Internal energy Knowledge-Based Study

Study notes

1. What is Internal Energy?

   In thermal physics, the internal energy ($U$) of a system is the total energy stored within its microscopic components. Every substance is made of atoms or molecules that are in perpetual motion and subject to intermolecular forces.

   Internal energy is defined as the sum of two microscopic contributions:

   • Microscopic kinetic energy: The energy associated with the random translational, rotational, and vibrational motions of individual particles.
   • Microscopic potential energy: The energy stored in the mutual interactions and attractive bonds between particles.

   It is vital to distinguish internal energy from macroscopic mechanical energy. A moving container of gas has macroscopic kinetic energy due to its overall velocity and gravitational potential energy due to its position; however, these bulk mechanical forms are not part of the gas's internal energy.

   How Phase and Interactions Influence Internal Energy

   The balance between microscopic kinetic and potential energy varies significantly depending on the state of matter:

   • Solids: Particles oscillate about fixed positions. Intermolecular forces are strong, meaning microscopic potential energy makes a significant contribution alongside vibrational kinetic energy.
   • Liquids: Particles move past one another while remaining bound by strong electrostatic interactions, possessing both kinetic and significant intermolecular potential energy.
   • Gases: Particles are far apart and move rapidly. In an ideal gas, intermolecular forces are assumed to be negligible except during brief elastic collisions. Consequently, the internal energy of an ideal gas consists entirely of the kinetic energy of its particles.

   For a monatomic ideal gas of $n$ moles, the internal energy depends solely on temperature:

   $$U = \frac{3}{2}nRT$$

   where $R$ is the universal gas constant and $T$ is the absolute temperature in kelvins.

   Modifying the Internal Energy of a System

   Internal energy is a state function; its value depends strictly on the current state of the system (such as its temperature, pressure, and volume), not on the historical pathway taken to reach that state.

   The internal energy of a closed system changes when energy is transferred across its boundary. According to the First Law of Thermodynamics:

   $$\Delta U = Q - W$$

   • $\Delta U$: Change in internal energy ($\text{J}$).
   • $Q$: Heat added to the system ($Q > 0$ when absorbed, $Q < 0$ when released).
   • $W$: Work done by the system on its surroundings ($W > 0$ during expansion, $W < 0$ during compression).

   Note: When work is defined as work done on the system ($W_{\text{on}}$), the relationship is written as $\Delta U = Q + W_{\text{on}}$.

   Distinguishing Heat, Temperature, and Internal Energy

   A common error in thermal physics is treating heat, temperature, and internal energy interchangeably:

   • Temperature ($T$): A measure of the average translational kinetic energy per particle in a substance.
   • Heat ($Q$): Energy in transit between systems due to a temperature difference. A body does not "contain" heat; it contains internal energy.
   • Internal Energy ($U$): The total microscopic kinetic and potential energy of all particles in the system.

   For example, a large glacier at $0\text{ }^\circ\text{C}$ has a lower temperature than a hot cup of tea at $90\text{ }^\circ\text{C}$, but the glacier has a substantially greater internal energy because of its enormous mass and total number of particles.