CHEM122 Thermodynamics

Fundamental Laws of Thermodynamics

Overview of the Laws

• 1st Law of Thermodynamics: Energy cannot be created or destroyed, only transformed. This principle emphasizes the conservation of energy in all processes.

• 2nd Law of Thermodynamics: Entropy, a measure of disorder, is always increasing in an isolated system, indicating that energy transformations are not 100% efficient.

• 3rd Law of Thermodynamics: As temperature approaches absolute zero, the entropy of a perfect crystal approaches zero, establishing a baseline for entropy measurements.

Implications of the Laws

• The first law implies that energy input must equal energy output in any closed system, leading to the concept of energy balance in chemical reactions.

• The second law introduces the concept of irreversibility in natural processes, explaining why certain reactions occur spontaneously while others do not.

• The third law provides a theoretical limit for the efficiency of thermodynamic processes, particularly in cryogenics and low-temperature physics.

Fundamental Concepts of Thermodynamics

Work and Energy Calculations

• Work (w) is defined as the product of pressure (P) and change in volume (ΔV), expressed as w = -P × ΔV.

• Example calculation: For a pressure of 1 atm and a volume change of 100 cm³, w = -1 × 100 atm cm³ = -101325 Pa × 100 × 10⁻⁶ m³ = -10.1 J.

• The relationship between pressure, volume, and energy is crucial in thermodynamics, where 1 Pa m³ = 1 N m² m³ = 1 Nm = 1 J.

The First Law of Thermodynamics

• The first law states that the change in internal energy (ΔE) of a system is equal to the heat added to the system (q) plus the work done on the system (w): ΔE = q + w.

• Internal energy (E) is the sum of kinetic and potential energies of all molecules in the system, highlighting the microscopic nature of energy.

• In an adiabatic process, where no heat is exchanged (q = 0), the change in internal energy is equal to the work done: ΔE = w = ΔE_final - ΔE_initial.

Conservation of Energy

• For an isolated system, where no heat or work is exchanged (q = 0, w = 0), the change in internal energy is zero: ΔE = 0, illustrating the principle of conservation of energy.

• In cyclic processes, the work done by a closed system equals the heat withdrawn from the surroundings, reinforcing the idea that energy is conserved over cycles.

Enthalpy and Heat Capacity

Enthalpy in Thermodynamics

• Enthalpy (H) is defined as H = E + PV, where P is pressure and V is volume, and is particularly useful for processes at constant pressure.

• The change in enthalpy (ΔH) can be expressed as ΔH = ΔE + PΔV, linking it to heat flow at constant pressure: q_p = ΔE + PΔV.

Heat Capacity Definitions

• Heat capacity is the amount of heat energy required to raise the temperature of a substance by one degree Celsius, indicating a substance's ability to store energy.

• Molar heat capacity refers to the heat required to change the temperature of one mole of a substance by 1 degree, while specific heat capacity refers to the heat required to change the temperature of 1 g of a substance by 1 degree. For example, the specific heat capacity of water is 4.18 J g⁻¹ °C⁻¹.

Heat Capacity at Constant Volume and Pressure

• At constant volume, all heat energy increases the internal energy of the gas: q = C_V ΔT, where C_V is the heat capacity at constant volume.

• At constant pressure, some heat energy is used to do work (expand the gas), leading to the equation: q = C_P ΔT = ΔE + PΔV, where C_P is the heat capacity at constant pressure.

Reaction Enthalpies and Hess's Law

Reaction Enthalpies

• The change in enthalpy for a reaction (ΔH_r) is defined as the difference between the enthalpies of products and reactants: ΔH_r = H_products - H_reactants.

• Exothermic reactions release heat (ΔH_r < 0), while endothermic reactions absorb heat (ΔH_r > 0).

Standard Enthalpies of Formation

• The standard enthalpy of formation (ΔH°_f) is the change in enthalpy when forming one mole of a compound from its elements in their standard states, with ΔH°_f = 0 for elements in their most stable form.

• Example: ΔH°_f(O₂(g)) = 0, ΔH°_f(C(graphite)) = 0, ΔH°_f(O₃(g)) = +142.7 kJ/mol.

Hess's Law

• Hess's Law states that the total enthalpy change for a reaction is the same, regardless of the number of steps taken, emphasizing that enthalpy is a state function.

• This allows for the calculation of reaction enthalpies by breaking down reactions into steps involving standard enthalpies of formation.

Entropy and the Second Law of Thermodynamics

Understanding Entropy

• Entropy is a measure of disorder or randomness in a system, with higher entropy indicating greater disorder.

• The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time, leading to the concept of spontaneous processes.

Spontaneity and Natural Processes

• Spontaneous processes occur without external intervention and are driven by the natural tendency of systems to increase their entropy.

• The direction of spontaneous processes is determined by the balance of energy and entropy changes, guiding the understanding of thermodynamic feasibility.

Second Law of Thermodynamics

Overview of the Second Law

• The second law states that the driving force for a spontaneous process is an increase in the entropy of the universe.

• Entropy is a thermodynamic property that helps predict how a system will change under various conditions such as compression, expansion, heating, or cooling.

• Rudolf Clausius, who introduced the concept in 1854, noted that the entropy of the universe tends to a maximum, indicating that all matter is becoming more disordered.

Entropy Defined

• Entropy is a measure of randomness or disorder within a system, often described as moving from order to disorder.

• The total entropy of an entire system can never decrease; it must always increase during any change.

• This principle applies universally, suggesting a natural tendency towards greater disorder in all physical processes.

Entropy and Statistics

Probability in Entropy

• Consider a system with two identical flasks connected by a narrow neck, where the probability of a molecule being in either flask is 0.5.

• If four molecules are placed in the system, the probability of all four being in one flask can be calculated using the binomial distribution.

Binomial Distribution Example

• The probability of 'v' successes in 'n' trials is given by the formula: P(v successes in n trials) = (n! / (v!(n-v)!)) * (0.5^n).

• For example, the probabilities for four molecules in the left flask are: 0 in left (0.0625), 1 in left (0.2500), 2 in left (0.3750).

• As more molecules are added, the distribution of probabilities narrows, indicating a higher likelihood of equal distribution.

Impact of Increasing Molecules

• As the number of molecules increases, the probability distribution becomes narrower, making extreme distributions less likely.

• For instance, with 1000 atoms, the chance of a distribution of 450:550 is less than 0.5% compared to a 500:500 distribution.

Enthalpy of Reaction and Hess's Law

Understanding Enthalpy Changes

• The change in enthalpy (ΔHr) for a reaction is calculated as ΔHr = Hproducts – Hreactants.

• Exothermic reactions release heat (ΔHr < 0), while endothermic reactions absorb heat (ΔHr > 0).

• Enthalpy is a state function, meaning ΔHr is independent of the pathway taken during the reaction.

Example Calculation of ΔHr

• For the reaction N2 + 2O2 → 2NO2, the enthalpy changes for the steps are: N2 + O2 → 2NO (ΔHr = +180 kJ) and 2NO + O2 → 2NO2 (ΔHr = -112 kJ).

• Therefore, the overall ΔHr for the reaction is calculated as 180 + (-112) = 68 kJ.

Standard Enthalpy of Formation

• The standard enthalpy of formation (ΔHf) is crucial for calculating reaction enthalpies.

• For example, the formation of CH4(g) from its elements has ΔHf = -75 kJ/mol, while O2(g) has ΔHf = 0 kJ/mol.

• The overall reaction enthalpy for CH4(g) + 2O2(g) → CO2(g) + 2H2O(l) can be calculated using the standard enthalpies of formation for each compound.

Understanding Entropy

Definition and Significance of Entropy

• Entropy is a measure of disorder or randomness in a system, reflecting the number of microscopic configurations that correspond to a thermodynamic system's macroscopic state.

• It is a fundamental concept in thermodynamics, indicating the direction of spontaneous processes; systems tend to evolve towards states of higher entropy.

• The relationship between entropy (S) and heat (q) is expressed as ΔS ∝ q, indicating that heat transfer influences the entropy change in a system.

• Entropy increases when heat flows from a hot body to a cold body, demonstrating the second law of thermodynamics: spontaneous processes increase the total entropy of the universe.

• The formula ΔS ≈ q/T illustrates that the change in entropy is proportional to the heat transferred divided by the temperature at which the transfer occurs.

• Historical context: The concept of entropy was introduced by Rudolf Clausius in the 19th century, fundamentally altering the understanding of energy transformations.

Entropy and Heat Transfer

• Heat transfer occurs spontaneously from hot to cold bodies, leading to an increase in the entropy of the colder body.

• The spontaneous nature of heat transfer is a key principle in thermodynamics, emphasizing that energy disperses naturally.

• The equation ΔS ∝ 1/T indicates that the change in entropy is inversely related to temperature, meaning that at lower temperatures, the same amount of heat transfer results in a larger change in entropy.

• Example: When a hot solution transfers heat to a cold solution, the entropy of the cold solution increases significantly, while the hot solution's entropy decreases.

• The concept of entropy is crucial in understanding irreversible processes, where the total entropy of the system and surroundings increases.

• Case study: The melting of ice in warm water demonstrates how heat transfer increases the entropy of the system.

Isothermal Processes in Ideal Gases

Isothermal Expansion and Compression

• An isothermal process occurs at constant temperature, meaning the internal energy of an ideal gas remains unchanged (ΔE = 0).

• During isothermal expansion, the gas does work on the surroundings, and the heat absorbed equals the work done (q = -w).

• In a hypothetical scenario, if a gas expands against zero external pressure, the work done is zero, and the process is adiabatic.

• The relationship between pressure (P), volume (V), and temperature (T) for an ideal gas is given by the ideal gas law: PV = nRT.

• Example: If a gas expands from volume V1 to V2, the work done can be calculated using w = -PexΔV, where Pex is the external pressure.

• Visual Aid: A diagram illustrating the isothermal expansion and compression of an ideal gas can enhance understanding.

Work Done in Isothermal Processes

• The work done during isothermal expansion can be calculated using the formula w = -PexΔV, where Pex is the external pressure and ΔV is the change in volume.

• In a one-step expansion, if the gas expands from V1 to V2, the work done is negative, indicating energy is transferred out of the system.

• In a two-step expansion, the total work done is the sum of the work done in each step, demonstrating that multiple steps can yield more work output.

• The concept of pathway dependence in work is crucial; the work done is not solely a function of the initial and final states but also of the path taken.

• Example: Comparing one-step and two-step expansions shows that the two-step process yields more work output, emphasizing the importance of process design in thermodynamic systems.

• Table: A comparison of work done in one-step vs. two-step expansions can clarify the differences.

Pathway Dependence of Work

Work in Expansion vs. Compression

• The work required for compression is generally greater than the work obtained from expansion, highlighting the inefficiencies in energy transfer.

• In a two-step compression, the total work required can be calculated as the sum of the work done in each step, which is often greater than the work done in expansion.

• The relationship between work done in expansion and compression illustrates the non-conservative nature of work in thermodynamic processes.

• Example: In a two-step compression from 4V1 to V1, the work required is significantly higher than the work produced during expansion, demonstrating the energy costs associated with compression.

• The concept of work being a pathway-dependent function is critical in understanding thermodynamic cycles and efficiency.

• Case Study: Analyzing the Carnot cycle can provide insights into the efficiency of different thermodynamic processes.

Understanding Entropy in Isothermal Processes

Definition and Concept of Entropy

• Entropy is a measure of disorder or randomness in a system, often associated with the second law of thermodynamics. It quantifies the amount of energy in a physical system that is not available to do work.

• In the context of an ideal gas, entropy can be understood through the lens of isothermal processes, where temperature remains constant during expansion or compression.

• The concept of entropy is crucial in understanding the efficiency of thermodynamic processes, particularly in reversible and irreversible scenarios.

• An infinite step isothermal expansion implies that the gas expands in infinitely small increments, allowing for maximum work output.

• The relationship between entropy and work can be illustrated through the formula for work done during expansion: ( w = -\int P_{ex} dV ).

• The hypothetical nature of infinite steps highlights the theoretical limits of work extraction from a gas.

Isothermal Expansion of an Ideal Gas

• In an isothermal expansion, the external pressure is adjusted in infinitesimally small increments, closely matching the internal gas pressure (( P \approx P_{ex} )).

• The work done during this process can be expressed as: ( w = -\int P_{ex} dV = -\int nRT/V dV ).

• For a reversible process, the work done can be calculated as: ( w_{rev} = -nRT \ln(V_2/V_1) ), where ( V_2 ) is the final volume and ( V_1 ) is the initial volume.

• For example, if ( V_2 = 4V_1 ), the work done is: ( w_{rev} = -nRT \ln(4) = -1.39nRT ).

• This demonstrates that the work done is dependent on the logarithmic ratio of the volumes, emphasizing the relationship between volume change and work output.

• The concept of maximum work output is critical in thermodynamics, as it sets a benchmark for real-world applications.

Isothermal Compression of an Ideal Gas

• Isothermal compression is the reverse process of expansion, where the gas is compressed while maintaining a constant temperature.

• The work done during compression can also be expressed similarly: ( w = -\int P_{ex} dV ).

• For a specific case where the final volume is ( V_{final} = 1/4 V_1 ), the work done is: ( w_{rev} = -nRT \ln(1/4) = 1.39P_1V_1 ).

• This indicates that work is done on the gas, converting energy into potential work.

• The relationship between work and heat during these processes is crucial, as it illustrates the conversion of energy forms.

• The theoretical maximum work is achieved only under reversible conditions, which are idealized and not achievable in real processes.

Work and Heat in Isothermal Cycles

Summary of Work and Heat in Cycles

• In an isothermal cycle, the change in internal energy (( \Delta E )) is zero, leading to the relationship: ( \Delta E = q + w ).

• The work and heat exchanged during a cycle can be summarized in a table format, illustrating the differences in energy transfer for various steps.

• For example, in a one-step expansion followed by a one-step compression, the work input and output can be quantified as follows: | Process | Work Input | Work Output | Heat Input | Heat Output | |:-------------- |:---------- |:----------- |:---------- |:----------- | | 1-Step Exp. | 3.00P1V1 | 0.75P1V1 | 3.00P1V1 | 0.75P1V1 | | 2-Step Exp. | -2.00P1V1 | 1.00P1V1 | 2.00P1V1 | -1.00P1V1 | | 4-Step Exp. | -1.39P1V1 | 1.39P1V1 | 1.39P1V1 | -1.39P1V1 | | ∞ (Reversible) | -1.39P1V1 | 1.39P1V1 | 1.39P1V1 | -1.39P1V1 |

• This table illustrates how the number of steps affects the efficiency of work and heat transfer in isothermal processes.

Consequences of Isothermal Processes

• The inescapable conclusion from isothermal expansion and compression cycles is that work is converted into heat, demonstrating the second law of thermodynamics.

• In finite-step cycles, energy transitions from ordered (work) to disordered (heat) forms, emphasizing the irreversibility of real processes.

• The theoretical maximum work is only achievable under reversible conditions, which are ideal and not practically realizable.

• The concept of entropy plays a crucial role in understanding these transformations, as it quantifies the degree of disorder in energy states.

• The implications of these processes are significant in various applications, including engines and refrigerators, where efficiency is paramount.

• Understanding these principles is essential for advancing thermodynamic theory and practical applications.

Fundamental Concepts of Thermodynamics

Reversible vs. Irreversible Processes

• A reversible cyclic process returns both the system and surroundings to their original conditions, allowing for maximum efficiency and no net change in the universe.

• An irreversible cyclic process, while returning the system to its original state, results in permanent changes to the surroundings, indicating energy dissipation and loss of efficiency.

Key Thermodynamic Principles

• Spontaneity: A process is spontaneous if it occurs without external intervention, typically associated with an increase in the entropy of the universe.

• Entropy (S): A measure of disorder or randomness in a system, which tends to increase in spontaneous processes, as stated by the second law of thermodynamics.

Gibbs Free Energy

• Gibbs Free Energy (G): Defined as G = H - TS, where H is enthalpy, T is temperature, and S is entropy. It indicates the maximum reversible work obtainable from a thermodynamic system at constant temperature and pressure.

• A process is spontaneous if ΔG < 0, which corresponds to an increase in the entropy of the universe (ΔSuniv > 0).

Gibbs Free Energy and Chemical Reactions

Overview of Gibbs Free Energy

• Gibbs free energy (G) is a thermodynamic potential that measures the maximum reversible work obtainable from a thermodynamic system at constant temperature and pressure.

• The change in Gibbs free energy (ΔG) during a reaction indicates the spontaneity of the process: if ΔG < 0, the reaction is spontaneous; if ΔG > 0, it is non-spontaneous; if ΔG = 0, the system is at equilibrium.

• The standard free energy of formation (ΔGf°) is the change in free energy when one mole of a compound is formed from its elements in their standard states.

Calculating Gibbs Free Energy Changes

• The formula for calculating the change in Gibbs free energy for a reaction is: ΔG°(reaction) = ΣΔGf°(products) - ΣΔGf°(reactants).

• Example: For the reaction C(graphite) → C(diamond), ΔG° was calculated as +3 kJ, indicating that the reaction is spontaneous in the reverse direction (diamond to graphite).

• Hess's Law can be applied to combine known ΔG° values for various reactions to find the ΔG° for a target reaction.

Dependence on Pressure and Concentration

• The Gibbs free energy of a reaction mixture changes as the reaction proceeds, influenced by pressure and concentration, which affect entropy (S).

• The relationship is given by: ΔG = ΔG° + RTln(Q), where Q is the reaction quotient.

• For example, at 25 °C with specific pressures of CO and H2, ΔG was calculated to be -34 kJ, indicating a spontaneous reaction.

Thermodynamics and Equilibrium

Gibbs Free Energy and Equilibrium Constants

• At equilibrium, ΔG = 0, and the relationship between ΔG° and the equilibrium constant (K) is given by: ΔG° = -RTln(K).

• This indicates that the standard Gibbs free energy change is directly related to the position of equilibrium for a reaction.

Temperature Dependence of Equilibrium Constants

• The temperature dependence of K can be analyzed using the equation: ΔG° = ΔH° - TΔS°. This shows how changes in temperature affect the spontaneity of reactions.

• A plot of ln(K) versus 1/T can be used to determine ΔH° and ΔS° from the slope and intercept, respectively.

Summary of Key Concepts

Spontaneity and Temperature Effects

• Spontaneous processes are characterized by an increase in the entropy of the universe: ΔS_universe = ΔS_system + ΔS_surroundings > 0.

• The relationship ΔG = ΔH - TΔS helps to understand how enthalpy (ΔH) and entropy (ΔS) changes influence spontaneity at different temperatures.

Conditions for Spontaneity

• A process is spontaneous at all temperatures if ΔH is negative and ΔS is positive.

• A process is spontaneous at high temperatures if ΔH is positive and ΔS is also positive, where the entropy effect dominates.

• Conversely, a process is spontaneous at low temperatures if ΔH is negative and ΔS is negative.