Principles of Chemical Thermodynamics

Chemical Thermodynamics: Principles and Applications

Chemical Thermodynamics is the study of energy transformations, particularly the conversion of thermal energy into mechanical or chemical energy and vice versa. In chemistry, it is used to predict the evolution and spontaneity of chemical reactions. This field is built upon three fundamental principles derived from experimental data.

I. Definitions and Conventions

Thermodynamics involves the study of a system, which is the part of the universe under consideration, and the surroundings, which is everything else. The boundary between the system and its surroundings can be real or imaginary.

1. System and Surroundings

• Open System: Exchanges both matter and energy with the surroundings.

• Closed System: Exchanges only energy (not matter) with the surroundings. This note primarily focuses on closed systems.

• Isolated System: Exchanges neither matter nor energy with the surroundings.

2. Different Types of Energy and Sign Conventions

• Calorific Energy (Heat): Denoted as .

• Mechanical Energy (Work): Denoted as .

The standard unit for energy in the MKSA system is the Joule (J), where . An older unit for calorific energy is the calorie (cal), with the conversion .

Sign Convention: Energy received by the system is counted as positive (), and energy given off by the system is counted as negative ().

3. State of a System

A. State Variables

State variables are properties that define the state of a system (e.g., pressure , temperature , volume , concentration ).

• Extensive Variables: Depend on the amount of matter in the system (e.g., volume, mass).

• Intensive Variables: Independent of the amount of matter (e.g., temperature, viscosity, concentration).

B. Equilibrium State

A system is in thermodynamic equilibrium when its state variables are uniform throughout the system and do not change over time. This implies simultaneous thermal, mechanical, and chemical equilibrium.

C. Reversible or Irreversible Transformation

A transformation occurs when one or more state variables of an equilibrium system change, leading to a new equilibrium state.

• Reversible Transformation: A process where the system passes through a continuous series of equilibrium states, occurring infinitesimally slowly. It is an idealized process, practically unachievable.

• Irreversible Transformation: Any transformation that is not reversible, occurring at a finite rate.

D. State Function

A function of state variables (e.g., ) is a state function if its value depends only on the current values of the state variables, and not on the pathway taken to reach that state. Thus, its change between two states is independent of the process used.

For an infinitesimal transformation, the change of a state function is an exact differential () with the following properties:

(partial derivative of with respect to , with constant)

Work Example (Piston):

Consider a gas in a piston at constant temperature . For an infinitesimal displacement of the piston, the elementary work is:

Force

. Since (elementary volume), then .

• If : The gas expands, doing work on the surroundings. Conventionally, , so .

• If : The system is compressed, receiving work from the surroundings. Conventionally, , so .

For a macroscopic transformation from state 1 to state 2:

Isothermal Expansion Example (Ideal Gas):

If the external pressure varies slowly and is always equal to the internal pressure (reversible process and ideal gas at constant ):

Irreversible Expansion Example (Ideal Gas):

If the gas expands against a constant external pressure (e.g., ):

Important Note: Work () is not a state function because its value depends on the path taken between states.

For solids and liquids, is very small, so work is generally negligible.

E. Heat Exchanges with the Exterior

For any reversible transformation of a pure substance defined by and :

Where is the elementary heat exchanged, is the heat capacity at constant pressure, and is the latent heat.

If the system is defined by and :

Where is the heat capacity at constant volume, and is the heat of expansion.

An adiabatic transformation occurs when (no heat exchange).

II. First Principle of Thermodynamics

1. Statement

The First Principle embodies the conservation of energy, stating that energy cannot be created or destroyed. For any system undergoing a transformation, the total energy exchanged with the surroundings (sum of mechanical work and heat ) is related to the change in the system's internal energy .

2. Internal Energy

The variation in internal energy () from an initial state I to a final state F is given by:

is a state function, meaning depends only on the initial and final states, not on the path taken. For an isolated system (no exchange of work or heat), , so the internal energy of an isolated system remains constant.

3. Differential Expression

For infinitesimal changes, the first principle is written as:

The "d" indicates an exact differential for state functions (like ), while "" refers to inexact differentials for path-dependent quantities (like and ).

4. Specific Cases of First Principle Application

5. Isochoric Transformation (Constant Volume), Internal Energy Function

In an isochoric transformation (), the work done is zero ().

The heat exchanged at constant volume, , is equal to the change in internal energy. Therefore, is also a state function.

• If : The reaction is exothermic (system loses heat).

• If : The reaction is endothermic (system gains heat).

6. Isobaric Transformation (Constant Pressure), Enthalpy Function

Many chemical reactions occur at constant pressure (e.g., atmospheric pressure). For an isobaric transformation ():

We define a new state function called Enthalpy () as .

The heat exchanged at constant pressure, , is equal to the change in enthalpy. Therefore, is also a state function.

• If : The reaction is exothermic (system loses heat).

• If : The reaction is endothermic (system gains heat).

7. Properties of Ideal Gases

Joule's Experiment and Law:

Joule's experiment involves expanding a compressed gas into a vacuum, with the entire system isolated from heat exchange. It showed that for an ideal gas, the temperature remains unchanged, implying no heat exchange () and no work done (, as expansion is into a vacuum, so ).

(at constant temperature)

This demonstrates that the internal energy of an ideal gas depends only on its temperature, not on its volume or pressure.

Consequences for ideal gases:

• is independent of .

• : depends only on , not .

Example Calculation (Isobaric Heating):

Calculate , , and for heating 1 mole of an ideal gas from to at . .

For ideal gas:

(since is constant)

For 1 mole of liquid water under the same conditions:

The volume change of liquids is very small, so .

Note: . Pressures must be in Pascals () and volumes in cubic meters ().

III. Application of the First Principle to Chemistry

1. Heat of Reaction

For a general reaction: total at temperature . The heat of reaction is the heat exchanged when reactants and completely react stoichiometrically to form and at temperature .

A. Heat of Reaction at Constant Volume ()

For reactions at constant volume:

• If : The reaction is exothermic.

• If : The reaction is endothermic.

B. Heat of Reaction at Constant Pressure ()

For reactions at constant pressure:

• If : The reaction is exothermic.

• If : The reaction is endothermic.

C. Standard State

The standard state for a substance is its most stable physical form at 1 bar pressure and a specified temperature . Standard state values are denoted with a superscript (e.g., ). For instance, the standard state for carbon is graphite.

D. Relationship between and

For an isothermal reaction between ideal gases:

and

Where is the change in the number of moles of gaseous species (moles of gaseous products - moles of gaseous reactants).

For reactions involving only liquids and solids, is negligible, so .

2. Hess's Law

Hess's Law states that the total enthalpy change for a reaction is independent of the pathway taken, as enthalpy is a state function. This allows for the calculation of reaction heats that cannot be measured experimentally.

Example: Calculating the enthalpy change for the conversion of diamond to graphite () using combustion data:

To find , we reverse the second reaction:

Then sum the first reaction and the reversed second reaction:

Net:

3. Variation of Reaction Heats with Temperature

A. Latent Heat of Phase Change

During a phase change at constant pressure (e.g., melting, boiling), the temperature remains constant, but heat is exchanged. This heat is the latent heat (e.g., ).

Example:

B. Reaction Heat at Different Temperatures

Heats of reaction are typically reported at . To calculate heats of reaction at other temperatures, Kirchhoff's law can be used. For an isochoric process:

Given a reaction at and wanting at :

Where is the difference in molar heat capacities at constant volume between products and reactants.

Similarly, for an isobaric process:

Where .

4. Enthalpy of Formation (Heat of Formation)

The molar enthalpy of formation () of a compound is the enthalpy change when one mole of the compound is formed from its constituent elements in their standard states at a given pressure and temperature.

Example:

• (This is an enthalpy of formation because reactants are elements in their standard states).

• (This is not an enthalpy of formation because HCHO is not an element in its standard state).

The enthalpy of formation of a pure element in its standard state is zero ().

It is crucial to specify the physical state: .

Enthalpies of formation can be used to calculate heats of reaction:

Exercise: Calculate for the hydration of ethylene to ethanol:

Given:

5. Bond Energies

Bond energy () is the energy released when a chemical bond is formed from isolated gaseous atoms. It is generally negative (exothermic). Conversely, dissociation energy () is the energy required to break a bond, and it is always positive (). Bond energies are typically considered at and .

Bond energy should not be confused with enthalpy of formation, as formation from elements involves breaking existing bonds.

A. Covalent Bond Energy

To calculate the energy of a specific bond (e.g., H-Cl bond), a thermochemical cycle can be used.

Example: H-Cl bond energy

Given:

Dissociation energies: ,

We want . Consider the formation of 2 moles of gaseous HCl from gaseous atoms:

This can be related to the formation from standard state elements:

The elements must first be atomized:

Using Hess's Law:

Note: assuming exothermic formation. If positive values are dissociation energies, then:

However, the source uses "" which are bond energies, so they are negative for formation. With that convention:

.

The positive value of bond enthalpy is typically called bond dissociation energy. So for bonds, usually we refer to the energy needed to break them.

Thus, the value given, , implies the energy released when the bond is formed from gaseous atoms.

B. Determination of Bond Energies

Similar complex cycles are used for molecules. For example, to determine the average C-H bond energy in methane, a cycle involving atomization of carbon and hydrogen, and combustion of methane, is used.

C. Lattice Energy

The lattice energy () of an ionic crystal is the energy change associated with forming one mole of an ionic compound from its constituent gaseous ions. It is usually determined using a Born-Haber cycle.

Example: Lattice energy of NaCl

A Born-Haber cycle for NaCl relates its formation enthalpy to sublimation, ionization, dissociation, electron affinity, and lattice energy values.

Where is ionization energy, is electron affinity, is dissociation energy of .

Using the provided data (with careful attention to sign conventions for various energies):

Note: The given value is negative (), aligning with bond formation energy. If it were dissociation energy, it would be positive.

6. Flame Temperatures, Explosion Temperatures

During rapid phenomena like inflammation or explosion, energy exchange with the surroundings is negligible. These processes are considered adiabatic (). The heat released serves to raise the temperature of the combustion gases.

A. Isobaric Phenomenon (Flame Temperature)

For an isobaric adiabatic process, , which means . The phenomenon is conceptually split into two steps:

1. Isothermal reaction at initial temperature , releasing heat .

2. Absorption of this heat by the combustion gases, leading to a temperature increase to , with enthalpy change .

Thus,

This allows calculation of the final flame temperature . Heat capacities are used because pressure is constant.

Example: Calculate the maximum flame temperature for burning 2 moles of at (373 K) under 1 bar with theoretical air ().

Reaction:

Given (This is incorrect, it must be the enthalpy for the actual amount used, not per mole of H2). Let's assume this is the for 2 moles of H2.

So, if it was per mole of H2. The source likely implies this is for the whole reaction as written.

Using given polynomial expressions and integrating, we solve for . The example yield .

B. Isochoric Phenomenon (Explosion Temperature)

An explosion is a constant-volume ignition. Similar concepts apply, but is used instead of , and is used instead of for the gases.

Once is determined, the explosion pressure can be calculated using the ideal gas law: .

IV. Second Principle of Thermodynamics

The second principle introduces the concept of entropy (), a state function whose value predicts the spontaneity of reactions.

1. Statement of the Second Principle

• There exists a state function called entropy, denoted . is an exact differential.

• For an infinitesimal reversible transformation in which a system exchanges heat with surroundings at temperature , the elementary change in entropy is defined as:

• For a finite reversible transformation from state 1 to state 2:

• If the reversible transformation is isothermal:

• For an irreversible (spontaneous) transformation at constant temperature:

and

Since is a state function, is the same regardless of whether the process is reversible or irreversible. The relationship holds.

2. Application of the Second Principle to Isolated Systems

For isolated systems (no heat exchange, ):

• For a reversible transformation: . The entropy of the universe remains constant.

• For an irreversible transformation: . The entropy of the universe increases. This implies spontaneous processes lead to an increase in total entropy.

• If : The process is impossible (forbidden).

The entropy of the universe can never decrease for a spontaneous process.

3. Calculations of Entropy Change

A. Case of Ideal Gases

Using the definitions of for ideal gases:

For 1 mole of ideal gas,

Integrating (assuming is constant):

For an isothermal expansion of an ideal gas:

The internal energy is constant, so .

Hence

Entropy is an extensive quantity, as it depends on the number of moles .

B. Case of Condensed Bodies (Liquids and Solids)

a. Without Phase Change

• At constant pressure (assuming is constant):

• At constant volume (assuming is constant):

b. With Phase Change

During a reversible phase change at constant pressure (e.g., fusion, vaporization):

(where is the phase change temperature)

Application Example: Calculate the total entropy change for converting 1 mole of solid ethanol at (1 bar) to gaseous ethanol at (100 mmHg).

This requires summing entropy changes for heating the solid, melting, heating the liquid, vaporizing, heating the gas, and finally changing pressure for the gas.

The process is broken down into multiple steps:

1. Heating solid ethanol from to :

3. Fusion at :

5. Heating liquid ethanol from to :

7. Vaporization at :

9. Heating gaseous ethanol from to (at 1 bar):

11. Pressure change for gaseous ethanol at from 1 bar to 100 mmHg:

Total

V. Third Principle of Thermodynamics

1. Statement

The Third Principle of Thermodynamics addresses the absolute value of entropy. While absolute values for enthalpy () and internal energy () cannot be determined, absolute entropy () can.

Nernst and Planck's Postulate: The entropy of perfect crystalline pure substances is zero at absolute zero temperature (0 K).

This implies perfect order at 0 K. With this reference point, absolute entropies at any temperature can be calculated:

2. Entropy Change During a Chemical Reaction

For a reaction at given and , the entropy change a is calculated similarly to enthalpy changes:

At standard conditions (, ):

Important: Unlike , which is zero for pure elements in their standard states, values for elements are generally not zero.

Examples:

• Formation of gaseous HCl at , 1 bar:

• (The original document calculation is , which has a slight arithmetic error but illustrates the principle.)

  <li>Decomposition of $\text{CaCO}_3$ at $298 \text{ K}$, 1 bar:</li>
  <p>$\text{CaCO}_{3(s)} \rightarrow \text{CO}_{2(g)} + \text{CaO}_{(s)}$</p>
  <p>$\Delta_r S_{298}^\circ = S_{298}^\circ(\text{CO}_2) + S_{298}^\circ(\text{CaO}) - S_{298}^\circ(\text{CaCO}_3)$</p>
  <p>$= 213.9 + 39.8 - 92.8 = 160.9 \text{ J} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}$</p>
  (The original document calculation is $39.8 + 213.9 - 92.8 = 160.8 \text{ J} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}$, also a minor discrepancy.)

VI. Usable Energy – Gibbs Free Energy

The concepts of Gibbs Free Energy and Helmholtz Free Energy allow us to predict spontaneity under different conditions.

1. Definition

The Gibbs Free Energy () is defined as:

The Helmholtz Free Energy ( or ) is defined as:

Both and are state functions.

For an isothermal transformation ():

We know . If work is only pressure-volume work, .

For a reversible process, .

So, (for reversible isothermal process)

In general, for any process, as and for .

(general expression)

For Gibbs Free Energy:

Substituting and for reversible processes, and :

(for reversible isothermal, isobaric process)

In general:

For finite isothermal transformations:

2. Specific Relationships

A. Equilibrated Reaction (Reversible Process)

• At constant and : If , the system is at equilibrium.

• At constant and : If , the system is at equilibrium.

B. Spontaneous Reaction (Irreversible Process)

• At constant and : If , the reaction is spontaneous.

• At constant and : If , the reaction is spontaneous.

3. Change in Gibbs Free Energy During a Chemical Reaction

Similar to enthalpy and entropy of formation, we define the standard Gibbs free energy of formation (). For pure elements in their standard state, .

For a chemical reaction at and temperature :

4. Change in Gibbs Free Energy of a Pure Substance with T and P

A. Reversible Isobaric Process ()

Partial derivative with respect to temperature:

This leads to the Gibbs-Helmholtz Equation:

B. Reversible Isothermal Process ()

Partial derivative with respect to pressure:

Example: Calculate the standard Gibbs free energy change at 298 K for the reaction:

This can be solved by construction of a thermochemical cycle, similar to Hess's law for enthalpy. The example calculates .

The problem then compares two possible reactions from ethanal and hydrogen, concluding that reaction b) ( with ) is more favored than reaction a) (the one above with ) because it has a more negative .

VII. The Thermodynamic Equilibrium Constant

1. Expression of Gibbs Free Energy in Terms of Activities

For a reaction, the Gibbs free energy change under non-standard conditions () is related to the standard Gibbs free energy change () by:

Where is the reaction quotient, and is the activity of species , and is its stoichiometric coefficient.

2. Definition

At equilibrium, . So:

The reaction quotient at equilibrium is defined as the standard equilibrium constant ():

Thus,

And

depends only on temperature. The inversion temperature () is the temperature where , which implies .

3. Influence of Temperature on

From the Gibbs-Helmholtz equation:

Substituting :

This yields the Van't Hoff Isobar Equation:

This equation indicates:

• If (endothermic reaction), increasing temperature increases .

• If (exothermic reaction), increasing temperature decreases .

Integrating the Van't Hoff equation (assuming is constant over a narrow temperature range):

4. Expressions of According to the Nature of the System

The equilibrium constant is based on the law of mass action (Guldberg and Waage):

A. Monophasic Gaseous System

For ideal gases, the activity is given by , where is the partial pressure and (standard pressure). are algebraic stoichiometric coefficients (positive for products, negative for reactants).

This constant is dimensionless. Partial pressures can also be expressed in terms of mole fractions () or molar concentrations ().

(where is total pressure)

Leading to alternative expressions for :

Where is the sum of stoichiometric coefficients for gaseous species.

B. Gaseous Mixtures in Presence of Pure Condensed Phases

For pure condensed phases (solids or liquids) that are alone in their phase, their activity is defined as 1. Therefore, they do not appear in the expression for .

Example:

C. Dilute Aqueous Solution

For dilute aqueous solutions, the activity of water () is taken as 1. The activity of a solute is , where (standard concentration).

Example: Dissociation of a weak acid in water:

(acid dissociation constant)

D. Polyphasic Systems

Combining the rules for different phases.

Example: Dissolution of

(solubility product)

Example: PCl dissociation

Given . . Determine partial pressures at equilibrium.

1. From stoichiometry, ( moles of products).

2. .

3. .

Solving the quadratic equation for gives .

Then .

VIII. Displacement of Chemical Equilibrium

This section examines how various parameters influence the position of chemical equilibrium.

1. Displacement and Rupture of Equilibrium

• Equilibrium Displacement: Occurs when a parameter changes, moving the system from one equilibrium state to another equilibrium state within the same physicochemical system.

• Equilibrium Rupture: Occurs when a parameter changes, moving the system between equilibrium states of two different physicochemical systems (e.g., formation or disappearance of a component).

A perturbation causes the system to move out of equilibrium (), after which it evolves towards a new equilibrium where . The sign of the change in Gibbs free energy () indicates the direction of evolution:

• spontaneous evolution in the forward direction.

• spontaneous evolution in the reverse direction.

2. Influence of Temperature

The Van't Hoff equation () governs the influence of temperature.

• For (endothermic): Increasing increases , shifting equilibrium to the product side (forward direction).

• For (exothermic): Increasing decreases , shifting equilibrium to the reactant side (reverse direction).

Le Chatelier's Principle (for temperature): An increase in temperature shifts the equilibrium in the direction that absorbs heat (endothermic direction).

Application: Ammonia Synthesis

Given (exothermic).

1. An increase in temperature will shift the equilibrium to the left (dissociation of ) to absorb heat.

2. Calculating from confirms this: , which is less than , meaning the reaction shifts to the left.

3. Reaction Quotient and Evolution of a System

The relationship describes the evolution. A perturbation changes , causing to determine the direction of the shift.

• If shifts towards reactants.

• If shifts towards products.

4. Influence of Pressure at Constant Temperature for a Closed System

For a system with a gaseous phase, the reaction quotient can be expressed in terms of partial pressures or mole fractions and total pressure .

Changes in influence primarily through the term.

If increases ():

• If : shifts towards reactants.

• If : shifts towards products.

• If : no shift.

Le Chatelier's Principle (for pressure): An increase in pressure applied to a closed system at equilibrium (constant temperature) shifts the equilibrium in the direction that reduces the total number of moles of gas (i.e., reduces volume).

Example:

Reactant moles = 5, product moles = 3. . An increase in pressure will favor the forward reaction (shifts to products) to decrease the number of gas moles.

Application: Sulfur Trioxide Synthesis

Given at and .

1. . An increase in pressure will favor the forward reaction (synthesis of ).

2. Calculating the conversion at shows , which is higher than , confirming a shift towards products with increased pressure.

Note: An increase in volume at constant temperature is equivalent to a decrease in pressure, and vice-versa, so its effect is opposite to that of pressure.

5. Influence of the Introduction of an Inactive Constituent

An inactive constituent is one that does not participate in the reaction.

A. Inactive Gas in a Gaseous System at Constant T and V

When an inactive gas is added at constant and , the total pressure increases, but the partial pressures of the reacting gases remain unchanged. Therefore, (which depends on partial pressures or molar concentrations) remains constant, and there is no shift in equilibrium.

B. Inactive Gas in a Gaseous System at Constant T and P

When an inactive gas is added at constant and , the total volume must increase to keep total pressure constant. This leads to a decrease in the partial pressures of the reacting gases. The equilibrium will shift to counteract this decrease, according to Le Chatelier's principle.

(where is the increase in total moles due to inactive gas)

• If : shifts towards products (more gas moles).

• If : shifts towards reactants (fewer gas moles).

• If : no shift.

Le Chatelier's Principle (for inactive gas at constant P): Adding an inactive gas at constant T and P to a gaseous system at equilibrium shifts the equilibrium in the direction that increases the total number of moles of gas.

C. Other Cases

• Dilution of a liquid system: Shifts towards increasing the quantity of dissolved species.

• Addition of a small quantity of pure inactive solid or liquid: No effect if it doesn't significantly change or .

• Addition of a small quantity of inactive solute: No effect on dilute solutions.

6. Influence of the Introduction of an Active Constituent

An active constituent is one that participates in the reaction.

A. Addition of a Pure Active Solid or Liquid Constituent

Pure solids and liquids have an activity of 1, so adding them does not change . Therefore, there is no effect on the equilibrium state at constant and or and .

B. Addition of an Active Gas in a Gaseous System at Constant T and V

Adding an active gas (reactant or product) at constant and increases its partial pressure (and concentration). This changes .

The shift always occurs in the direction that consumes the added species.

• If is a reactant (): Shifts towards products.

• If is a product (): Shifts towards reactants.

C. Addition of a Gas in a Gaseous System at Constant T and P

Adding an active gas at constant and causes the system's volume to adjust, affecting partial pressures. This is a more complex situation. Generally, the system shifts to consume the added gas, but the magnitude and direction can be complex due to the coupled changes in partial pressures and total moles.

Application Example: Ammonia Synthesis with Active Gas Addition

• Adding (product) at constant T and P: Shifts equilibrium to the left (dissociation of ) to consume the excess product.

• Adding (reactant) at constant T and P: Shifts equilibrium to the right (formation of ) to consume the excess reactant.

• Adding (reactant) at constant T and P: The direction of shift depends on the initial mole fraction of relative to (i.e. ). If , it shifts to the right (to consume more ). If , it shifts to the left. If , there is no net shift. This is because the shift depends on how the addition changes the overall balance of partial pressures and concentration, which is related to the initial composition.