Question: When does isothermal process occur?
About the article "Thermodynamics", section "Commonly considered thermodynamic processes".

Candidate passages:
Isothermal processes can occur in any kind of system, including highly-structured machines, and even living cells.
Various parts of the cycles of some heat engines are carried out isothermally and may be approximated by a Carnot cycle.
Phase changes, such as melting or evaporation, are also isothermal processes.
In Isothermal non flow Process, the work done by compressing the perfect gas (Pure Substance) is a negative work, as work is done on the system, as result of compression, the volume will decrease, and temperature will try to increase.
To maintain the temperature at constant value (as the process is isothermal) heat energy has to leave the system and enter the environment.
The amount of energy entering the environment is equal to the work done (by compressing the perfect gas) because internal energy does not change.
The thermodynamic sign convention is that heat entering the environment is also negative.
Thence -Q = W. In equation of work, the term nRT can be replaced by PV of any state of an ideal gas.
The product of pressure and volume is in fact, 'Moving Boundary Work'; the systems boundaries are compressed.
For Expansion the same theory is applied.
As per Joule's Law for the perfect gas, Internal energy is the function of absolute temperature.
In an Isothermal process the temperature is constant.
Hence, the internal energy is constant, and the net change in internal energy is ZERO.
Within the perfect, or ideal gas, there are no inter-molecular forces and the gas particles are infinitesimal.
However, for a real pure substance there is a component of internal energy corresponding to the energy used in overcoming inter-molecular forces.
In an Isothermal process, when the volume of the gas changes, the average distance between each molecule changes as well.
So if the real pure gas undergoes an Isothermal process, there is a net change in internal temperature consistent with this component of internal energy. 
As in the case of advection turning, when there is a cross-isothermal component of the geostrophic wind, a sharpening of the temperature gradient results.
The thermal wind causes a deformation field and frontogenesis may occur. 
The following is a list of the four processes that occur between the four stages of the ideal Ericsson cycle: Process 1 -> 2: Isothermal compression.
The compression space is assumed to be intercooled, so the gas undergoes isothermal compression.
The compressed air flows into a storage tank at constant pressure.
In the ideal cycle, there is no heat transfer across the tank walls.
Process 2 -> 3: Isobaric heat addition.
From the tank, the compressed air flows through the regenerator and picks up heat at a high constant-pressure on the way to the heated power-cylinder.
Process 3 -> 4: Isothermal expansion.
The power-cylinder expansion-space is heated externally, and the gas undergoes isothermal expansion.
Process 4 -> 1: Isobaric heat removal.
Before the air is released as exhaust, it is passed back through the regenerator, thus cooling the gas at a low constant pressure, and heating the regenerator for the next cycle.  
Compression of a gas increases its temperature, often referred to as the "heat of compression".
formula_1 where  formula_2 so  formula_3 in which "p" is pressure, "V" is volume, "n" takes different values for different compression processes (see below), and 1 & 2 refer to initial and final states.
Adiabatic - This model assumes that no energy (heat) is transferred to or from the gas during the compression, and all supplied work is added to the internal energy of the gas, resulting in increases of temperature and pressure.
Theoretical temperature rise is:  formula_4 with "T"1 and "T"2 in degrees Rankine or kelvins, "p"2 and "p"1 being absolute pressures and "k" = ratio of specific heats (approximately 1.4 for air).
The rise in air and temperature ratio means compression does not follow a simple pressure to volume ratio.
This is less efficient, but quick.
Adiabatic compression or expansion more closely model real life when a compressor has good insulation, a large gas volume, or a short time scale (i.e., a high power level).
In practice there will always be a certain amount of heat flow out of the compressed gas.
Thus, making a perfect adiabatic compressor would require perfect heat insulation of all parts of the machine.
For example, even a bicycle tire pump's metal tube becomes hot as you compress the air to fill a tire.
The relation between temperature and compression ratio described above means that the value of n for an adiabatic process is k (the ratio of specific heats).
Isothermal - This model assumes that the compressed gas remains at a constant temperature throughout the compression or expansion process.
In this cycle, internal energy is removed from the system as heat at the same rate that it is added by the mechanical work of compression.
Isothermal compression or expansion more closely models real life when the compressor has a large heat exchanging surface, a small gas volume, or a long time scale (i.e., a small power level).
Compressors that utilize inter-stage cooling between compression stages come closest to achieving perfect isothermal compression.
However, with practical devices perfect isothermal compression is not attainable.
For example, unless you have an infinite number of compression stages with corresponding intercoolers, you will never achieve perfect isothermal compression.
For an isothermal process, n is 1, so the value of the work integral for an isothermal process is:  formula_5 When evaluated, the isothermal work is found to be lower than the adiabatic work.
Polytropic - This model takes into account both a rise in temperature in the gas as well as some loss of energy (heat) to the compressor's components.
This assumes that heat may enter or leave the system, and that input shaft work can appear as both increased pressure (usually useful work) and increased temperature above adiabatic (usually losses due to cycle efficiency).
Compression efficiency is then the ratio of temperature rise at theoretical 100 percent (adiabatic) vs. actual (polytropic).
Polytropic compression will use a value of n between 0 (a constant-pressure process) and infinity (a constant volume process).
For the typical case where an effort is made to cool the gas compressed by an approximately adiabatic process, the value of n will be between 1 and k.  
In thermodynamics, the work involved when a gas changes from state "A" to state "B" is simply  formula_2 For an isothermal, reversible process, this integral equals the area under the relevant pressure-volume isotherm, and is indicated in purple in the figure (at the bottom right-hand of the page) for an ideal gas.
Again, "p" = "nRT" / "V" applies and with "T" being constant (as this is an isothermal process), we have:  formula_3 By convention, work is defined as the work the system does on its environment.
If, for example, the system expands by a piston moving in the direction of force applied by the internal pressure of a gas, then the work is counted as positive, and as this work is done by using internal energy of the system, the result is that the internal energy decreases.
Conversely, if the environment does work on the system so that its internal energy increases, the work is counted as negative.
It is also worth noting that, for many systems, if the temperature is held constant, the internal energy of the system also is constant, and so formula_4.
From First Law of Thermodynamics, formula_5, so it follows that formula_6 for this same isothermal process.
When no heat flows into or out of the gas because its container is at the same temperature, then there is no work done.
Thus, work=0 which means external pressure on any moving surface is zero.
This is called free expansion.