isothermal process
An isothermal process is a change that happens while the temperature stays constant. The prefix iso- means 'same' and -thermal means 'heat or temperature', so isothermal literally means 'same temperature'. Picture a gas trapped under a piston, sitting in a large water bath that never changes temperature. Compress or expand the gas slowly enough that any heat generated or absorbed leaks in or out to the bath, and the gas stays at the bath's temperature the whole time.
For an ideal gas the internal energy depends only on temperature, so if T does not change then ΔU = 0. The first law ΔU = Q - W then becomes 0 = Q - W, that is Q = W: every joule of heat that flows in comes straight back out as work done by the gas (during expansion), and every joule of work done on the gas leaves as heat (during compression). On a graph of pressure against volume, an isothermal path for an ideal gas follows P V = constant (Boyle's law), a smooth curve called an isotherm.
Isothermal steps appear in the Carnot cycle and in slow, well-cooled compressions and expansions. The catch is the word 'slowly': to hold the temperature truly constant the process must be gentle enough that heat has time to flow and keep the system in step with the bath. A fast compression heats the gas before the heat can escape, so real rapid changes are closer to adiabatic than isothermal.
A gas in a cylinder held in a 300 K water bath is slowly expanded. It absorbs 400 J of heat from the bath and, because ΔU = 0, does exactly 400 J of work lifting the piston. Its temperature reads 300 K from start to finish.
At constant temperature, heat in equals work out for an ideal gas.
ΔU = 0 for an isothermal process is exact only for an ideal gas, whose internal energy depends on temperature alone; for real gases and liquids the internal energy shifts a little even at constant temperature.