CHE 130: Thermodynamics – Core Concepts and Applications

Internal Energy, U

Definition and Components

Internal energy (U) refers to the total energy contained within a system due to the microscopic motions and interactions of its molecules. It is a fundamental concept in thermodynamics and is crucial for understanding energy changes in chemical and physical processes.

• Constituents: Includes kinetic energy of translation, rotation, and vibration, as well as potential energy from intermolecular forces.

• Measurement: Internal energy cannot be measured directly; only changes in internal energy (ΔU) are meaningful in thermodynamic analysis.

1st Law of Thermodynamics

Conservation of Energy

The First Law of Thermodynamics states that energy cannot be created or destroyed, only transformed from one form to another. The total energy of an isolated system remains constant.

• Mathematical Statement:

$\Delta (\text{energy of system}) + \Delta (\text{energy of surroundings}) = 0$

• Implication: Any energy lost by the system is gained by the surroundings, and vice versa.

Energy Balance for Closed Systems

Application of the First Law

For a closed system (no mass enters or leaves), the change in internal energy equals the net energy transferred as heat and work.

• Equation:

$\Delta U = Q + W$

• Q: Heat added to the system

• W: Work done on the system

Reversible Process

Characteristics and Significance

A reversible process is an idealized thermodynamic process that occurs infinitely slowly, allowing the system to remain in equilibrium at all times.

• Is frictionless and never more than differentially removed from equilibrium.

• Traverses a succession of equilibrium states.

• Driven by infinitesimal differences in driving forces (e.g., pressure, temperature).

• Can be reversed at any point by a differential change in external conditions.

• When reversed, retraces its forward path, restoring the initial state of system and surroundings.

• In reality, all natural processes are irreversible, but reversible processes are useful for theoretical analysis.

Equilibrium

Definition and Criteria

Equilibrium is a static condition where there is no net tendency for change within the system.

• Absence of driving forces: No gradients in temperature, pressure, or chemical potential.

• Mechanical, thermal, and chemical equilibrium must all be satisfied.

• For a system in equilibrium, the number of independent variables (degrees of freedom) must be fixed to establish its intensive state.

Phase Rule

Gibbs Phase Rule

The Gibbs Phase Rule provides a relationship between the number of phases, components, and degrees of freedom in a system at equilibrium.

• Formula:

$F = 2 - \pi + N$

• F: Degrees of freedom (number of intensive variables that can be changed independently without changing the number of phases in equilibrium)

• π: Number of phases present

• N: Number of chemical species (components)

Example Applications

• Liquid water in equilibrium with its vapor: $F = 2 - 2 + 1 = 1$

• Liquid water in equilibrium with a mixture of water vapor and nitrogen: $F = 2 - 2 + 2 = 2$

• Liquid solution of alcohol in water in equilibrium with its vapor: $F = 2 - 2 + 2 = 2$

Enthalpy

Definition and Equations

Enthalpy (H) is a thermodynamic property defined as the sum of the internal energy and the product of pressure and volume.

• Definition:

$H \equiv U + PV$

• Differential form:

$dH = dU + d(PV)$

• Upon integration:

$\Delta H = \Delta U + \Delta (PV)$

Heat Capacity

Definition and Types

Heat capacity is the amount of heat required to change the temperature of a body by one degree. It is an important property for understanding how substances respond to heat transfer.

• At constant volume:

$C_v = \left( \frac{\partial U}{\partial T} \right)_V$

• At constant pressure:

$C_p = \left( \frac{\partial H}{\partial T} \right)_P$

• The greater the heat capacity, the smaller the temperature change for a given quantity of heat transferred.

Constant-Volume and Constant-Pressure Processes

Equations for Reversible Processes

• At constant volume (V):

$dU = C_v dT$

$Q = nC_v \int_{T_1}^{T_2} C_v dT$

• At constant pressure (P):

$dH = C_p dT$

$Q = nC_p \int_{T_1}^{T_2} C_p dT$

• These equations are used to calculate heat transfer in processes where either volume or pressure is held constant.

Example: Ideal Gas Processes

Application of Thermodynamic Principles

Consider an ideal gas (where $PV/T$ is constant) undergoing a series of processes. The following example illustrates the calculation of work, heat, and changes in internal energy and enthalpy for different process paths.

• Given: 1 mole of air, $V = 0.0247$ m3, $T = 298.15$ K, $P = 1$ bar.

• Processes:

  1. Cooling at constant pressure followed by heating at constant volume

  2. Heating at constant volume followed by cooling at constant pressure

• Key Equations Used:

$Q = nC_v (T_2 - T_1)$ (at constant V) $Q = nC_p (T_2 - T_1)$ (at constant P) $\Delta U = Q$ (at constant V, for reversible process) $\Delta H = Q$ (at constant P, for reversible process)

• Result: The property changes (ΔU and ΔH) for the overall change in state are the same for both paths, but the heat (Q) and work (W) depend on the path taken.

Summary Table: Key Thermodynamic Quantities

Quantity	Symbol	Definition/Formula	Units
Internal Energy	U	Microscopic energy of molecules	Joules (J)
Enthalpy	H	$H = U + PV$	Joules (J)
Heat Capacity (at constant V)	$C_v$	$C_v = \left( \frac{\partial U}{\partial T} \right)_V$	J/K or J/(mol·K)
Heat Capacity (at constant P)	$C_p$	$C_p = \left( \frac{\partial H}{\partial T} \right)_P$	J/K or J/(mol·K)
Degrees of Freedom	F	$F = 2 - \pi + N$	Dimensionless

References

• Smith, J.M., Van Ness, H.C., Abbott, M.M. (2005). Introduction to Chemical Engineering Thermodynamics, 7th ed. NY: McGraw Hill.

Additional info: These notes are foundational for physical chemistry and chemical engineering, and while not specific to organic chemistry, thermodynamics is essential for understanding reaction energetics, phase behavior, and equilibrium in organic systems.