Editing Isobaric process

{{Short description|Thermodynamic process in which pressure remains constant}}
{{refimprove article|date=October 2016}}

{{Thermodynamics|cTopic=[[Thermodynamic system|Systems]]}}
In [[thermodynamics]], an '''isobaric process''' is a type of [[thermodynamic process]] in which the [[pressure]] of the [[Thermodynamic system|system]] stays constant: Δ''P''&nbsp;=&nbsp;0. The [[heat]] transferred to the system does [[work (thermodynamics)|work]], but also changes the [[internal energy]] (''U'') of the system.  This article uses the physics sign convention for work, where positive work is [[work (thermodynamics)#Sign convention|work done by the system]].  Using this convention, by the [[first law of thermodynamics]],

[[Image:Isobaric process plain.svg|thumb|250px|The yellow area represents the work done]]

:<math> Q = \Delta U + W\, </math>

where ''W'' is work, ''U'' is internal energy, and ''Q'' is heat.<ref>{{cite web|title=First Law of Thermodynamics|url=https://www.grc.nasa.gov/www/k-12/airplane/thermo1.html|website=www.grc.nasa.gov|access-date=19 October 2017}}</ref> Pressure-[[volume]] work by the closed system is defined as:

:<math>W = \int \! p \,dV \,</math>

where Δ means change over the whole process, whereas ''d'' denotes a differential. Since pressure is constant, this means that

:<math> W = p \Delta V\, </math>.

Applying the [[ideal gas law]], this becomes

:<math> W = n\,R\,\Delta T</math>

with ''R'' representing the [[gas constant]], and ''n'' representing the [[amount of substance]], which is assumed to remain constant (e.g., there is no [[phase transition]] during a [[chemical reaction]]). According to the [[equipartition theorem]],<ref>{{cite web|url=http://www.insula.com.au/physics/1221/L9.html|title=Lecture 9 (Equipartition Theory)|first=Peter|last=Eyland|website=www.insula.com.au}}</ref> the change in internal energy is related to the temperature of the system by

:<math> \Delta U = n\,c_{V,m}\,\Delta T</math>,

where ''c<sub>V, m</sub>'' is molar [[heat capacity]] at a [[constant volume]].

Substituting the last two equations into the first equation produces:

:<math>\begin{align}
Q &= n\,c_{V,m}\,\Delta T + n\,R\,\Delta T \\
Q &= n\Delta T(c_{V,m}+R) \\
Q &= n\Delta T c_{P,m}
\end{align}
</math>
where ''c<sub>P</sub>'' is molar heat capacity at a constant [[pressure]].

==Specific heat capacity==
To find the molar specific heat capacity of the gas involved, the following equations apply for any general gas that is calorically perfect.  The property ''γ'' is either called the [[adiabatic index]] or the [[heat capacity ratio]].  Some published sources might use ''k'' instead of ''γ''.

Molar isochoric specific heat:
:<math>c_V = \frac{R}{\gamma - 1}</math>.

Molar isobaric specific heat:
:<math>c_p = \frac{\gamma R}{\gamma - 1}</math>.
The values for ''γ'' are ''γ''&nbsp;=&nbsp;{{sfrac|7|5}} for [[diatomic]] gases like [[Atmosphere of Earth#Composition|air and its major components]], and ''γ''&nbsp;=&nbsp;{{sfrac|5|3}} for [[monatomic gases]] like the [[noble gas]]es. The formulas for specific heats would reduce in these special cases:

Monatomic:
:<math>c_V = \tfrac32 R</math> and <math>c_P = \tfrac52 R</math>

Diatomic:
:<math>c_V = \tfrac52 R</math> and <math>c_P = \tfrac72 R</math>

An isobaric process is shown on a ''P''–''V'' diagram as a straight horizontal line, connecting the initial and final [[thermostatic]] states. If the process moves towards the right, then it is an expansion. If the process moves towards the left, then it is a compression.

==Sign convention for work==
The motivation for the specific [[work (thermodynamics)#Sign convention|sign conventions]] of [[thermodynamics]] comes from early development of heat engines. When designing a heat engine, the goal is to have the system produce and deliver work output. The source of energy in a heat engine, is a heat input.

* If the volume compresses (Δ''V''&nbsp;= final volume − initial volume <&nbsp;0), then ''W''&nbsp;<&nbsp;0. That is, during isobaric [[Compression (physics)|compression]] the gas does negative work, or the environment does positive work. Restated, the environment does positive work on the gas.
* If the volume expands (Δ''V''&nbsp;= final volume − initial volume >&nbsp;0), then ''W''&nbsp;>&nbsp;0. That is, during isobaric [[Thermal expansion|expansion]] the gas does positive work, or equivalently, the environment does negative work. Restated, the gas does positive work on the environment.
* If heat is added to the system, then ''Q''&nbsp;>&nbsp;0. That is, during isobaric expansion/heating, positive heat is added to the gas, or equivalently, the environment receives negative heat. Restated, the gas receives positive heat from the environment.
* If the system rejects heat, then ''Q''&nbsp;<&nbsp;0. That is, during isobaric compression/cooling, negative heat is added to the gas, or equivalently, the environment receives positive heat. Restated, the environment receives positive heat from the gas.

==Defining enthalpy==
An [[isochoric process]] is described by the equation ''Q''&nbsp;=&nbsp;Δ''U''.  It would be convenient to have a similar equation for isobaric processes. Substituting the second equation into the first yields

:<math> Q = \Delta U + \Delta (p\,V) = \Delta (U + p\,V) </math>

The quantity ''U''&nbsp;+&nbsp;''pV'' is a state function so that it can be given a name. It is called [[enthalpy]], and is denoted as ''H''.  Therefore, an isobaric process can be more succinctly described as

:<math> Q = \Delta H \,</math>.

Enthalpy and isochoric specific heat capacity are very useful mathematical constructs, since when analyzing a process in an [[Thermodynamic system#Open system|open system]], the situation of zero work occurs when the fluid flows at constant pressure.  In an open system, enthalpy is the quantity which is useful to use to keep track of energy content of the fluid.

== Examples of isobaric processes ==
The [[Reversible process (thermodynamics)|reversible expansion]] of an ideal gas can be used as an example of an isobaric process.<ref>{{Cite book|last=Gaskell, David R., 1940-|title=Introduction to the thermodynamics of materials|date=2008|publisher=Taylor & Francis|isbn=978-1-59169-043-6|edition=5th|location=New York|pages=32|oclc=191024055}}</ref>  Of particular interest is the way heat is converted to work when expansion is carried out at different working gas/surrounding gas  pressures.

[[File:Example of Isobaric Process.png|thumb|Isobaric expansion of 2 cubic meters of air at 300 Kelvin to 4 cubic meters, causing the temperature to increase to 600 Kelvin while the pressure remains the same.|alt=|333x333px]]

In the first process example, a cylindrical chamber 1 m<sup>2</sup> in area encloses 81.2438&nbsp;mol of an [[Ideal gas|ideal diatomic gas]] of molecular mass 29 g mol<sup>−1</sup> at 300 K. The surrounding gas is at 1 atm and 300 K, and separated from the cylinder gas by a thin piston.  For the limiting case of a massless piston, the cylinder gas is also at 1 atm pressure, with an initial volume of 2 m<sup>3</sup>. Heat is added slowly until the gas temperature is uniformly 600 K, after which the gas volume is 4 m<sup>3</sup> and the piston is 2 m above its initial position.  If the piston motion is sufficiently slow, the gas pressure at each instant will have practically the same value (''p''<sub>sys</sub> = 1 atm) throughout.

For a thermally perfect diatomic gas, the molar specific heat capacity at constant pressure (''c<sub>p</sub>'') is <sup>7</sup>/<sub>2</sub>R or 29.1006 J mol<sup>−1</sup> deg<sup>−1</sup>.  The molar heat capacity at constant volume (''c<sub>v</sub>'') is <sup>5</sup>/<sub>2</sub>R  or 20.7862 J mol<sup>−1</sup> deg<sup>−1</sup>.  The ratio <math>\gamma</math> of the two heat capacities is 1.4.<ref name=":0">{{Cite web|url=https://ccrma.stanford.edu/~jos/pasp/Heat_Capacity_Ideal_Gases.html|title=Heat Capacity of Ideal Gases|website=ccrma.stanford.edu|access-date=2018-10-05}}</ref>

The heat ''Q'' required to bring the gas from 300 to 600 K is

:<math>Q = {\Delta\Eta} = n\,c_p\,\Delta\Tau  = 81.2438\times 29.1006\times 300 = 709,274\text{ J}</math>.

The increase in [[internal energy]] is

:<math>\Delta\ U = n\,c_v\,\Delta\Tau  = 81.2438\times 20.7862\times 300 = 506,625\text{ J} </math>

Therefore, <math>W = Q - \Delta U = 202,649\text{ J} = nR\Delta\Tau </math>

Also

<math>W = {p\Delta\nu} = 1~\text{atm} \times 2\text{m3} \times 101325\text{Pa} = 202,650\text{ J} </math>, which of course is identical to the difference between Δ''H'' and Δ''U''.

Here, work is entirely consumed by expansion against the [[Surroundings (thermodynamics)|surroundings]].  Of the total heat applied (709.3 kJ), the work performed (202.7 kJ) is about 28.6% of the supplied heat.

[[File:Isobaric Process Exaple 2.png|thumb|Isobaric expansion of a gas pressurized to 2 atmospheres by a 10,333.2 kg mass. Like before, the gas doubles in volume and temperature while remaining at the same pressure. |alt=|320x320px]]
The second process example is similar to the first, except that the massless piston is replaced by one having a mass of 10,332.2&nbsp;kg, which doubles the pressure of the cylinder gas to 2 atm.  The cylinder gas volume is then 1 m<sup>3</sup> at the initial 300 K temperature.  Heat is added slowly until the gas temperature is uniformly 600 K, after which the gas volume is 2 m<sup>3</sup> and the piston is 1 m above its initial position.  If the piston motion is sufficiently slow, the gas pressure at each instant will have practically the same value (''p''<sub>sys</sub> = 2 atm) throughout.

Since enthalpy and internal energy are independent of pressure,

: <math>Q = {\Delta\Eta} = 709,274\text{ J}</math> and  <math>\Delta U = 506,625\text{ J}</math>.
:<math>W = {p\Delta V} = 2~\text{atm} \times 1~\text{m3}\times 101325\text{Pa} = 202,650\text{ J} </math>

As in the first example, about 28.6% of the supplied heat is converted to work.  But here, work is applied in two different ways: partly by expanding the surrounding atmosphere and partly by lifting 10,332.2&nbsp;kg a distance ''h'' of 1 m.<ref>{{Cite book|last=DeVoe, Howard.|title=Thermodynamics and chemistry|date=2001|publisher=Prentice Hall|isbn=0-02-328741-1|location=Upper Saddle River, NJ|pages=58|oclc=45172758}}</ref>

:<math>W_{\rm lift} = 10\,332.2~\text{kg} \times 9.80665~\text{m/s²}\times1\text{m} = 101,324\text{ J} </math>
:

Thus, half the work lifts the piston mass (work of gravity, or “useable” work), while the other half expands the surroundings.

The results of these two process examples illustrate the difference between the fraction of heat converted to usable work (''mg''Δ''h)'' vs. the fraction converted to pressure-volume work done against the surrounding atmosphere. The usable work approaches zero as the working gas pressure approaches that of the surroundings, while maximum usable work is obtained when there is no surrounding gas pressure.  The ratio of all work performed to the heat input for ideal isobaric gas expansion is

:<math>\frac{W}{Q} = \frac{nR\Delta\Tau}{nc_p\Delta\Tau}   = \frac{2}{5} </math>

==Variable density viewpoint==
A given quantity (mass ''m'') of gas in a changing volume produces a change in [[density#Changes in density|density]] ''ρ''. In this context the [[ideal gas law]] is written

:<math> R(T\,\rho) = M P </math>

where ''T'' is [[thermodynamic temperature]] and ''M'' is [[molar mass]]. When R and M are taken as constant, then pressure ''P'' can stay constant as the density-temperature [[Quadrant (plane geometry)|quadrant]] {{nowrap|(''ρ'',''T'')}} undergoes a [[squeeze mapping]].<ref>{{Cite book|last=Olver, Peter J.|title=Classical invariant theory|date=1999|publisher=Cambridge University Press|isbn=978-1-107-36236-9|location=Cambridge, UK|pages=217|oclc=831669750}}</ref>

==Etymology==
The adjective "isobaric" is derived from the [[Ancient Greek|Greek]] words ἴσος (''isos'') meaning "equal", and βάρος (''baros'') meaning "weight."

==See also==
* [[Adiabatic process]]
* [[Cyclic process]]
* [[Isochoric process]]
* [[Isothermal process]]
* [[Polytropic process]]
* [[Isenthalpic process]]

==References==
{{reflist}}

[[Category:Thermodynamic processes]]
[[Category:Atmospheric thermodynamics]]