The terms "thermal energy" and "internal energy" mean the same thing. "Thermal energy" is more colloquial while "internal energy" is the technical term.

In terms of the molecular model of matter, the internal energy of a body or system is the internal mechanical energy of the molecules constituting the body or system. If a body is at rest, this "internal mechanical energy" of the molecules is the sum the kinetic energy of the molecules and the potential energy of interaction between the molecules.

If a body has "macroscopic" motion, which could be translational motion or rotational motion or both, not all of the kinetic energy of its molecules contributes to the internal energy. E.g., if the body is carrying out translational motion, the macroscopic kinetic energy associated with such motion, equal to M/2 V², must be subtracted from the total kinetic energy of the molecules to get the internal kinetic energy. Here M is the body's mass and V the speed of the body's center of mass.

The molecular model cannot be used to measure the internal energy of a body because one never knows the positions and velocities of the molecules inside the body. One needs a macroscopic definition of the internal energy to be able to measure internal energy.

Macroscopically, the internal energy of a body or system is defined as follows. The definition assumes that the body or system is thermally isolated from its environment. To imagine thermal isolation, think of a thick padding of styrofoam surrounding the body or system. (In a careful theoretical development, one needs to define thermal isolation differently.) Then only work-like interactions between the body or system and its environment are possible.

Suppose an amount of work W is done on the body or system in a work-like interaction with the environment and that the body or system is at rest, both before and after this interaction has process, then the change DU in the body's or system's internal energy resulting from the interaction is defined as

DU = W,if body at rest.(1)

Examples of work-like interaction are rubbing the surface of an object, stirring a liquid (if the system consists of a container with a liquid inside it), and compressing a gas in a cylinder.

Comment 1. Underlying this definition of internal energy is the principle of energy conservation. The principle implies that if energy is transferred to a system by means of work, then it must be possible to define an energy for the system such that the system's energy increases by an amount equal to the energy transferred.

Comment 2. Eq.(1) defines the change in internal energy of the body or system in a process that takes the body or system from an initial state to a final state. One is free to assign an arbitrary value to the internal energy for a suitable initial state. Eq.(1) then determines the internal energy associated with any other state of the body or system.

Comment 3. One must check experimentally that the same amount of work, no matter in what form the work is done, always brings about the same final system state starting from a given initial system state. Only if this is true, is it possible to use Eq.(1) to assign a unique internal energy value to a given state of a system.

The first quantitative experiments like this were done by J.P. Joule in the 19th century. The SI-unit of energy, the joule, is named after him.

The fact that, given thermal isolation, the same amount of work is required to take a system from one state to another, no matter what the process is in which the work is delivered, is called the first law of thermodynamics.

Comment 4. If the body or system on which work is being done changes its macroscopic kinetic energy during the process of work-like interaction with the environment, Eq.(1) must be generalized to

W = DKE + DU(2)

where DKE is the change in the body's or system's kinetic energy.