🧤Physical Chemistry I Unit 4 – Heat Capacity and Thermochemistry

Key Concepts and Definitions

• Heat capacity quantifies the amount of heat required to change the temperature of a substance by a specific amount
• Specific heat capacity is the heat capacity per unit mass of a substance (J/g·K or J/kg·K)
• Molar heat capacity is the heat capacity per mole of a substance (J/mol·K)
• Thermochemistry studies the energy changes associated with chemical reactions and physical transformations
• Enthalpy (H) is a thermodynamic quantity representing the total heat content of a system at constant pressure
• Calorimetry measures the heat transfer during physical and chemical processes
• Endothermic processes absorb heat from the surroundings (positive enthalpy change)
• Exothermic processes release heat to the surroundings (negative enthalpy change)

Fundamentals of Heat Capacity

• Heat capacity (C) is defined as the amount of heat (q) required to change the temperature (ΔT) of a substance: $C = \frac{q}{\Delta T}$
• The heat capacity of a substance depends on its mass, chemical composition, and physical state
• Substances with higher heat capacities require more energy to increase their temperature compared to those with lower heat capacities
• Heat capacity is an extensive property, meaning it depends on the amount of substance present
  • Doubling the mass of a substance doubles its heat capacity
• Specific heat capacity (c) is an intensive property, independent of the amount of substance: $c = \frac{C}{m}$
• Molar heat capacity (Cm) is another intensive property, expressed per mole of substance: $Cm = \frac{C}{n}$
• The heat required to change the temperature of a substance can be calculated using: $q = mc\Delta T$ or $q = nCm\Delta T$

Types of Heat Capacity

• Heat capacity at constant volume (Cv) measures the heat required to change the temperature of a system while keeping its volume constant
  • Cv is typically used for gases in closed systems where no work is done
• Heat capacity at constant pressure (Cp) measures the heat required to change the temperature of a system while maintaining constant pressure
  • Cp is commonly used for solids, liquids, and gases in open systems where work can be done
• For an ideal gas, the molar heat capacities are related by: $Cp - Cv = R$, where R is the ideal gas constant (8.314 J/mol·K)
• The ratio of heat capacities (γ) for an ideal gas is given by: $\gamma = \frac{Cp}{Cv}$
  • γ is an important parameter in adiabatic processes and the study of gas behavior
• The heat capacity of a substance can change with temperature, especially near phase transitions
• Dulong-Petit law states that the molar heat capacity of a solid element at high temperatures is approximately 3R (24.9 J/mol·K)

Measuring Heat Capacity

• Calorimetry is the primary method for measuring heat capacity experimentally
• A calorimeter is an insulated device that measures the heat exchanged during a physical or chemical process
• In a typical calorimetry experiment, a known amount of heat is added to a sample, and the resulting temperature change is measured
• The heat capacity of the sample can be calculated using: $C = \frac{q}{\Delta T}$, where q is the heat added and ΔT is the temperature change
• Bomb calorimetry is used to measure the heat of combustion of a substance in a constant-volume calorimeter
  • The sample is ignited in a sealed "bomb" filled with oxygen, and the heat released is measured
• Differential scanning calorimetry (DSC) measures the heat flow to a sample as a function of temperature
  • DSC can determine phase transition temperatures, enthalpies of fusion and vaporization, and heat capacities
• Modulated DSC (MDSC) separates the reversible and irreversible heat flow components, providing more detailed information about a sample's thermal properties

Introduction to Thermochemistry

• Thermochemistry is the study of heat and energy associated with chemical reactions and physical transformations
• The first law of thermodynamics states that energy cannot be created or destroyed, only converted from one form to another
  • In a closed system, the change in internal energy (ΔU) equals the heat (q) added to the system minus the work (w) done by the system: $\Delta U = q - w$
• The enthalpy (H) of a system is defined as: $H = U + PV$, where U is the internal energy, P is the pressure, and V is the volume
• The change in enthalpy (ΔH) for a process at constant pressure is equal to the heat exchanged: $\Delta H = q_p$
• Thermochemical equations represent chemical reactions and include the enthalpy change as a stoichiometric quantity
  • For example, the combustion of methane: $CH_4 (g) + 2O_2 (g) \rightarrow CO_2 (g) + 2H_2O (l)$ $\Delta H = -890$ kJ/mol
• Hess's law states that the overall enthalpy change for a reaction is independent of the pathway and is the sum of the enthalpy changes for the individual steps

Laws of Thermochemistry

• The first law of thermochemistry is the application of the first law of thermodynamics to chemical systems
  • The change in internal energy (ΔU) of a system is equal to the heat (q) added to the system minus the work (w) done by the system: $\Delta U = q - w$
• The second law of thermochemistry states that the enthalpy change for a reaction is equal to the sum of the enthalpy changes for the individual steps (Hess's law)
  • This allows the calculation of enthalpy changes for reactions that are difficult to measure directly
• Kirchhoff's law describes the temperature dependence of enthalpy changes
  • The enthalpy change of a reaction at a given temperature (T2) can be calculated from the enthalpy change at another temperature (T1) using: $\Delta H_{T2} = \Delta H_{T1} + \int_{T1}^{T2} \Delta C_p dT$
• The third law of thermochemistry states that the entropy of a perfect crystal at absolute zero is zero
  • This provides a reference point for calculating absolute entropies and Gibbs free energies of substances

Enthalpy and Enthalpy Changes

• Enthalpy (H) is a state function that represents the total heat content of a system at constant pressure
• The change in enthalpy (ΔH) is the heat exchanged with the surroundings during a process at constant pressure: $\Delta H = q_p$
• Standard enthalpy of formation (ΔHf°) is the enthalpy change when one mole of a compound is formed from its constituent elements in their standard states at 1 atm pressure and a specified temperature (usually 298 K)
  • By convention, the standard enthalpy of formation for elements in their standard states is zero
• Standard enthalpy of combustion (ΔHc°) is the enthalpy change when one mole of a substance is completely burned in excess oxygen at standard conditions
• Hess's law allows the calculation of enthalpy changes for reactions using known enthalpy changes of other reactions
  • For example, the enthalpy of formation of CO(g) can be calculated using the enthalpies of combustion of C(s) and CO(g): $\Delta H_f^\circ (CO) = \Delta H_c^\circ (C) - \Delta H_c^\circ (CO)$
• Bond dissociation enthalpies can be used to estimate the enthalpy change of a reaction by considering the bonds broken and formed

Calorimetry and Its Applications

• Calorimetry is the experimental measurement of heat transfer during physical and chemical processes
• The basic principle of calorimetry is the conservation of energy: the heat lost by a hot object equals the heat gained by a cold object
• In a coffee cup calorimeter, a reaction takes place in a solution inside an insulated container, and the temperature change of the solution is measured
  • The heat of the reaction can be calculated using: $q_{rxn} = -q_{soln} = -mc\Delta T$, where m is the mass of the solution, c is its specific heat capacity, and ΔT is the temperature change
• Bomb calorimetry measures the heat of combustion of a substance in a constant-volume calorimeter
  • The heat capacity of the calorimeter is determined using a calibration reaction with a known enthalpy change (e.g., benzoic acid combustion)
• Calorimetry can be used to determine the enthalpy changes of various processes, such as:
  • Heats of solution and dilution
  • Heats of neutralization (acid-base reactions)
  • Heats of fusion and vaporization
  • Heats of combustion and formation
• Calorimetric data can be used to calculate other thermodynamic quantities, such as entropy changes and Gibbs free energy changes

Real-World Applications

• Heat capacity and thermochemistry principles are crucial in designing and optimizing industrial processes, such as chemical manufacturing and energy production
• In the food industry, calorimetry is used to determine the caloric content of food products and to study the thermal properties of ingredients during processing
• Thermal energy storage systems, such as phase change materials (PCMs), rely on the high heat capacities and latent heats of certain substances to store and release thermal energy efficiently
  • PCMs are used in building materials, solar thermal systems, and temperature-regulating textiles
• In materials science, calorimetry techniques (DSC, MDSC) are employed to characterize the thermal properties of polymers, composites, and other materials
  • This information is valuable for product development, quality control, and failure analysis
• Thermochemical data is essential for understanding and predicting the behavior of chemical reactions in various fields, such as:
  • Combustion and fuel technology
  • Environmental chemistry and pollution control
  • Biochemistry and metabolic processes
  • Pharmaceutical development and drug stability
• Calorimetry is used in battery research to study the thermal behavior of battery materials and to optimize battery design for safety and performance