KAHIBARO
Table of Contents
Energy Inside Matter
When we heat a cup of water, warm a metal rod, or cool a gas, energy is being transferred and redistributed inside matter. Two central ideas help describe this process, heat and internal energy. They are closely related, but they are not the same thing.
Internal energy is the energy stored within a system because of the microscopic motion and interactions of its particles. Heat is energy transferred between systems because of a temperature difference.
A system can gain or lose energy by heating, and that changes its internal energy. But heat is not something a body simply contains in the same way that it contains mass. Heat refers to energy in transit.
Internal energy is energy stored in a system.
Heat is energy transferred into or out of a system because of a temperature difference.
A system does not "contain heat". It contains internal energy.
What Internal Energy Means
Matter is made of atoms and molecules that are always moving, vibrating, rotating, and interacting with one another. Even an object that looks perfectly still has enormous microscopic activity inside it.
The internal energy, usually written as $U$, is the total microscopic energy of the system. This includes the kinetic energy of random molecular motion and the potential energy associated with intermolecular forces and atomic arrangements.
For example, in a gas, molecules move in many directions with many different speeds. Their random kinetic energy contributes strongly to internal energy. In a solid, atoms vibrate around fixed positions, and both their vibrational kinetic energy and interaction potential energy contribute to internal energy.
Internal energy does not include every kind of energy the object may have. If a box slides across the floor, its bulk motion gives it kinetic energy, but that is not part of its internal energy. Likewise, if the box is high above the ground, it has gravitational potential energy, but that also is not part of its internal energy.
Microscopic Picture
A useful way to think about internal energy is to separate motion into two kinds. One is the motion of the object as a whole. The other is the random motion of particles inside it.
If a train moves at high speed, the train has macroscopic kinetic energy. But the air inside the train, the metal in the wheels, and the passengers' bodies all also have internal energy due to microscopic particle motion. These are different forms of energy.
For an ideal gas, the internal energy depends only on the random kinetic energy of its molecules. That means it depends only on temperature, not on volume or pressure by themselves.
For an ideal gas, internal energy depends only on temperature.
What Heat Means
Heat is the transfer of energy from one system to another because they have different temperatures. If a hot object touches a cold object, energy flows from the hotter one to the colder one. That transferred energy is heat.
Suppose a hot spoon is placed in cool water. The spoon loses energy, the water gains energy, and the transfer occurs because of the temperature difference. We describe that transfer as heat flowing from the spoon to the water.
The symbol often used for heat is $Q$. If energy enters a system by heating, $Q$ is positive for that system. If energy leaves the system by heating, $Q$ is negative.
This sign convention is useful when tracking energy changes.
Internal Energy and Heat Are Different
Because both heat and internal energy are measured in joules, beginners often confuse them. The difference is easier to see with an example.
Imagine a metal block sitting on a table.
If the block is hot, it has a large internal energy compared with a colder state.
If the block touches an ice cube, energy is transferred from the block to the ice cube. That transfer is heat.
So internal energy is a property of the state of the system, while heat describes a process.
| Quantity | Meaning | Symbol | Is it stored in the system? |
|---|---|---|---|
| Internal energy | Microscopic energy inside the system | $U$ | Yes |
| Heat | Energy transferred due to temperature difference | $Q$ | No |
Internal energy is a state quantity.
Heat is a transfer quantity.
Do not use them interchangeably.
Changing Internal Energy
A system's internal energy can change in different ways. In this chapter, the key idea is that heating changes internal energy. If energy enters the system as heat, the internal energy usually increases. If energy leaves as heat, the internal energy usually decreases.
In simple situations where no other kind of energy transfer is involved, the change in internal energy is equal to the heat added:
$$
\Delta U = Q
$$
This relation is especially useful when we focus only on heating and cooling without discussing work yet.
For example, if $500 \ \text{J}$ of heat is added to a substance and no other energy transfer occurs, then its internal energy increases by $500 \ \text{J}$.
If $200 \ \text{J}$ of heat leaves the substance, then
$$
Q = -200 \ \text{J}
$$
and
$$
\Delta U = -200 \ \text{J}
$$
so the internal energy decreases by $200 \ \text{J}$.
In situations where heating is the only energy transfer,
$$
\Delta U = Q
$$
Positive $Q$ means energy enters the system.
Negative $Q$ means energy leaves the system.
Relation to Temperature
Temperature and internal energy are related, but they are not identical.
Temperature tells us how hot or cold a system is, and microscopically it is related to the average kinetic energy of particles. Internal energy includes that microscopic kinetic energy, but it can also include microscopic potential energy.
Because of this, a change in internal energy often changes temperature, but not always in the same way for every substance. Also, some processes can change internal energy without changing temperature, especially during phase changes, which are treated elsewhere.
As a basic rule, when a substance is heated and remains in the same phase, its internal energy usually increases and its temperature usually rises.
Examples from Everyday Life
A pan on a stove gains energy by heat transfer from the burner. The internal energy of the pan increases, so its temperature rises.
An ice cube in warm air gains heat from the surroundings. Its internal energy increases.
A hot drink left on a table loses heat to the cooler room. Its internal energy decreases.
Rubbing your hands together can make them warmer, but that situation involves more than heating by temperature difference alone. It brings in other energy transfer mechanisms, which belong to later thermodynamics topics.
A Simple Particle Model
Consider two objects, one hot and one cold, placed in contact. In the hotter object, particles on average have greater microscopic energy. Through collisions and interactions at the boundary, energy is transferred to the colder object. This continues until thermal equilibrium is reached.
Thermal equilibrium means there is no net heat transfer between them. At that point, they have the same temperature.
The arrow does not mean a substance called heat is stored in the hot object. It means energy is being transferred from one system to the other.
Units
Both heat and internal energy are forms of energy, so both are measured in joules, abbreviated $\text{J}$.
$$
1 \ \text{J} = 1 \ \text{N} \cdot \text{m}
$$
In thermal physics, other units are sometimes used in practice, such as calories, but the SI unit is the joule.
A calorie is defined historically as the amount of energy needed to raise the temperature of a certain amount of water by a certain amount. In SI form,
$$
1 \ \text{cal} \approx 4.186 \ \text{J}
$$
Comparing Key Ideas
| Idea | Description |
|---|---|
| Internal energy $U$ | Energy stored microscopically in a system |
| Heat $Q$ | Energy transferred because of temperature difference |
| Heating | A process that can change internal energy |
| Thermal equilibrium | State in which there is no net heat transfer |
A First Energy Balance View
A good habit in thermodynamics is to define the system clearly. Once the system is chosen, ask whether energy is entering or leaving it.
If the system absorbs heat, then $Q > 0$ and its internal energy increases, provided no other energy transfer needs to be considered.
If the system releases heat, then $Q < 0$ and its internal energy decreases.
For example, let the system be a cup of tea. If the tea cools by giving $150 \ \text{J}$ to the room, then for the tea,
$$
Q = -150 \ \text{J}
$$
and in the simple picture of this chapter,
$$
\Delta U = -150 \ \text{J}
$$
Common Misconceptions
A very common mistake is to say that an object "has heat inside it." A better statement is that the object has internal energy.
Another common mistake is to assume that a high temperature always means a large internal energy. Temperature alone does not tell the whole story. A tiny spark can have a very high temperature but very little total internal energy because it contains very little matter. A large bucket of warm water can have much more internal energy overall.
High temperature does not always mean large total internal energy.
Internal energy depends on both the microscopic energy per particle and the amount and nature of the substance.
Final Picture
Heat and internal energy are foundational ideas in thermodynamics. Internal energy is the microscopic energy contained in matter. Heat is the transfer of energy caused by a temperature difference. Heating changes internal energy, and in simple cases,
$$
\Delta U = Q
$$
Understanding this distinction is essential for everything that follows in thermodynamics.