Internal Energy — Core Principles

Core Principles

Internal energy ($U$) is the total energy stored within a thermodynamic system at the microscopic level, excluding the system's bulk kinetic and potential energies. It comprises the kinetic energies of molecular motion (translational, rotational, vibrational) and the potential energies from intermolecular forces, chemical bonds, and electronic configurations.

Internal energy is a state function, meaning its value depends only on the system's current state (e.g., temperature, pressure, volume) and not on the path taken to reach that state. The First Law of Thermodynamics defines the change in internal energy ($Delta U$) as the sum of heat ($q$) added to the system and work ($w$) done on the system: $Delta U = q + w$.

For processes at constant volume, $Delta U = q_v$. For ideal gases, internal energy depends solely on temperature, expressed as $Delta U = n C_v Delta T$. Understanding internal energy is crucial for analyzing energy transformations in chemical reactions and physical processes.

Important Differences

vs Enthalpy ($H$)

Aspect	This Topic	Enthalpy ($H$)
Definition	Internal Energy ($U$): Total energy contained within a system, excluding bulk kinetic/potential energy.	Enthalpy ($H$): Defined as $H = U + PV$, where $P$ is pressure and $V$ is volume. It accounts for internal energy plus the energy required to make space for the system at constant pressure.
Primary Use Case	Change in internal energy ($ Delta U $) is equal to heat exchanged at constant volume ($q_v$). Relevant for bomb calorimetry.	Change in enthalpy ($ Delta H $) is equal to heat exchanged at constant pressure ($q_p$). Relevant for most chemical reactions in open containers.
Mathematical Relation (First Law)	$ Delta U = q + w $ (general form)	$ Delta H = Delta U + Delta (PV) $. For constant pressure, $ Delta H = Delta U + P Delta V $.
Dependence for Ideal Gas	Depends only on temperature ($ Delta U = n C_v Delta T $).	Depends only on temperature ($ Delta H = n C_p Delta T $).
Measurement	Measured directly as $q_v$ in a bomb calorimeter.	Measured directly as $q_p$ in a coffee-cup calorimeter.

Internal energy ($U$) and enthalpy ($H$) are both state functions crucial in thermodynamics, but they serve different purposes depending on the conditions of a process. Internal energy represents the total microscopic energy within a system, and its change ($ Delta U $) directly equals the heat exchanged at constant volume ($q_v$). Enthalpy, defined as $H = U + PV$, is particularly useful for processes occurring at constant pressure, where its change ($ Delta H $) equals the heat exchanged ($q_p$). While $ Delta U $ accounts for energy changes within the system, $ Delta H $ also includes the energy associated with the work of expansion or compression against the surroundings at constant pressure. For ideal gases, both $U$ and $H$ depend solely on temperature.