Internal Energy

Definition and Concept

Thermodynamic Internal Energy

Internal energy (U): total microscopic energy contained in a thermodynamic system. Includes kinetic and potential energies of molecules, atoms, ions. Excludes macroscopic kinetic and potential energy.

System Perspective

System: defined volume or mass under study. Internal energy is property of system's internal configuration and motion. Independent of environment except via energy exchange.

Significance

Central to energy conservation, thermodynamic analysis, phase changes, chemical reactions. Basis for first law formulations and state property calculations.

"Internal energy is the sum of all energies associated with the microscopic components of a system." -- Callen, Thermodynamics

First Law of Thermodynamics

Energy Conservation Principle

First law: energy cannot be created or destroyed, only transformed. Change in internal energy equals net heat added minus net work done.

Mathematical Expression

ΔU = Q - W, where ΔU: change in internal energy, Q: heat added, W: work done by system.

Implications

Quantifies energy exchange in chemical, physical, biological processes. Establishes internal energy as central thermodynamic state function.

"The internal energy change of a system is equal to the heat supplied to the system minus the work done by the system." -- Mayer

Components of Internal Energy

Kinetic Energy of Particles

Translational: linear motion of molecules. Rotational: spinning about axes. Vibrational: oscillations of atoms in molecules.

Potential Energy of Particles

Intermolecular forces: van der Waals, electrostatic interactions. Chemical bonds: bond energies, electronic configurations.

Nuclear Energy

Energy stored in atomic nuclei. Typically negligible in chemical thermodynamics but essential in nuclear reactions.

Internal Energy as a State Function

Definition of State Function

Property depends only on current system state, not path taken. Internal energy depends on variables like temperature, pressure, volume.

Implications for Thermodynamics

Energy changes depend on initial and final states alone. Enables calculation of energy changes via state variables.

Contrast with Path Functions

Heat and work: path dependent. Internal energy: path independent. Facilitates use in energy balance equations.

"Internal energy, being a state function, is uniquely defined by the thermodynamic state of the system." -- Atkins

Measurement and Units

Units of Internal Energy

SI unit: joule (J). Also calorie (cal), electronvolt (eV) used at molecular scale.

Calorimetry Techniques

Measure heat transfer at constant volume or pressure to infer ΔU. Bomb calorimeter for constant volume measurements.

Indirect Measurement Methods

Calculate from thermodynamic tables, equations of state, spectroscopy data.

Energy Changes in Processes

Isothermal Processes

Temperature constant. ΔU = 0 for ideal gases. Energy exchange via heat and work balanced.

Adiabatic Processes

No heat exchange (Q=0). ΔU = -W. Internal energy changes solely from work.

Isochoric and Isobaric Processes

Constant volume: W=0, ΔU=Q. Constant pressure: ΔH=Q, relates to ΔU via ΔH=ΔU+PΔV.

Isothermal: ΔU = 0Adiabatic: ΔU = -WIsochoric: ΔU = Q (W=0)Isobaric: ΔH = Q = ΔU + PΔV

Relation to Heat and Work

Heat (Q)

Energy transfer due to temperature difference. Increases or decreases internal energy depending on direction.

Work (W)

Energy transfer via force acting over distance. Includes expansion, compression, electrical work.

Sign Conventions

Q positive when heat added to system. W positive when system does work on surroundings.

"Internal energy changes represent the net effect of heat added and work done." -- Zemansky

Mathematical Formulation

General Differential Form

dU = δQ - δW. Exact differential for U, inexact for heat and work.

For Ideal Gas

U depends only on temperature. dU = nCv dT, where Cv is molar heat capacity at constant volume.

Thermodynamic Potentials

Internal energy related to enthalpy (H), Helmholtz (A), Gibbs (G) energies by Legendre transforms.

dU = TdS - PdVU = U(S,V)For ideal gas: ΔU = nCvΔT

Applications in Thermodynamics

Chemical Reactions

Calculate reaction enthalpies and internal energy changes. Basis for enthalpy of formation.

Phase Transitions

Analyze latent heat, energy required for phase changes. Internal energy changes reflect molecular rearrangements.

Engine Cycles

Evaluate work output and efficiency. Internal energy changes track energy conversion in cycles like Carnot, Otto.

Limitations and Assumptions

Classical Assumptions

Neglects relativistic effects, nuclear energy except in nuclear reactions. Assumes macroscopic equilibrium.

Non-Equilibrium Systems

Internal energy poorly defined in transient or non-uniform systems. Requires statistical mechanics for microscopic interpretation.

Quantum Effects

At low temperatures or small scales, discrete energy levels affect internal energy calculations.

Experimental Techniques

Calorimetry

Direct measurement of heat changes to determine ΔU. Types include constant volume and constant pressure calorimeters.

Spectroscopy

Energy level transitions provide insight into molecular internal energy states.

Thermodynamic Tables

Tabulated values of U from experiments and theoretical calculations used for engineering and scientific purposes.

References

• Callen, H.B., Thermodynamics and an Introduction to Thermostatistics, 2nd ed., Wiley, 1985, pp. 101-130.
• Atkins, P.W., Physical Chemistry, 10th ed., Oxford University Press, 2014, pp. 150-175.
• Zemansky, M.W., Heat and Thermodynamics, 7th ed., McGraw-Hill, 1997, pp. 45-60.
• Reif, F., Fundamentals of Statistical and Thermal Physics, McGraw-Hill, 1965, pp. 220-245.
• Moran, M.J., Shapiro, H.N., Fundamentals of Engineering Thermodynamics, 7th ed., Wiley, 2010, pp. 80-110.