Isothermal Process:

• An isothermal process is a change of a system, in which the temperature remains constant: ΔT = 0. This typically occurs when a system is in contact with an outside thermal reservoir (heat bath), and the change occurs slowly enough to allow the system to continually adjust to the temperature of the reservoir through heat exchange.
• In contrast, an adiabatic process is where a system exchanged no heat with its surroundings (Q=0). In other words, in an isothermal process, the value ΔT = 0 but Q ≠ 0, while in an adiabatic process, ΔT ≠ 0 but Q=0.

• A change which occurs where no heat enters of leaves the system. It occurs when the work is done fast. Q=0 because no heat is transferred therefore U=W.

Isothermal Expansion and Compression:

Work Done and Heat transferred:

• In an isothermal process, the temperature is constant. Applying the forst law of thermodynamics to this closed process.

dU = dQ - dW

• For an ideal gas, the internal energy is a function of temperature only, and since the temperature is constant, then dU is zero and

dQ = dW = PdV

• Using the ideal gas law and integrating between the start and end of the process

• This equation tells us that if we do some work on a gas to compress it, the same amount of energy will appear as heat transferred from the gas as it is compressed.

Entropy Change:

• The Entropy change comes from the equation which incorporates the forst and second laws. The energy balance is the first law, and the heat transfer is expressed as an entropy change which is a statement of the second law.

dU = TdS - PdV

• dU is zero because the process is isothermal and the working fluid is an ideal gas, so that

TdS = PdV

• Substituting for the pressure from the ideal gas law for the pressure

• And finally integrating between the start and end process

• Isothermal compression is shown above P-V and T-S diagram. Note that as the gas is compressed heat is given out and that as it expands heat is absorbed.