Thermodynamic terms and concepts

A system is the part of the universe singled out for study; everything else is the surroundings, and the boundary between them decides what can cross.

• Open system — matter and energy both cross the boundary (a beaker of hot tea, uncovered).
• Closed system — energy crosses, matter does not (a sealed but thermally conducting flask of gas).
• Isolated system — neither matter nor energy crosses (an ideal, perfectly insulated thermos).

The condition of a system at any instant is described by measurable macroscopic variables — pressure $P$, volume $V$, temperature $T$, amount $n$ — called state variables. A quantity built from these is a state function if its value depends only on the system's current state, never on how that state was reached. Internal energy $U$, enthalpy $H$, entropy $S$, and Gibbs free energy $G$ are all state functions. For any state function $X$ carried from an initial state 1 to a final state 2,

where: $X_1$ and $X_2$ are the values of the property in the initial and final states, and $\Delta X$ is written only for quantities that qualify as state functions. This equation is deceptively simple — its whole content is that the path in between is irrelevant, however tortuous. A path function carries no such guarantee. Heat $q$ and work $w$ are the two path functions used throughout the chapter: the same overall change of state can be reached while absorbing different amounts of heat or performing different amounts of work, depending on the route (reversible or irreversible, single-step or gradual). Because a path function is not pinned down by the endpoints alone, it is never written with a "$\Delta$" — there is no such thing as $\Delta q$ or $\Delta w$, only $q$ and $w$ for a specified process. Properties also split by how they respond to the size of the sample:

• Extensive properties scale with the amount of matter — volume, mass, internal energy, enthalpy, entropy, heat capacity. Doubling the substance (intensive conditions unchanged) doubles these.
• Intensive properties do not scale with amount — temperature, pressure, density, molar volume, and any quantity already expressed "per mole" or "per gram." Notably, the ratio of two extensive properties is always intensive: density = mass / volume.

Finally, a process earns a special name from whichever variable is held fixed, or which exchange is forbidden: isothermal ($\Delta T = 0$), isobaric ($\Delta P = 0$), isochoric ($\Delta V = 0$), and adiabatic ($q = 0$).

A student's instinct is to treat "$\Delta$" as something you can slap onto any quantity — but the definition of a state function is exactly what licenses that notation, and the reasoning behind it is worth walking through once. Take a gas expanding from $2\ \text{L}$ to $6\ \text{L}$. Do it two ways: in one giant step against a fixed opposing pressure, or in many tiny steps with the opposing pressure adjusted almost continuously (closer to reversible). In both cases the volume goes from $2\ \text{L}$ to $6\ \text{L}$, so because $V$ is a state function, $\Delta V = 6 - 2 = 4\ \text{L}$ regardless of which route was taken — the path is simply irrelevant to $\Delta V$. Work is different. For an expansion against an external pressure, the work done equals the area swept under the $P$–$V$ curve for the actual path followed. A single large step against one constant pressure sweeps a different area than a staircase of small steps, even though both staircases start and finish at the same $V$. So the two processes can do different amounts of work, $w_1 \neq w_2$, despite having identical $\Delta V$. That is the operational meaning of "path function": fix the endpoints and $w$ is still not determined, because it depends on the shape of the journey between them, not just on where it starts and ends. The same reasoning applies to heat $q$ — an isothermal expansion and an adiabatic one can connect very different pairs of states while exchanging very different amounts of heat. This is also why insulation and isolation are not synonyms. Insulating a container blocks heat, but if the vessel is still open at the top, matter can enter or leave — the system is open, not isolated. True isolation requires blocking both channels at once: a sealed and insulated, rigid container.

A reaction is carried out in a stoppered, vacuum-jacketed steel vessel containing 2 mol of gas. Classify the system, name the type of process, and sort volume, pressure, molar mass, and total internal energy as extensive or intensive.

Step 1. Check the boundary. The vessel is stoppered, so no matter crosses; it is vacuum-jacketed (insulated), so no heat crosses either; being rigid steel, no expansion work is possible. Neither matter nor energy crosses the boundary, so this is an isolated system.

Step 2. Since $q = 0$ throughout, any process happening inside is also adiabatic — note that this is a separate label (about heat exchange) layered on top of "isolated" (about the boundary as a whole).

Step 3. Classify each quantity. Volume of the 2 mol of gas is extensive (it would double if the amount of gas doubled at the same conditions). Pressure is intensive (it does not depend on how much gas you have, only on $n$, $V$, $T$ together through the ideal gas law). Molar mass is intensive — it is a ratio, mass divided by moles, and ratios of two extensive quantities are always intensive. Total internal energy $U$ is extensive.

Step 4. Suppose the reaction takes the system from a state with internal energy $U_A$ to a state with internal energy $U_B$. Because $U$ is a state function, $\Delta U = U_B - U_A$ no matter what sequence of intermediate steps the reaction actually followed inside the vessel — this is the same guarantee proven above for $\Delta V$, now applied to $U$.

Answer: isolated (and therefore adiabatic) system; volume and internal energy are extensive; pressure and molar mass are intensive; $\Delta U = U_B - U_A$ regardless of the reaction's internal path.

Competitive stretch

If the same vessel instead had a small pinhole allowing gas to slowly leak out while staying thermally insulated, the classification changes: matter can now cross the boundary, so the system becomes open (matter and, indirectly, the energy it carries can leave), even though no heat crosses as heat. The lesson is to check matter and energy transfer separately, not to assume one blocked channel decides the whole classification.

• Writing "$\Delta q$" or "$\Delta w$" — heat and work are path functions; only state functions take a "$\Delta$."
• Assuming insulated automatically means isolated — insulation blocks heat only; if matter can still enter or leave, the system is open, not isolated.
• Confusing closed with isolated — a closed system can still exchange heat (a sealed can in ice water), it just cannot exchange matter.
• Calling temperature extensive because "more substance feels hotter" — temperature is intensive; mixing two samples at $300\ \text{K}$ gives $300\ \text{K}$, not $600\ \text{K}$.

Ask the tutor: If a described setup does not clearly say what can or cannot cross the boundary, ask the AI tutor to help you reason through the classification step by step.