Complete energy balance

Let us put the pieces together. Consider a gas that receives heat $Q$ and that can expand, doing work $L_\text{gas}$ on the piston and receiving work $L_\text{atm}$ from the atmosphere. The energy balance of the gas between the initial state $A$ and the final state $B$ is:

Law — First law with atmosphere

$\ev{E_{\text{int},B} - E_{\text{int},A} = Q - L_\text{gas} + L_\text{atm}}$ $Q$ is positive if it enters the gas; $L_\text{gas}$ is the work done by the gas (energy that leaves); $L_\text{atm}$ is the work done by the atmosphere on the gas (often negative during expansion).

The reading is that of a ledger: the change in internal energy of the gas is everything that comes in minus everything that goes out. Heat $Q$ comes in and (sometimes) the work of the atmosphere; the work that the gas does on the piston goes out. The signs must be handled carefully, but the logic is energy conservation applied to the gas.

For an ideal gas there is a powerful shortcut: since $E_\text{int} = \frac{f}{2}nRT$, the change in internal energy is $\Delta E_\text{int} = \frac{f}{2}nR\,\Delta T$. The energy balance thus becomes directly an equation for temperatures, without needing to know any other details of the gas.

With these tools — internal energy, equation of state, work of the gas and of the atmosphere — a great many thermodynamic problems can already be solved without ever talking about reversible or irreversible processes. That is the programme of the next chapter: solving thermodynamics problems with energy conservation alone.

Collegamenti

Argomenti: Teoria cinetica dei gas · Termodinamica Concetti: Energia interna · Primo principio della termodinamica Competenze: Conservazione dell’energia

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