Thermodynamics

1. Introduction to Thermodynamics

Thermodynamics is the study of energy transformations. It deals with energy changes of macroscopic systems and is concerned with the initial and final states of a system in equilibrium, not the rate of change.

2. Thermodynamic Terms

System and Surroundings

A system is the part of the universe under observation. The surroundings are the rest of the universe that can interact with the system.

Universe = System + Surroundings

Types of Systems

Open System: Exchanges both energy and matter.
Closed System: Exchanges energy but not matter.
Isolated System: Exchanges neither energy nor matter.

State of the System

The state of a system is described by its macroscopic properties. State functions/variables (e.g., Pressure, Volume, Temperature) depend only on the current state, not how it was reached.

3. Internal Energy (U)

Internal energy (U) represents the total energy of the system. It is a state function and can change through heat (q) or work (w).

An adiabatic process is one with no heat transfer (q = 0). For such a process, the change in internal energy is equal to the adiabatic work:

∆U = w_ad

Work (w) Sign Convention

Positive (w > 0): Work done ON the system.

Negative (w < 0): Work done BY the system.

Heat (q) Sign Convention

Positive (q > 0): Heat transferred TO the system.

Negative (q < 0): Heat transferred FROM the system.

First Law of Thermodynamics

The change in internal energy is the sum of heat and work. This is the law of conservation of energy.

∆U = q + w

For an isolated system, q=0 and w=0, so ∆U = 0. The energy of an isolated system is constant.

4. Applications - Work

For pressure-volume work, such as gas in a cylinder, the work done is:

w = -p_ex ∆V

A reversible process proceeds infinitely slowly through equilibrium states. For an isothermal reversible expansion/compression of an ideal gas:

w_rev = -nRT ln(V_f / V_i)

Free expansion is expansion into a vacuum (p_ex = 0), so no work is done (w=0).

5. Enthalpy (H)

Enthalpy (H), often called heat content, is a state function defined as:

H = U + pV

The change in enthalpy (∆H) is equal to the heat absorbed or released at constant pressure (q_p):

∆H = q_p

Exothermic reactions: Release heat, ∆H is negative.
Endothermic reactions: Absorb heat, ∆H is positive.

The relationship between enthalpy and internal energy is:

∆H = ∆U + ∆n_gRT

where ∆n_g is the change in moles of gas.

6. Heat Capacity, Extensive & Intensive Properties

Heat Capacity (C)

The heat required to raise a system's temperature is determined by its heat capacity.

q = C ∆T

Extensive and Intensive Properties

Extensive Properties: Depend on the amount of matter (e.g., mass, volume, enthalpy).
Intensive Properties: Do not depend on the amount of matter (e.g., temperature, density, pressure).

For an ideal gas, the relationship between molar heat capacity at constant pressure (C_p) and constant volume (C_V) is:

C_p - C_V = R

7. Measurement of ∆U and ∆H: Calorimetry

Calorimetry is the science of measuring heat changes.

∆U is measured at constant volume in a bomb calorimeter.
∆H is measured at constant pressure in a simpler calorimeter open to the atmosphere.

8. Enthalpy Changes in Reactions

The Standard Enthalpy of Reaction (∆_rH⁰) is the enthalpy change when a reaction occurs with all substances in their standard states (pure form at 1 bar).

Standard Enthalpy of Formation (∆_fH⁰) is the enthalpy change when one mole of a compound is formed from its elements in their most stable reference states. By convention, ∆_fH⁰ for an element in its reference state is zero.

∆_rH⁰ = Σ ∆_fH⁰(products) - Σ ∆_fH⁰(reactants)

Hess’s Law of Constant Heat Summation

Since enthalpy is a state function, the total enthalpy change for a reaction is the same whether it occurs in one step or multiple steps. This allows for the calculation of enthalpy changes for reactions that are difficult to measure directly.

Other Enthalpy Types

Enthalpy of Combustion (∆_cH⁰): For complete combustion of 1 mole of a substance.
Enthalpy of Atomization (∆_aH⁰): To break 1 mole of bonds to get gaseous atoms.
Bond Enthalpy: Energy to break 1 mole of covalent bonds in the gas phase.
Lattice Enthalpy (∆_latticeH⁰): To separate 1 mole of an ionic compound into gaseous ions. Determined indirectly via a Born-Haber cycle.
Enthalpy of Solution (∆_solH⁰): When 1 mole of a substance dissolves in a solvent.

9. Spontaneity

A spontaneous process has the potential to proceed without external assistance. It is not necessarily fast. A decrease in enthalpy (exothermic reaction) favors spontaneity, but it is not the only factor.

Entropy (S) and the Second Law

Entropy (S) is a measure of the randomness or disorder of a system. It is a state function. The gaseous state has the highest entropy, and a crystalline solid has the lowest.

                Second Law of Thermodynamics: For a spontaneous process, the total entropy of the system and its surroundings always increases. 

                ∆Stotal = ∆Ssystem + ∆Ssurroundings > 0

Gibbs Energy (G)

To combine the effects of enthalpy and entropy, we use the Gibbs energy (G), also a state function.

G = H - TS

The change in Gibbs energy, ∆G, determines spontaneity at constant temperature and pressure:

∆G = ∆H - T∆S

If ∆G < 0, the process is spontaneous.
If ∆G > 0, the process is non-spontaneous.
If ∆G = 0, the system is at equilibrium.

Gibbs Energy and Equilibrium

The standard Gibbs energy change (∆_rG⁰) is related to the equilibrium constant (K) by the equation:

∆_rG⁰ = -RT ln K = -2.303 RT log K

This equation links the thermodynamic favorability of a reaction to the extent to which it proceeds.