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Chapter 13. Thermodynamics

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1. System and Surroundings

2. Thermodynamics First Law

3. Thermodynamics Second Law

4. Thermodynamics Third Law

5. Gibbs Free Energy

a. Advanced Thermodynamics

1. System and Surroundings

⑴ Definition of System

① System: The subject of interest, reactants and products in chemical reactions

② Surroundings: Everything else except the system

③ Universe: Refers collectively to the system and surroundings

⑵ Classification of Systems

① Open system: Matter and energy can enter and exit

② Closed system: Only energy can enter and exit

③ Isolated system: No interaction with the surroundings

⑶ Conditions of a System

① Temperature

○ Celsius temperature: Temperature scale where the freezing point of water is set at 0 ℃ and the boiling point at 100 ℃ under normal atmospheric pressure

○ Fahrenheit temperature: Temperature scale where the freezing point of water is set at 32 ℉ and the boiling point at 212 ℉ under normal atmospheric pressure

○ Absolute temperature: Temperature scale where the freezing point of water is set at 273.15 K and the boiling point at 373.15 K under normal atmospheric pressure

○ Rankine temperature: Temperature scale where the freezing point of water is set at 0 ℉ and the boiling point at 180 ℉ under normal atmospheric pressure

② Pressure

③ STP (standard temperature and pressure): 0 ℃, 1 atm

○ Volume of 1 mole of gas = 22.4 L

④ SATP (standard ambient temperature and pressure): 25 ℃, 1 bar

○ Volume of 1 mole of gas = 24.79 L

⑤ NTP (normal temperature and pressure): 20 ℃, 1 atm

⑥ ATP (actual temperature and pressure): Actual temperature and pressure

⑦ Measurement of Temperature and Pressure

○ Bayard-Alpert pressure gauge: Measures current by ionizing gas, used for pressure measurement under low pressure conditions

○ Capacitance manometer: Used for producing high-quality tires

⑶ Thermodynamics

① The science that describes energy, spontaneity, and more about reactants and products

② Laws of Thermodynamics

○ First Law of Thermodynamics: Law of conservation of energy, internal energy of an isolated system remains constant

○ First kind of perpetual motion machine: A machine that creates energy from nothing

○ Second Law of Thermodynamics: Law of increasing entropy, predicts spontaneity and direction of reactions

○ Second kind of perpetual motion machine: A machine that performs energy conversions that are impossible

○ Third Law of Thermodynamics: At absolute zero, the entropy of a perfectly crystalline substance is zero

○ Zeroth Law of Thermodynamics: Definition of equality of temperatures

○ Necessity of the law: There is no definition of temperature; it’s merely a relative scale that indicates the direction of energy flow

○ Contents: In thermal equilibrium, T_A = T_B, T_B = T_C → T_A = T_C

⑷ State Functions and Path Functions

① State function: A physical quantity related only to the present state

○ Example: Internal energy

② Path function: A physical quantity influenced by the process of change in state

○ Example: Heat, work

2. Thermodynamics First Law

⑴ Energy

① Work = Force × Distance

② Energy: The ability to do work

⑵ Internal Energy (e): The total kinetic energy of all particles composing the system

① Internal energy of a gas is a state function with respect to temperature

○ Demonstrating that internal energy is a function of temperature link

○ Reason for internal energy being a state function: Kinetic energy is a state function

② Energy equipartition law

○ Definition: Each degree of freedom of a gas molecule has an average energy of ½ kBT

○ Degrees of freedom: The number of independent ways a gas molecule can have energy

○ Translational energy

○ Rotational energy

○ Vibrational energy

③ Monatomic Gas

○ Has 3 degrees of freedom: Translational energy in x, y, z axes

○ Average kinetic energy of a single molecule in a monoatomic gas

④ Diatomic Gas

○ Has 5 degrees of freedom: Translational energy in x, y, z axes; Rotational energy in x, y axes

○ Average kinetic energy of a single molecule in a diatomic gas

⑤ Angular Triatomic Gas

○ Has 6 degrees of freedom: Translational energy in x, y, z axes; Rotational energy in x, y, z axes

○ Average kinetic energy of a single molecule in an angular triatomic gas

⑥ Polyatomic Gas

○ Has 7 degrees of freedom: Translational energy in x, y, z axes; Rotational energy in x, y, z axes; Vibrational energy

○ Average kinetic energy of a single molecule in a polyatomic gas

○ Variation in molar heat capacities with temperature for polyatomic gases

Figure 1. Variation in molar heat capacities with temperature for polyatomic gases

⑦ Dulong and Petit’s Law

○ Assumes atoms in a solid vibrate harmonically around their equilibrium positions

○ Holds only in classical statistical mechanics, not in quantum statistical mechanics

⑶ Work (w): The energy transferred to the surroundings when a gas expands

① Definition: Energy with a direction

○ In physics, work done on the surroundings is defined as positive; in chemistry, work done on the surroundings is defined as negative. Here, the latter convention is adopted

○ Physics is more interested in the interaction between the system and surroundings, while chemistry focuses more on the system itself

② Isothermal Reversible Expansion

③ Constant External Pressure

④ Free Expansion: External pressure is 0, thus w = 0

⑷ Heat (q)

① Definition: Energy without a direction

② First Law of Thermodynamics: Relationship between internal energy, work, and heat

○ Note that q = q in.

③ Terms related to Heat

○ Exothermic reaction

○ Endothermic reaction

○ Adiabatic wall

○ Diathermic wall

④ Heat Capacity

○ Heat: Quantity of energy in heat. Q (unit: cal, J, etc.)

○ Heat capacity: Amount of heat required to raise the temperature by one unit. C (unit: cal/℃, J/K, etc.)

○ Specific heat capacity: Heat capacity per unit mass. C_s = C/m (unit: kJ/kg·℃)

○ Molar heat capacity: Heat capacity per mole. C_m = C/n (unit: kJ/mol·℃)

○ Molar constant-volume heat capacity: Heat transferred at constant volume per 1 ℃

○ Molar constant-pressure heat capacity: Heat transferred at constant pressure per 1 ℃

○ Relationship between constant-volume and constant-pressure heat capacities

○ Molar heat capacity of a gas: For temperature T (unit: K) and heat capacity Cp (unit: cal/g-mol·K)

Table 1. Molar heat capacities of gases (Source: Smith and Van Ness)

○ (Note) The reason for the unit being cal/g-mol·K is to calculate based on the total mass of fuel + air as 1 g

⑤ Calorimeters

○ Bomb calorimeter (constant-volume calorimeter): Measures heat by maintaining constant volume, setting work to 0, and measuring heat

○ Simple calorimeter: Measures heat by maintaining constant pressure

○ Flow calorimeter: Measures steam density during flow process at constant enthalpy

⑸ Enthalpy

① Overview

○ Definition: Internal energy with added concept of work

○ Enthalpy can be defined even when pressure is not constant

○ Enthalpy is equal to the heat transferred at constant pressure

○ Enthalpy is equal to the heat transferred without expansion

○ Originally thought enthalpy was only defined under constant pressure, but this understanding changes after looking at the following equation

② Characteristics

○ State function: Enthalpy is only a function of temperature. It remains constant if the initial and final temperatures are the same

○ Enthalpy has no absolute value, so the change in enthalpy (ΔH) is important

○ If a reaction proceeds in reverse, the sign of ΔH is reversed

○ Magnitude property: Proportional to the quantity of substance

○ Measured with simple calorimeter at constant pressure: C_P is the heat capacity at constant pressure, but dh = mC_PdT holds even when pressure is not constant

③ Reaction Enthalpy

○ Formation enthalpy (ΔH_f): Enthalpy change when 1 mole of a substance is formed from stable elements

○ Formula 1. Formation enthalpy of a diatomic gas (e.g., O₂(g)) = 0

○ Formula 2. Formation enthalpy of a metallic crystal (e.g., Mg(s)) = 0

○ Decomposition enthalpy (-ΔH_f): Enthalpy change when 1 mole of a substance is decomposed into stable elements

○ Bond dissociation enthalpy (ΔH_D): Enthalpy change when a substance is decomposed into gaseous atoms that make up the substance

○ Reaction where bonds are broken is endothermic

○ Combustion enthalpy (ΔH_C): Enthalpy change when 1 mole of a substance is completely combusted

○ Solvation enthalpy

○ ΔH_sol = ΔH_lattice + ΔH_hydration

○ ΔH_sol: MN(s) → M⁺(aq) + N^-(aq)

○ ΔH_lattice (＞ 0): MN(s) → M⁺(g) + N^-(g)

○ Absolute value of lattice enthalpy is larger for smaller metal ions and larger charges on individual ions

○ ΔH_hydration (＜ 0): M⁺(g) + N^-(g) → M⁺(aq) + N^-(aq)

○ Lattice energy: Refers to the enthalpy change when each constituent element is split into gaseous monoatomic ions.

○ Example 1. Lattice energy of NaF(s) ΔH ⇔ NaF(s) → Na⁺(g) + F^-(g), ΔH

○ Example 2. Lattice energy of Na₂O(s) ΔH ⇔ Na₂O(s) → 2Na⁺(g) + O^2-(g), ΔH

○ Standard formation enthalpy

○ Standard bond dissociation enthalpy

○ Standard combustion enthalpy

④ Born-Haber Cycle

○ Born-Haber Cycle: A hypothetical reaction pathway used to explain the formation of ionic compounds.

○ Hess’s Law: Since enthalpy is a state function, if the reactants and products are the same, the ΔH is the same regardless of the pathway taken.

⑤ Other

○ Slope on the heating curve

○ Solid: Since intermolecular forces between molecules are almost not broken, heat is transferred as kinetic energy of motion.

○ Liquid: Some energy is used to break intermolecular forces between molecules. The slope is smallest.

○ Gas: No intermolecular forces between molecules. Heat is transferred as kinetic energy of motion. The slope is largest.

○ Relationship with internal energy

○ Gas reactions: If there is no change in the gas molar ratio between reactants and products, ΔH = ΔU.

○ Reactions in liquids and solids can be considered at constant pressure, and work is almost negligible, so ΔH = ΔU.

3. Second Law of Thermodynamics

⑴ Overview

① Entropy: A state function that represents disorder.

② Reversible process: A process with a total entropy change of 0.

③ The second law of thermodynamics can be expressed in four different ways.

⑵ Expression 1

① Spontaneous change: Change that occurs without external influence.

② Law: Spontaneous changes occur in the direction of increasing the universe’s entropy.

③ Surroundings’ entropy ΔS_surroundings: Independent of whether the reaction is reversible or irreversible.

○ The application of the second law of thermodynamics is possible only when an isolated system is formed by the system of interest and its surroundings.

○ Calculation of ΔS_surroundings: Independent of reversible · irreversible reactions. Assumes constant temperature and pressure in the surroundings.

○ Although ΔS_surroundings is not as simple as described above, it is important to note that it is composed of values derivable from the system.

④ Conclusion: If ΔS_total > 0, spontaneous; if ΔS_total < 0, non-spontaneous.

⑶ Expression 2: Clausius Inequality

① Clausius’s expression: Heat flows from hot to cold. It cannot spontaneously flow from cold to hot.

② Formulation (with equality condition for reversible processes)

③ Proof

○ To do work on the system, the external pressure must be increased along with the rise in external pressure.

○ Therefore, the reversible condition does more work on the surroundings than the irreversible condition.

○ If the initial and final states of reversible and irreversible processes are the same (applying physical representation),

○ Hence, the following equation is obtained.

④ Derivation of the Second Law of Thermodynamics: If the system is isolated from the surroundings, dq = 0, so dS ≥ 0.

⑤ Free Expansion

○ Free expansion and isothermal expansion have the same entropy change intuitively.

○ In free expansion, there is no heat exchange.

○ In free expansion, since it’s not a reversible process, the above formula cannot be applied.

⑷ Expression 3: Carnot Engine

① Kelvin-Planck’s expression: It’s impossible to create a heat engine that converts all absorbed heat into work.

② In other words, an engine with 100% efficiency cannot be created.

③ Carnot engine efficiency is 70%: With 100 units of heat, only 70 units can be converted into work.

⑸ Expression 4: Statistical Mechanical Definition

① Defined by Boltzmann’s equation, also known as Boltzmann entropy.

② State number W: Number of ways the molecules of the system can be arranged while keeping the total energy constant.

○ Positional probability: The greater the disorder of molecular arrangement, the more cases there are in that direction for the reaction to proceed.

○ Molecular motion states: More cases for molecular arrangement occur at different energy levels, preferred at higher temperatures → Entropy increases with higher temperatures.

○ Absolute value of entropy and definition of entropy at 0 can be defined → Third Law of Thermodynamics.

③ Equivalence of Thermodynamic and Statistical Mechanical Definitions

○ Assume that the microscopic state number given to a single molecule is proportional to its volume.

○ For N molecules, the total state number is represented as follows

○ The statistical entropy during isothermal expansion of an ideal gas from V₁ to V₂ is as follows:

○ Ultimately, the definition of statistical entropy is equivalent to the definition of thermodynamic entropy.

○ It’s conjectured that Boltzmann performed reverse calculations to extract profound meaning from thermodynamic definition.

④ Classical Statistical Mechanics and Quantum Statistical Mechanics

○ Classical Statistical Mechanics: Initiated by Boltzmann’s thermodynamic concepts.

○ Quantum Statistical Mechanics: Characterized by quantum mechanical concepts such as orbital theory.

⑹ Application of Entropy

① Application 1: General Thermodynamic Process

② Application 2: Predicting Entropy Change in Gas Reactions

○ Reaction entropy ΔS: In reactions where the number of gas molecules increases, ΔS > 0.

③ Application 3: Expansion in Vacuum

○ ΔT = 0, W = 0 → Q = 0

○ Thermodynamic definition doesn’t apply in this situation.

○ Approaching the problem with the concept of state number from statistical mechanics is similar to isothermal expansion.

④ Application 4: Entropy of Phase Transition

○ Under constant pressure, phase transition temperature T_transition is constant, and Q_p = ΔH.

○ Standard molar entropy of phase transition ΔS°: Entropy change during transition at standard conditions (1 bar).

○ Trouton’s rule: Most liquids have a standard vaporization entropy of about 85 J/K.

○ Water has a relatively large ΔS°_vap (∵ liquid water has low entropy).

Table 2. Standard Vaporization Entropy of Liquids at Normal Boiling Points

⑤ Application 5: The reason time flows in one direction is related to the second law of thermodynamics.

4. Third Law of Thermodynamics

⑴ Nernst’s Heat Theorem

① Expression 1: Entropy change associated with physical or chemical changes approaches 0 as temperature approaches absolute zero.

② Expression 2: The entropy of all perfect crystalline substances is 0 at T = 0.

⑵ Residual Entropy

① The entropy actually measured at T = 0.

② Example 1: Residual entropy of CO: Random orientation allows relatively free motion, increasing residual entropy.

○ Considering two CO molecules, CO ··· CO and CO ··· OC are two possible arrangements. According to Boltzmann’s statistical definition, the absolute entropy is k ln (2×2) = k ln 4. When expanded to 1 mole (6.02 × 10²³ molecules), the predicted residual entropy is S = N_Akln2 = 5.76 J/K. The actual measured value is 4.6 J/K, which is quite similar.

③ Example 2: Residual entropy of HCl: Only one type of arrangement is possible, making the residual entropy close to 0.

⑶ Standard Molar Entropy

① Absolute zero: Defines the entropy of a pure solid crystal as 0 at 0 K.

② Calculation of standard molar entropy of substances

③ Characteristic 1: S_m(s) < S_m(ℓ) < S_m(g): Entropy becomes more disordered as it goes from solid to gas.

④ Characteristic 2: Molar entropy of heavy elements is higher than that of light elements.

⑤ Characteristic 3: Larger and more complex chemical species have higher molar entropy than smaller and simpler ones.

⑥ Standard Molar Entropy at 25°C

Table 3. Standard Molar Entropy of Substances at 25°C (J/K · mol)

⑷ Standard Reaction Entropy

5. Gibbs Free Energy (G)

⑴ Definition: A state function that indicates the spontaneity of a reaction.

① Applying the second law of thermodynamics requires considering both the system and the surroundings, which is cumbersome.

② Gibbs free energy requires only the properties of the system to judge the spontaneity of the reaction, making it easier.

○ (Note) If properties of the system and surroundings are considered separately, the calculation process becomes more complicated.

③ Meaning 1: ΔG < 0 indicates a spontaneous reaction, ΔG > 0 indicates a non-spontaneous reaction, ΔG = 0 indicates equilibrium.

④ Meaning 2: For a constant temperature and pressure, the maximum reversible work obtainable from a reaction.

⑤ Limitation: The change in the surroundings’ entropy is not that simple.

⑵ Proof

Since at constant temperature and pressure, dq = dH and dq_surroundings = -dH, the equation changes as follows.

Multiplying both sides by -T yields the Gibbs free energy equation.

Since spontaneous reactions have dS_total > 0,

⑶ Physical Interpretation: ΔG = Maximum non-expansion work

① Electrochemistry and Gibbs Free Energy (where E = reduction potential)

⑷ Relationship between ΔG and Reaction Rate

① Formula

○ The above equation is valid only for isothermal processes: However, to go beyond this, the van ‘t Hoff equation is used.

○ ΔG° represents the change in free energy under standard conditions, but there are cases where an arbitrary reference state is used to solve problems.

○ When ΔG = 0, the reaction quotient Q can be replaced by the equilibrium constant K.

② Proof

○ If the number of moles is n and the Gibbs free energy per mole is G_m, then ΔG is defined as follows:

○ Example of gases (for isothermal processes)

○ The change in free energy is equivalent to the maximum non-expansion work.

○ If the initial state is the standard state (i.e., P₁ = 1), it is as follows:

○ Dividing both sides by n gives:

○ Since free energy has no absolute value, it is calculated as a change from an arbitrary reference point.

○ Example of solution (for isothermal processes)

○ According to the osmotic pressure formula, pressure and concentration have the following relationship (where C is molarity):

○ Thus, the following relational expression is obtained:

○ Finally, the following conclusion is obtained:

○ The concept of activity can unify cases of gases and solutions.

○ Applying it to chemical reactions yields:

③ Interpretation

○ Relationship between free energy change ΔG_r at any point and standard free energy change ΔG_r°

○ Standard formation free energy ΔG_r°: Change in free energy when one mole of product is formed under standard conditions

○ ΔG_r°＜0: Forward reaction is spontaneous under standard conditions, K＞1

○ ΔG_r°＞0: Reverse reaction is spontaneous under standard conditions, K＜1

○ ΔG_r° = 0: Equilibrium is reached under standard conditions, K = 1

○ Relationship between reaction quotient Q and equilibrium constant K

○ Q＞K: Reverse reaction proceeds

○ Q = K: Equilibrium

○ Q＜K: Forward reaction proceeds

○ Reaction path - Free energy curve

Figure 2. Reaction path - Free energy curve

○ The slope of the reaction path - free energy curve is ΔG_r

○ At the initial stage of the reaction where there are no products, the forward reaction proceeds, so ΔG_r ＜ 0

○ As the reaction progresses, G_r increases (∵ Q increases) and reaches equilibrium where ΔG_r = 0 (forward reaction = reverse reaction)

○ Equilibrium point is where the slope is zero

④ Application: van’t Hoff equation

○ Introduction: To go beyond isothermal process constraints, linear approximation is used to estimate changes in free energy

○ Assumption: ΔH° and ΔS° are constant with respect to temperature

○ Formula

○ Point. In the above equation, it is written as TΔS instead of Δ(TS), reflecting some constraints of the isothermal process

○ Example

⑸ Relationship between free energy change ΔG and reaction rate

① Brønsted-Evans-Polanyi (BEP) principle: Free activation energy is proportional to the free energy of the reaction

② The above principle seems to be an empirical rule, like the Hammond postulate.

Input: 2019-01-09 13:33

Modified: 2023-01-05 00:35