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Here an elastic fluid means a gas or vapour. Such a fluid is mobile, occupies the accessible space of its container, and is appreciably compressible. Its pressure acts normally on a confining surface. In static equilibrium the stress at a point is isotropic: a fluid has no equilibrium resistance to a sustained shear deformation. Pressure can still vary with height through hydrostatic equilibrium; isotropy does not mean the same pressure at every position.

Reducing a confined gas's volume at fixed temperature normally raises its pressure. Heating at fixed volume normally raises the pressure, and gases exchange material through diffusion. Their particles are in continual thermal motion, whose momentum transfer gives a microscopic account of gas pressure in the kinetic theory of gases.

For a dilute gas away from condensation, the approximate ideal gas law isIt contains Boyle's law at fixed temperature and the corresponding temperature-expansion laws. These simple proportionalities are approximations, not exact properties of every elastic fluid at every density. Intermolecular interactions matter in a dense gas, and a vapour can condense; the classification does not promise that every sample remains gaseous under arbitrary compression or cooling.

The strong evidence is the reproducible conversion between mechanical work and thermal changes. In Joule's paddle-wheel experiment, descending masses drive vanes in a liquid, and a measured input of mechanical energy produces a measured rise in temperature. Allowing for the vessel and thermal losses gives a quantitative mechanical equivalent. Sustained frictional heating, including Rumford's cannon-boring experiment, also challenged a simple account in which rubbing merely released a fixed stored supply of a material heat fluid.^[ref]

Electrical and compressional experiments reinforce the connection. A resistor warms under Joule heating, with electrical input $\int VIdt$. A gas compressed with little heat transfer can rise in temperature, showing that mechanical work can increase its internal energy. Conversely an expanding gas can perform work while cooling. Such experiments connect several kinds of change through a conserved energy balance.

The kinetic theory of gases supplies a microscopic model. Translational molecular motion produces momentum transfer to walls; for a dilute monatomic ideal gas,Thus increasing molecular kinetic energy is reflected in temperature and pressure changes. These formulas describe a specified model, not a proof that every substance's internal energy consists only of translational motion. Rotations, vibrations and intermolecular potential energy also matter.

In modern terminology, heat is energy transferred because of a temperature difference. The body has internal energy, and microscopic motion accounts for part of it. Calling heat “a form of motion” captures the historical rejection of a conserved material caloric, but should not obscure that distinction between stored energy and an energy-transfer process.