System and Surrounding

Every thermodynamic system in a definite state can be fully described using just four variables: pressure $P$, volume $V$, temperature $T$, and the amount of substance $n$. This immediately splits every measurable quantity into two very different behavioural classes.

State functions are properties whose value is completely fixed by the current state of the system, and does not care in the slightest HOW that state was arrived at. Pressure, volume, temperature, internal energy $U$, enthalpy $H$ and free energy $G$ are all state functions. If a system moves from state A to state B, $\Delta(\text{state function})$ is exactly the same number no matter whether the journey went directly, via ten intermediate steps, or via some wildly circuitous reversible/irreversible mixture -- only the endpoints matter. …

Internal energy, symbol $U$, is a characteristic property of a system equal to the total energy possessed by every one of its constituent atoms, ions and molecules. Concretely, the energy of the whole collection of molecules making up a system is the sum of several distinct contributions:

where $U_t$ is translational energy (molecules moving through space), $U_v$ is vibrational energy (bonds stretching/bending), $U_r$ is rotational energy (molecules tumbling), $U_b$ is bond energy (energy stored in the chemical bonds themselves), $U_e$ is electronic energy, and $U_i$ is the energy arising from intermolecular interactions between neighbouring molecules.

In practice, thermodynamics never deals with the absolute value of $U$ -- there is no experimental way to measure it, and no need to. What is always measured and used is the change in internal energy, $\Delta U$, between two states.

Why internal energy matters. It is internal energy that physically distinguishes two different structural forms of the same substance. Graphite ($C_{graphite}$) and diamond ($C_{diamond}$) are both pure carbon, yet they differ because their atoms are bonded into different structures, giving them genuinely different internal energies.

Characteristics of $U$:

• It is an extensive property -- it scales with the amount of substance present; doubling the amount of material doubles $U$.
• It is a state function -- it depends only on the current state variables $(T,P,V,n)$, never on the path taken to reach that state.
• For any change of state, $\Delta U = U_f - U_i$. …

Heat ($q$). Heat is energy caught crossing the boundary that separates a system from its surroundings, driven by a temperature difference between the two; once that crossing has happened, the result shows up as a temperature change on one or both sides. Heat is fundamentally a path function -- how much of it flows depends on the route the process took. Its SI unit is the joule (J), though chemists also still commonly quote heat quantities in calories (cal); one calorie is defined as the heat required to raise the temperature of 1 gram of water by $1^\circ$C, in the vicinity of $15^\circ$C.

Sign convention for heat, symbol $q$: if heat flows INTO the system from the surroundings, the system's energy rises, so this is taken as positive, $+q$. If heat flows OUT of the system into the surroundings, the system's energy falls, so this is taken as negative, $-q$.

Work ($w$). Work is force ($F$) acting through a displacement ($x$): $-w=F\cdot x$ (7.1). The minus sign here is deliberate -- it flags that the work described is being done BY the system, at the cost of some of its own internal energy. Like heat, work is a path function; it only appears at the system's boundary, and only while the state is actually changing (surroundings, in thermodynamics, are assumed so vast that a system's actions never produce any measurable macroscopic change in them). Its SI unit is likewise the joule ($1\ \text{J} = 1\ \text{N}\!\cdot\!\text{m}$); for larger quantities the kilojoule (1 kJ = 1000 J) is used.

Sign convention for work, symbol $w$: if the system does the work (expanding, pushing against its surroundings), the system's own energy falls, so this is taken as negative, $-w$. If work is done ON the system (something compresses it), the system's energy rises, so this is taken as positive, $+w$.

Pressure-volume work. In elementary chemical thermodynamics the type of work almost always considered is the work of expanding or compressing a gas -- PV work, or expansion work. Picture a cylinder holding $n$ moles of ideal gas at internal pressure $P_{int}$ and volume $V_i$, closed by a frictionless piston of cross-sectional area $A$. Whenever the external pressure $P_{ext}$ differs from $P_{int}$, the piston moves until the two equalise.

Single-step (irreversible) compression, where $P_{ext}>P_{int}$ and the piston moves inward in one abrupt step to a final volume $V_f$: work is done ON the system (so $+w$). Combining $w=-F\Delta x$ (7.2) with $F=P_{ext}A$ (7.3) gives $w=-P_{ext}A\Delta x$; since $A\Delta x$ is exactly the volume change, $w=-P_{ext}(V_f-V_i)$ (7.4), which for a compression (where $V_f<V_i$, making $-\Delta V$ positive) simplifies to $w=P_{ext}\Delta V$ (7.5) -- a positive quantity, consistent with work being done on the system. …