understand that internal energy is determined by the state of the system and that it can be expressed as the sum of a random distribution of kinetic and potential energies associated with the molecules of a system

What is Internal Energy?

Internal energy, denoted by \$U\$, is the total energy stored inside a system.

It comes from two sources:

• Kinetic energy – the random motion of molecules (like dancers moving around).
• Potential energy – the forces between molecules (like the tension in a stretched spring).

Because the molecules are moving and interacting in a random way, \$U\$ can be written as a sum over all molecules:

\$U = \sumi \frac{1}{2} mi vi^2 + \sum{i

Internal Energy is a State Function 🏁

A state function depends only on the current state of the system, not on how it got there.

Think of it like the current temperature of a cup of tea – it tells you the tea’s state, no matter whether it was boiled, cooled, or stirred.

1. Change the pressure of a gas in a piston – the internal energy changes because the state (pressure, volume, temperature) changes.
2. Heat the gas – again, the state changes, so \$U\$ changes.
3. Do work on the gas – the state changes, so \$U\$ changes.

Key point: The path taken (how you changed the state) does not matter for \$U\$.

Random Distribution Analogy 🎲

Picture a crowded dance floor:

• Each dancer’s speed = kinetic energy.
• The distance between dancers and how they push or pull on each other = potential energy.
• The total excitement on the floor = internal energy.

Just like the dance floor’s excitement depends on how many dancers and how they interact, a gas’s internal energy depends on the number of molecules and their interactions.

Example: Heating a Cup of Tea ☕

When you heat tea, you add heat energy \$Q\$.

Some of that energy increases the kinetic energy of the water molecules (raising the temperature), and some changes the potential energy as the molecules move slightly farther apart.

The total change in internal energy is:

\$\Delta U = Q - W\$

(where \$W\$ is work done, e.g., by the tea expanding against atmospheric pressure).

Exam Tips 📚

• Remember that \$U\$ is a state function – you can use the first law of thermodynamics: \$\Delta U = Q - W\$.
• When a process is reversible, \$W\$ is the integral of pressure over volume: \$W = \int{Vi}^{V_f} P\,dV\$.
• For an ideal gas, use \$U = \frac{3}{2} nRT\$ (only kinetic part) and remember that \$U\$ does not depend on pressure.
• Practice drawing energy diagrams: label kinetic, potential, and total internal energy.

Quick Summary 📌

Internal energy is the sum of all random kinetic and potential energies of molecules.

It is a state function, so it depends only on the current state (temperature, pressure, volume) and not on how the system arrived there.

Use the first law of thermodynamics to relate changes in internal energy to heat added and work done.