Enthalpy and heat capacity are key concepts in energy balances. Enthalpy represents a system's total heat content, while heat capacity measures how much heat is needed to change a substance's temperature. These ideas are crucial for understanding energy flow in chemical processes.

Calculating enthalpy changes involves heat capacity, specific heat, and latent heat for phase changes. These concepts help engineers design heat exchangers, optimize processes, and perform accurate energy balance calculations. Understanding how heat capacity varies with temperature is essential for precise enthalpy calculations in real-world applications.

Enthalpy and Internal Energy

Definition and Relationship

• Enthalpy (H) represents the total heat content of a system at constant pressure
• Enthalpy is the sum of the internal energy (U) and the product of pressure (P) and volume (V): $H = U + PV$
• Changes in enthalpy (ΔH) equal the heat absorbed or released by a system at constant pressure: $ΔH = Q_p$
• Enthalpy is a state function, its value depends only on the initial and final states of the system, not on the path taken between those states

Pressure-Volume Work

• Pressure-volume work (W) is the work done by a system due to changes in volume against an external pressure
• For a constant-pressure process, the pressure-volume work is given by: $W = -PΔV$, where ΔV is the change in volume
• The negative sign indicates that work is done by the system on the surroundings when the volume increases (ΔV > 0)
• Pressure-volume work is path-dependent, unlike enthalpy, which is a state function

Calculating Enthalpy Changes

Heat Capacity and Specific Heat

• Heat capacity (C) is the amount of heat required to raise the temperature of a substance by one degree (Celsius or Kelvin)
• Specific heat capacity (c) is the heat capacity per unit mass of a substance: $c = C/m$
• Molar heat capacity (C_m) is the heat capacity per mole of a substance: $C_m = C/n$
• The change in enthalpy (ΔH) for a constant-pressure process is calculated using the heat capacity and the change in temperature (ΔT): $ΔH = C_p × ΔT$, where C_p is the heat capacity at constant pressure

Phase Changes and Latent Heat

• For phase changes (melting, vaporization), the enthalpy change equals the latent heat of the phase transition: $ΔH = ±L$
• Latent heat of fusion (L_f) is the energy required to melt a substance at its melting point (solid to liquid)
• Latent heat of vaporization (L_v) is the energy required to vaporize a substance at its boiling point (liquid to gas)
• During phase changes, the temperature remains constant while the substance absorbs or releases latent heat

Significance of Heat Capacity

Energy Balance Calculations

• Heat capacity determines the amount of heat required to change the temperature of a system
• In heat exchangers, the heat capacity of the fluids determines the heat transfer rate and outlet temperatures
• When mixing streams with different temperatures, the heat capacities are used to calculate the final mixture temperature
• Heat capacity data is essential for designing and sizing equipment (heat exchangers, reactors, distillation columns)

Process Design and Optimization

• Accurate heat capacity values are crucial for process simulation and optimization
• Heat integration strategies, such as pinch analysis, rely on heat capacity data to identify opportunities for energy recovery and minimize utility consumption
• In reactor design, heat capacity affects the temperature profile and heat removal requirements
• Distillation column design and operation depend on the heat capacities of the components to determine the energy requirements for vaporization and condensation

Temperature Dependence of Heat Capacity

Empirical Models

• Heat capacity is generally temperature-dependent, especially for gases and liquids
• The temperature dependence can be described using empirical equations (polynomial expressions, Shomate equation)
• For gases, the temperature dependence is often modeled using the ideal gas heat capacity, a function of temperature and molecular structure
• Examples of empirical models: $C_p(T) = a + bT + cT^2 + dT^3$, where a, b, c, and d are empirical constants specific to the substance

Impact on Enthalpy Calculations

• The temperature dependence of heat capacity affects the calculation of enthalpy changes, especially when the temperature range is large
• To accurately calculate enthalpy changes over a wide temperature range, the heat capacity must be integrated with respect to temperature: $ΔH = ∫(T_1 to T_2) C_p(T) dT$
• Neglecting the temperature dependence of heat capacity can lead to significant errors in enthalpy calculations, particularly in processes involving high temperatures or large temperature changes
• Example: In a gas-phase reaction with a large temperature increase, using a constant heat capacity value may underestimate the enthalpy change and lead to incorrect energy balance calculations