no well-defined path exists.

Connections

Topics: Termodinamica Concepts: Primo principio della termodinamica · Energia interna · Legge dei gas perfetti Objects: Gas ideale · Pistone e cilindro

Related exercises: Heating under the piston · Nitrogen heated in a rigid cylinder · Heat as a substance?