6.1 Laws of thermodynamics and state functions

Thermodynamics governs energy changes in chemical reactions. The laws of thermodynamics explain how energy is conserved, how entropy drives spontaneous change, and how these factors combine to predict whether a reaction will proceed on its own. This section covers the three laws, the distinction between state and path functions, and the key relationships among internal energy, enthalpy, entropy, and Gibbs free energy.

Laws of Thermodynamics

Laws of thermodynamics

First Law of Thermodynamics — Energy cannot be created or destroyed, only converted between forms (kinetic, potential, thermal, electrical, etc.). Mathematically, the change in internal energy of a system equals the heat added to the system minus the work done by the system:

$\Delta U = q - w$

This means the total energy of the universe (system + surroundings) stays constant during any process. If a system loses energy, the surroundings gain exactly that amount.

Second Law of Thermodynamics — The total entropy of the universe always increases during a spontaneous (irreversible) process. Heat flows spontaneously from hot objects to cold objects, never the reverse. A hot cup of coffee cools down in a cold room; the room never spontaneously heats the coffee further. This law also means no process can convert heat entirely into useful work; some energy is always "lost" to entropy increase.

Third Law of Thermodynamics — The entropy of a perfect crystal at absolute zero (0 K, or −273.15 °C) is exactly zero. At that temperature, particles are locked in a single arrangement with only one microstate. The Third Law also implies that reaching absolute zero is impossible in a finite number of steps; you can get closer and closer, but never quite there. This gives us a defined reference point for measuring absolute entropy values.

State vs. path functions

A state function depends only on the current state of the system (described by variables like pressure, volume, and temperature), not on how the system got there. Think of elevation: whether you hike up a mountain by a switchback trail or climb straight up a cliff, your change in elevation is the same.

State functions include:

• Internal energy ($U$)
• Enthalpy ($H$)
• Entropy ($S$)
• Gibbs free energy ($G$)

Their changes ($\Delta U$, $\Delta H$, $\Delta S$, $\Delta G$) are the same regardless of the path between initial and final states.

A path function depends on how the process occurs. The same start and end points can involve very different amounts of heat or work depending on the route taken.

Path functions include:

• Heat ($q$)
• Work ($w$)

Thermodynamic Processes and State Functions

Internal energy in thermodynamic processes

Internal energy ($U$) is the total energy stored in a system. It includes the kinetic energy of particle motion (translational, rotational, vibrational) and the potential energy from intermolecular forces. You can't measure $U$ directly, but you can measure changes in it:

$\Delta U = q - w$

Heat ($q$) is energy transferred between a system and surroundings because of a temperature difference.

• Positive $q$: heat flows into the system (e.g., melting ice absorbs heat)
• Negative $q$: heat flows out of the system (e.g., freezing water releases heat)

Work ($w$) is energy transferred when a force acts over a distance. In chemistry, this is most often pressure-volume (expansion/compression) work.

• Positive $w$: work done by the system on the surroundings (expansion)
• Negative $w$: work done on the system by the surroundings (compression)

Be careful here: some textbooks define the First Law as $\Delta U = q + w$ with the opposite sign convention for work (positive when done on the system). Check which convention your course uses and stick with it consistently.

Solving First Law problems:

1. Identify the system (e.g., gas in a cylinder) and surroundings (e.g., the atmosphere).
2. Determine the initial and final states (e.g., from 1 atm / 300 K to 2 atm / 400 K).
3. Determine the signs of $q$ and $w$ based on the direction of energy flow.
4. Plug values into $\Delta U = q - w$ and solve for the unknown quantity.

Relationships of thermodynamic state functions

Enthalpy ($H$) measures the heat content of a system at constant pressure. For any process at constant pressure:

$\Delta H = q_p$

• Negative $\Delta H$: exothermic (releases heat). Example: combustion of methane.
• Positive $\Delta H$: endothermic (absorbs heat). Example: dissolving ammonium nitrate in water.

Enthalpy is especially useful because most lab reactions happen at constant atmospheric pressure, so $\Delta H$ directly equals the heat you'd measure with a calorimeter under those conditions.

Entropy ($S$) quantifies the number of ways energy and matter can be arranged in a system (microstates). More microstates means higher entropy. For a reversible process:

$\Delta S = \frac{q_{rev}}{T}$

Entropy increases when gases expand, solids dissolve, or temperature rises. The Second Law requires that $\Delta S_{universe} > 0$ for any spontaneous process.

Gibbs free energy ($G$) combines enthalpy and entropy into a single criterion for spontaneity at constant temperature and pressure:

$\Delta G = \Delta H - T\Delta S$

• $\Delta G < 0$: spontaneous (e.g., iron rusting)
• $\Delta G > 0$: non-spontaneous (e.g., electrolysis of water requires energy input)
• $\Delta G = 0$: system is at equilibrium

How $\Delta H$, $\Delta S$, and temperature interact:

For the temperature-dependent cases (rows 3 and 4), the crossover temperature where $\Delta G = 0$ is:

$T = \frac{\Delta H}{\Delta S}$

Above or below that temperature, one term dominates and determines the sign of $\Delta G$. This is why water freezes below 0 °C (enthalpy wins) but evaporates above 100 °C at 1 atm (entropy wins).