# Isothermal process

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An isothermal process is a thermodynamic process in which the temperature of the system stays constant; ΔT = 0. This typically occurs when a system is in contact with an outside thermal reservoir, and the system changes slowly enough to allow it to adjust to the temperature of the reservoir. The opposite extreme in which a system exchanges no heat with its surroundings is known as an adiabatic process.

Assuming that the quantity n0 of moles of gas of the system remains constant, then the internal energy E of the system also remains constant:

[itex] \Delta E = n_0 R \Delta T = 0 [itex],

but this means, according to the ideal gas law, that

[itex] \Delta (P V) = 0 [itex]

so that

[itex] P_i V_i = P V = P_f V_f [itex]

where [itex] P_i [itex] is the pressure of the initial state, [itex] V_i [itex] is the volume of the initial state, [itex] P_f [itex] is the pressure of the final state, [itex] V_f [itex] is the volume of the final state, and P and V are the pressure and volume of an intermediate state of the isothermal process.

An isothermal process is shown as a hyperbolic line (T0 = constant) on a P-V (Pressure-Volume) diagram which asymptotically approaches both the V (abcissa) axis and the P (ordinate) axis. For an ideal gas, the line is called an isotherm and its equation is

[itex] P = {n_0 R T_0 \over V} [itex].

According to the first law of thermodynamics, the isotherm can be described also by the equation

[itex] Q = W [itex]

where W is work done by the system. 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 then performs on the surroundings, so that all the energy which comes into the system then comes right back out of the system so the internal energy (and thus the temperature) of the system remains constant.

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