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| ''</big>''{{Thermodynamics|cTopic=[[Thermodynamic system|Systems]]}}
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| <!-- [[Image:CarnotCycle1.png|300px|thumb|right|Carnot cycle for a heat engine on a [[temperature]]-[[entropy]] diagram. The horizontal lines are isothermal.]] -->
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| An '''isothermal process''' is a [[thermodynamic process|change]] of a ''system'', in which the [[temperature]] remains constant: Δ''T'' = 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 exchanges no heat with its surroundings (''Q'' = 0). In other words, in an isothermal process, the value Δ''T'' = 0 but Q ≠ 0, while in an adiabatic process, Δ''T'' ≠ 0 but Q = 0.
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| ==Details for an ideal gas==
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| [[image:Ideal gas isotherms.png|thumb|right|Several isotherms of an ideal gas on a p-V diagram]]
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| For the special case of a gas to which [[Boyle's law]] applies, the product ''pV'' is a constant if the gas is kept at isothermal conditions. However, in the cases where the product pv is an exponential term this does not comply. The value of the constant is ''nRT'', where n is the number of moles of gas present and ''R'' is the [[ideal gas constant]]. In other words, the [[ideal gas law]] ''pV'' = ''nRT'' applies. This means that
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| : <math> p = {n R T \over V} = {\text{constant} \over V} </math>
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| holds. The family of curves generated by this equation is shown in the graph presented at the bottom right-hand of the page. Each curve is called an isotherm. Such graphs are termed [[indicator diagram]]s and were first used by [[James Watt|James Watt and others]] to monitor the efficiency of engines. The temperature corresponding to each curve in the figure increases from the lower left to the upper right.
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| ==Calculation of work==
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| [[Image:Isothermal process.svg|thumb|right|The purple area represents "work" for this isothermal change]]
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| In thermodynamics, the [[Work (thermodynamics)|work]] involved when a gas changes from state ''A'' to state ''B'' is simply
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| :<math>W_{A\to B} = -\int_{V_A}^{V_B}p\,dV</math>
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| For an isothermal, [[Reversible process (thermodynamics)|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:
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| :<math>W_{A\to B} = -\int_{V_A}^{V_B}p\,dV = -\int_{V_A}^{V_B}\frac{nRT}{V}dV = -nRT\int_{V_A}^{V_B}\frac{1}{V}dV = -nRT\ln{\frac{V_B}{V_A}}</math>
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| 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.
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| 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 <math> \Delta U = 0</math>. From [[First Law of Thermodynamics]], <math> \Delta U=Q+W</math>, so it follows that <math>-Q =W</math> for this same isothermal process.
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| When no heat flows into or out of the gas because the temperature is constant, then there is no work done. Thus, work=0 which means external pressure is zero. This is called [[free expansion]].
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| ==Applications==
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| Isothermal processes can occur in any kind of system, including highly-structured [[machines]], and even [[life|living]] cells. Various parts of the cycles of some [[heat engine]]s are carried out isothermally and may be approximated by a [[Carnot cycle]]. [[Phase changes]], such as [[melting]] or [[evaporation]], are also isothermal processes.
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| 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 [[Isobaric process#Sign convention for work|thermodynamic sign convention]] is that heat entering the environment is also negative. Thence -Q = W.
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| 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.
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| 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.<ref>Adkins 1983, p. 121.</ref>
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| ==See also==
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| * [[Adiabatic process]]
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| * [[Cyclic process]]
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| * [[Isobaric process]]
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| * [[Isochoric process]]
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| * [[Polytropic process]]
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| == Notes ==
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| {{Reflist}}
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| == References ==
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| *Adkins, C. J.(1983). ''Equilibrium Thermodynamics''. Cambridge University Press.
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| {{DEFAULTSORT:Isothermal Process}}
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| [[Category:Thermodynamic processes]]
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| [[Category:Atmospheric thermodynamics]]
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| {{Link FA|az}}
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