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Consider an Ideal Gas , in which the temperature depends only on the Internal Energy , which is a function of the Mean translational Kinetic Energy of the Molecule s, as given by a Boltzmann Distribution ; if the internal energy is constant, so is the temperature. Take the number of Moles ''n'' as a constant.

: \Delta U = n R \Delta T = 0 \,

but this means, according to the Ideal Gas Law , that

: \Delta (P V) = 0 \,

so that

: P_i V_i = P V = P_f V_f \,

where P_i and V_i are the pressure and volume of the initial state, P_f and V_f are the pressure and volume of the final state, and the Variable s ''P'' and ''V'' stand for the pressure and volume of any intermediate state during an isothermal process.

s of an ideal gas]]

Curves called ''isotherms'' appear as a Hyperbola s on a ''P''-''V'' (pressure-volume) diagram (''T'' = constant). Each one Asymptotically approaches both the ''V'' (abcissa) and ''P'' (ordinate) axes. This corresponds to a one- Parameter family of curves, a Function of ''T'', whose equation is

: P = {n R T \over V} \,

By the First Law Of Thermodynamics , the isotherms of an ideal gas are also determined by the condition that

: Q = W \,

where ''W'' is work done ''on'' the system. (While ''Q'' and ''W'' are incremental Quantities , they do not represent Differentials of State Function s.) This means that, during an isothermal process, all heat accepted by the system from its surroundings must have its Energy entirely converted to Work which it performs on the surroundings. That is, all the energy which comes into the system comes back out; the internal energy and thus the temperature of the system remain constant.

- Entropy diagram. The horizontal lines are isothermal.]]

Isothermal processes can occur in any kind of system, including highly structured Machines , and even Living cells. Various parts of the cycle of some Heat Engine s are carried out isothermally and may be Approximated by a Carnot Cycle .

















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