Interactive physics simulator
Internal Energy: U = KEmicro + PEmicro
Explore the microscopic energy stored within matter. Analyze how heat transfer and mechanical work alter particle motion, observe molecular phase changes, and study the First Law of Thermodynamics in real-time.
Microscopic Energy Lab
Interact with heating elements, piston volume, and friction strokes to observe how energy transfers into microscopic particle kinetic and potential states.
Live Telemetry
- Water Temp
- 20.0 °C
- Heat Added (Q)
- 0 J
- Microscopic KE
- 100 J
- Microscopic PE
- 400 J
- Internal Energy (U)
- 500 J
- Water State
- Liquid
- Volume (V)
- 5.0 L
- Pressure (P)
- 1.00 atm
- Temperature (T)
- 300.0 K
- Heat Input (Q)
- 0 J
- Work Done (W)
- 0 J
- Change (ΔU)
- 0 J
- Total energy (U)
- 370 J
- Block Temp
- 20.0 °C
- Rubbing Rate
- 0.0 Hz
- Work Input (W)
- 0 J
- Heat Loss (Q)
- 0 J
- Vibration Amp
- 0.05 nm
- Internal Energy (U)
- 1050 J
What is Internal Energy?
Internal Energy (symbolized by U) is the total thermodynamic energy stored within a system at the microscopic scale. It is the sum of the random kinetic energy and the potential energy of all the atoms, molecules, or particles that make up the substance.
Crucially, internal energy does not include the macroscopic kinetic energy of the system moving as a whole through space, nor does it include macroscopic potential energy due to external force fields (like the height of a beaker on a shelf). It is strictly the energy stored internally and microscopically inside the material boundary:
Microscopic Components of U
At the molecular level, internal energy is composed of two primary modes:
- Microscopic Kinetic Energy (KEmicro): The energy of motion. In liquids and gases, this includes molecules translating through space and rotating. In solids, it consists of atoms vibrating back and forth about their fixed lattice sites. Microscopic kinetic energy correlates directly with the macroscopic temperature of the substance.
- Microscopic Potential Energy (PEmicro): The energy of position and bonding. This is the energy associated with the attractive forces between molecules (intermolecular forces, like hydrogen bonds) and chemical bonds within molecules. Potential energy changes during phase transformations (melting, boiling) or chemical reactions.
The First Law of Thermodynamics
We cannot easily measure the absolute value of internal energy, but we can measure how it changes. The **First Law of Thermodynamics** (which states that energy is conserved) defines the change in internal energy (ΔU) as:
Where Q is the heat added to the system and W is the work done by the system on its surroundings. If heat is added (Q > 0) or work is done on the system (W < 0), the internal energy increases. If work is done by the system (W > 0) or heat leaves (Q < 0), the internal energy decreases.
Temperature vs. Internal Energy
While temperature and internal energy are related, they are not the same thing:
- Temperature measures only the average microscopic kinetic energy of the particles. It does not depend on the amount of substance or the bond potential energy.
- Internal Energy is an extensive property that depends on the mass (amount of substance) and accounts for both kinetic and potential energies. For example, a boiling beaker of water at 100°C has the same temperature as a single drop of boiling water, but the beaker has vastly more internal energy.
- During a phase change (like water boiling in the beaker tab), the temperature remains locked at 100°C, but the internal energy increases because the added heat goes into breaking bonds (increasing potential energy).
Solved Examples
A thermodynamic system absorbs 850 Joules of heat from its surroundings. At the same time, the system expands and performs 320 Joules of work on its surroundings. Calculate the change in the internal energy of the system.
- Identify the given values: Heat added to the system, Q = +850 J. Work done by the system, W = +320 J.
- State the First Law of Thermodynamics equation: ΔU = Q - W.
- Substitute the values into the equation: ΔU = 850 J - 320 J.
- Perform the subtraction: ΔU = 530 Joules.
- Conclude the result: The internal energy of the system increases by 530 Joules. The positive sign indicates that the thermal energy absorbed exceeded the mechanical work done.
Answer: ΔU = +530 J
A sample of gas is trapped inside a cylinder with a movable piston. An external force performs 450 Joules of work to compress the gas, while the cylinder is wrapped in insulation so that no heat can enter or leave (an adiabatic process). Find the change in the gas's internal energy and explain what happens to its temperature.
- Identify the given values: Since the process is adiabatic (insulated), heat transfer Q = 0 J. Work is done ON the system (compression), meaning work done by the system W = -450 J.
- Recall the First Law of Thermodynamics: ΔU = Q - W.
- Substitute the values: ΔU = 0 J - (-450 J).
- Solve the expression: ΔU = +450 Joules.
- Interpret the temperature change: Because the internal energy increases (+450 J) and the gas is ideal, this microscopic energy goes entirely into increasing the average kinetic energy of the molecules. Therefore, the temperature of the gas rises.
Answer: ΔU = +450 J (Temperature increases)
A beaker containing 10.0 grams of liquid water at 100°C absorbs 22,600 Joules of heat to completely vaporize into steam at 100°C. During this process, the steam expands against atmospheric pressure, doing 1,700 Joules of work. Calculate: (a) the change in internal energy, and (b) how much of this energy change is used to break molecular bonds versus pushing back the atmosphere.
- Identify the given parameters: Mass of water m = 10 g, Heat added Q = +22,600 J (latent heat of vaporization), Expansion work done W = +1,700 J.
- Calculate the total change in internal energy (ΔU) using the First Law: ΔU = Q - W = 22,600 J - 1,700 J = 20,900 J.
- Understand the microscopic significance: The internal energy change (ΔU = 20,900 J) represents the energy stored inside the water molecules. Since the temperature remains constant at 100°C, the average kinetic energy of the molecules is unchanged.
- Determine bond-breaking energy: The entire 20,900 J goes into increasing the microscopic potential energy (breaking the hydrogen bonds holding the liquid water molecules together). The remaining 1,700 J is lost to the environment as expansion work pushing the atmosphere.
Answer: ΔU = +20,900 J (Used entirely for breaking intermolecular bonds)
Common Mistakes
- Confusing U with macroscopic motion: Thinking that a water beaker's internal energy increases when you lift it or carry it across a room. Kinetic/potential energy of the container as a whole does not affect internal molecular bonds or random microscopic vibration.
- Assuming U is zero at 0°C or 0 Kelvin: Water molecules at 0°C still have translational and rotational kinetic energy, plus massive bond potential energy. Even at absolute zero (0 K), quantum mechanics guarantees a residual vibrational ground-state energy (zero-point energy).
- Incorrect signs for Work (W): Forgetting that work done on a gas (compression) reduces the volume, meaning work done by the gas is negative (W < 0), which makes the change in internal energy positive: ΔU = Q - (-W) = Q + Won.
Friction Work Conversion
When you slide a rough pad over a metal block, macroscopic mechanical work is performed against the force of friction. At the contact surface, microscopic collisions between surface atoms act like small hammers, exciting the atoms inside the crystal lattice.
This causes them to vibrate with greater amplitudes around their equilibrium positions. This mechanical work is converted directly into microscopic vibrational kinetic energy, raising the block's temperature and internal energy, as demonstrated in our infrared friction tab.
Practice Questions
1. If a system undergoes a complete cyclic process (returning to its exact initial state), what is the total change in its internal energy? Why?
The total change in internal energy is zero (ΔU = 0). Internal energy is a state function, meaning its value depends solely on the current state (temperature, pressure, volume, composition) of the system, not on the history or path taken. Since the initial and final states are identical in a cycle, the difference is zero.
2. Explain why temperature does not change when ice melts at 0°C, even though heat is continually added. What happens to the internal energy?
When ice melts, the added heat energy is not used to speed up the molecules (which would raise the temperature). Instead, it is used to break the rigid hydrogen bonds in the ice lattice, transforming it into liquid water. This increases the molecules' microscopic potential energy. As a result, the internal energy increases, while the temperature (microscopic kinetic energy) remains constant at 0°C.
3. A gas is allowed to expand into an evacuated chamber (a vacuum) without any heat exchange (free expansion). Does the internal energy of the gas change? Does its temperature drop?
In free expansion into a vacuum, the gas does no work because there is no external pressure to push against (W = 0). Since the chamber is insulated, no heat is exchanged (Q = 0). According to the First Law (ΔU = Q - W), ΔU = 0. For an ideal gas, since internal energy does not change, the temperature also remains completely constant.
4. What is the difference between thermal energy and internal energy?
Thermal energy is a subcomponent of internal energy, representing specifically the random, microscopic kinetic energy of the particles (translations, rotations, and vibrations) that scales directly with temperature. Internal energy is the complete microscopic energy, consisting of this thermal kinetic energy plus the microscopic potential energy stored in chemical bonds and intermolecular forces.
FAQ
Frequently Asked Questions
What is internal energy?
Internal energy (U) is the total energy stored within a thermodynamic system at the microscopic scale. It is the sum of the microscopic kinetic energies of the particles (due to translation, rotation, and vibration) and their microscopic potential energies (due to chemical bonds and intermolecular forces).
What is the formula for internal energy?
While absolute internal energy is difficult to calculate, the change in internal energy is defined by the First Law of Thermodynamics: ΔU = Q - W, where Q is heat added and W is work done by the system.
How is internal energy different from heat and temperature?
Temperature measures the average kinetic energy of the particles. Heat is the transfer of thermal energy due to a temperature difference. Internal energy is the total microscopic energy currently stored inside the system.
What is the SI unit of internal energy?
The SI unit of internal energy is the Joule (J), which is the standard unit for all energy and work in physics.
Is internal energy a state function or a path function?
Internal energy is a state function. Its value depends only on the current state of the system (such as temperature and pressure) and is completely independent of the path or process used to reach that state. In any closed cycle, the net change in internal energy is zero (ΔU = 0).
Can a system have internal energy at absolute zero?
Yes. According to quantum mechanics, particles at absolute zero (0 Kelvin) retain a non-zero ground-state energy known as zero-point energy. Thus, a system's internal energy approaches a minimum constant value, but never reaches absolute zero.
What is the difference between internal energy and thermal energy?
Thermal energy is specifically the kinetic energy of particles due to their random motion, which increases with temperature. Internal energy is the sum of this thermal kinetic energy plus the potential energy associated with chemical bonds and intermolecular attractions.
How does compressing a gas affect its internal energy?
Compressing a gas means performing mechanical work on it (W < 0). In an adiabatic compression (where no heat is exchanged, Q = 0), the work done directly increases the internal energy of the gas, causing its temperature to rise.
Why does water's internal energy increase during boiling if temperature remains constant?
During boiling, heat is absorbed (Q > 0) but temperature stays at 100°C because particle speeds do not increase. Instead, this energy (latent heat) is used to break intermolecular hydrogen bonds, which increases the microscopic potential energy, raising the total internal energy.
What are the microscopic components of internal energy?
The microscopic components are kinetic energy (from particle translation, rotation, and atomic vibration) and potential energy (from chemical bonds, atomic interactions, and intermolecular forces).
What is an ideal gas's internal energy?
For a monatomic ideal gas, there are no intermolecular forces, meaning microscopic potential energy is zero. Therefore, its internal energy depends entirely on temperature and is given by U = (3)/(2)nRT.
What happens to internal energy in an isothermal process?
In an isothermal process, the temperature remains constant. For an ideal gas, because U depends only on temperature, the change in internal energy is zero (ΔU = 0). This means any heat added is fully converted into work done by the gas (Q = W).