1.2 Basic concepts and definitions in thermodynamics

Thermodynamics: Core Concepts

Systems and Surroundings

A thermodynamic system is whatever part of the universe you've chosen to study. It could be a gas in a cylinder, a solution in a beaker, or a biological cell. Everything outside the system that can interact with it is called the surroundings.

The boundary separating system from surroundings can be a real physical barrier (like the walls of a container) or an imaginary surface you define for convenience. How that boundary behaves determines the type of system:

• Open system: exchanges both matter and energy with surroundings (e.g., an open beaker of water that can lose vapor and heat)
• Closed system: exchanges energy but not matter (e.g., a sealed piston that can expand or compress)
• Isolated system: exchanges neither energy nor matter (e.g., an ideal thermos, or the universe as a whole)

State Variables and Equilibrium

State variables (also called state functions) describe the condition of a system at equilibrium. Their values depend only on the current state, not on how the system got there. The key state variables are pressure ($P$), volume ($V$), temperature ($T$), and amount of substance ($n$).

Thermodynamic equilibrium means the macroscopic properties of the system aren't changing with time. There's no net flow of energy or matter between system and surroundings. A system must simultaneously be in thermal, mechanical, and chemical equilibrium.

State variables fall into two categories:

• Intensive properties don't depend on system size: temperature, pressure, density, molar heat capacity
• Extensive properties scale with the amount of material: volume, mass, entropy, enthalpy, internal energy

An extensive property becomes intensive when you divide by the amount of substance. For example, volume ($V$, extensive) becomes molar volume ($V_m = V/n$, intensive).

First Law of Thermodynamics: Applications

Conservation of Energy

The first law says that energy is conserved. For a closed system, the change in internal energy equals the heat added to the system minus the work done by the system:

$\Delta U = Q - W$

This sign convention (sometimes called the "physics convention") treats $Q > 0$ when heat flows into the system and $W > 0$ when the system does work on the surroundings. Some textbooks use $\Delta U = Q + W$ with $W > 0$ for work done on the system (the "chemistry convention"). Check which your course uses.

• Heat ($Q$): energy transfer driven by a temperature difference, flowing from hotter to cooler regions. Measured in joules (J) or calories (cal).
• Work ($W$): energy transfer through a force acting over a distance. In thermodynamics, this most often means pressure-volume work (a gas expanding or compressing against an external pressure). Measured in joules (J) or liter-atmospheres (L·atm), where $1 \text{ L·atm} = 101.325 \text{ J}$.

Neither heat nor work is a state function. They depend on the path taken between two states, not just the endpoints.

Thermodynamic Processes

Each named process holds one variable constant, which simplifies the first law:

• Isothermal (constant $T$): For an ideal gas, $\Delta U = 0$ because internal energy depends only on temperature. Therefore $Q = W$. Example: slow expansion of a gas in thermal contact with a heat bath.
• Adiabatic ($Q = 0$): No heat exchange with surroundings. $\Delta U = -W$, so the temperature changes as work is done. Example: rapid compression in a diesel engine heats the gas enough to ignite fuel.
• Isobaric (constant $P$): The heat transferred equals the enthalpy change: $Q_P = \Delta H$. Example: boiling water in an open container at atmospheric pressure.
• Isochoric (constant $V$): No pressure-volume work is done ($W = 0$). $Q_V = \Delta U$. Example: heating a gas in a sealed rigid container.

Heat, Work, and Internal Energy: Interrelationships

Internal Energy

Internal energy ($U$) is the total kinetic and potential energy of all particles in the system. This includes translational, rotational, vibrational, and electronic contributions. Because $U$ is a state function, $\Delta U$ between two states is the same regardless of path.

Two important consequences:

• In an isolated system, $\Delta U = 0$ (no heat or work crosses the boundary).
• In a cyclic process (system returns to its initial state), $\Delta U = 0$, so $Q = W$ over the full cycle.

Heat and Work

Heat capacity ($C$) quantifies how much heat is needed to raise a system's temperature by one degree:

• Specific heat capacity ($c$): heat capacity per unit mass, in units of J/(g·K)
• Molar heat capacity ($C_m$): heat capacity per mole, in units of J/(mol·K)

For an ideal gas, $C_m$ differs depending on whether heating occurs at constant volume ($C_{V,m}$) or constant pressure ($C_{P,m}$), and they're related by $C_{P,m} - C_{V,m} = R$.

Pressure-volume work for a process against external pressure $P_{ext}$:

$W = \int P_{ext} \, dV$

For a reversible expansion or compression, $P_{ext}$ equals the system pressure at every instant. For an irreversible process against a constant external pressure, $W = P_{ext} \Delta V$. The reversible process always yields the maximum work for an expansion (or minimum work input for a compression).

Enthalpy

Enthalpy ($H$) is defined as:

$H = U + PV$

It's a state function, and its real usefulness is that at constant pressure, $\Delta H = Q_P$. This makes enthalpy the natural quantity for tracking heat flow in most chemical reactions (which typically occur at atmospheric pressure).

Key applications:

• Standard enthalpy of formation ($\Delta H_f^\circ$): the enthalpy change when one mole of a compound forms from its elements in their standard states (1 bar, 25°C by convention). By definition, $\Delta H_f^\circ = 0$ for elements in their standard states.
• Standard enthalpy of reaction: $\Delta H_{rxn}^\circ = \sum \Delta H_f^\circ (\text{products}) - \sum \Delta H_f^\circ (\text{reactants})$
• Hess's law: Because enthalpy is a state function, $\Delta H$ for a reaction is independent of the pathway. You can combine known reactions to find $\Delta H$ for an unknown reaction.

Ideal Gas Law: Applications

Ideal Gas Equation

The ideal gas law relates four state variables:

$PV = nRT$

where $R = 8.314 \text{ J/(mol·K)} = 0.08206 \text{ L·atm/(mol·K)}$ is the universal gas constant.

The model assumes gas particles have negligible volume, experience no intermolecular forces, and undergo perfectly elastic collisions. Real gases approximate ideal behavior at high temperatures and low pressures (where particles are far apart).

For mixtures, Dalton's law states that the total pressure is the sum of each component's partial pressure:

$P_{total} = P_1 + P_2 + \cdots + P_n$

Each partial pressure is $P_i = x_i P_{total}$, where $x_i$ is the mole fraction of component $i$.

Gas Laws

These are all special cases of the ideal gas law with certain variables held constant:

• Boyle's law (constant $T$, $n$): $P_1V_1 = P_2V_2$. Doubling the volume halves the pressure.
• Charles's law (constant $P$, $n$): $V_1/T_1 = V_2/T_2$. Volume scales linearly with absolute temperature.
• Gay-Lussac's law (constant $V$, $n$): $P_1/T_1 = P_2/T_2$. Pressure scales linearly with absolute temperature.
• Combined gas law (constant $n$): $P_1V_1/T_1 = P_2V_2/T_2$

Always use absolute temperature (Kelvin) in these equations. Using Celsius will give wrong answers.

Kinetic Molecular Theory

The kinetic molecular theory (KMT) provides the microscopic basis for ideal gas behavior. Its postulates:

1. Gas particles are in constant, random motion.
2. Particle volumes are negligible compared to the container volume.
3. Collisions between particles and with walls are perfectly elastic (kinetic energy is conserved).
4. There are no attractive or repulsive forces between particles.

From KMT, the average translational kinetic energy per molecule depends only on temperature:

$KE_{avg} = \frac{3}{2}kT$

where $k = 1.38 \times 10^{-23} \text{ J/K}$ is the Boltzmann constant. This means at a given temperature, all ideal gases have the same average kinetic energy regardless of molecular identity.

The root-mean-square speed is:

$v_{rms} = \sqrt{\frac{3RT}{M}}$

where $M$ is the molar mass. Lighter molecules move faster: at 300 K, $v_{rms}$ for $H_2$ ($M = 0.002$ kg/mol) is about 1930 m/s, while for $O_2$ ($M = 0.032$ kg/mol) it's about 483 m/s.

Entropy: Spontaneity of Processes

Entropy and the Second Law

Entropy ($S$) quantifies the number of microstates (microscopic arrangements) available to a system. More microstates means higher entropy.

The second law of thermodynamics states that for any spontaneous process, the total entropy of the universe (system + surroundings) increases:

$\Delta S_{universe} = \Delta S_{system} + \Delta S_{surroundings} > 0$

For a reversible process, $\Delta S_{universe} = 0$. All real processes are irreversible, so the universe's entropy always increases.

For a reversible process, the entropy change is calculated as:

$\Delta S = \int \frac{dQ_{rev}}{T}$

For an irreversible process between the same two states, $\Delta S$ of the system is the same (it's a state function), but $\Delta S_{universe}$ is greater than zero.

The third law of thermodynamics states that the entropy of a perfect crystalline substance approaches zero as $T \to 0 \text{ K}$. At absolute zero, a perfect crystal has exactly one microstate. This provides the reference point for tabulating absolute (third-law) entropies.

Gibbs Free Energy

The Gibbs free energy combines enthalpy and entropy into a single criterion for spontaneity at constant $T$ and $P$:

$G = H - TS$

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

The sign of $\Delta G$ tells you everything:

Spontaneity depends on the competition between $\Delta H$ and $T\Delta S$:

• Exothermic + entropy increase ($\Delta H < 0$, $\Delta S > 0$): spontaneous at all temperatures
• Endothermic + entropy decrease ($\Delta H > 0$, $\Delta S < 0$): non-spontaneous at all temperatures
• Mixed signs: temperature determines which term dominates. For example, ice melting is endothermic ($\Delta H > 0$) but increases entropy ($\Delta S > 0$), so it becomes spontaneous above 273 K where $T\Delta S > \Delta H$.

The standard Gibbs energy change connects to the equilibrium constant:

$\Delta G^\circ = -RT \ln K$

A reaction with $K > 1$ has $\Delta G^\circ < 0$ and favors products under standard conditions.

Statistical Interpretation of Entropy

The Boltzmann equation gives entropy a molecular meaning:

$S = k \ln \Omega$

where $\Omega$ is the number of microstates accessible to the system. A gas has far more microstates than a solid at the same temperature, which is why gases have higher entropy.

This equation provides the molecular basis for the second law: systems evolve toward the macrostate with the greatest number of microstates because that state is overwhelmingly more probable. The free expansion of a gas into a vacuum is spontaneous because the number of accessible position microstates increases enormously.

For systems where microstates have unequal probabilities, the more general Gibbs entropy formula applies:

$S = -k \sum_i p_i \ln p_i$

where $p_i$ is the probability of microstate $i$. This reduces to the Boltzmann equation when all microstates are equally probable ($p_i = 1/\Omega$). At 0 K, a perfect crystal has one microstate ($p_1 = 1$), giving $S = 0$, consistent with the third law.