14 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Adiabatic process | 3/7 | https://en.wikipedia.org/wiki/Adiabatic_process | reference | science, encyclopedia | 2026-05-05T10:56:55.166101+00:00 | kb-cron |
=== Adiabatic expansion === Adiabatic expansion occurs when the pressure on an adiabatically isolated system is decreased, allowing it to expand in size, thus causing it to do work on its surroundings. When the pressure applied on a parcel of gas is reduced, the gas in the parcel is allowed to expand; as the volume increases, the temperature falls as its internal energy decreases. Adiabatic expansion occurs in the Earth's atmosphere with orographic lifting and lee waves, and this can form pilei or lenticular clouds. Due in part to adiabatic expansion in mountainous areas, snowfall actually occurs in some parts of the Sahara desert. Adiabatic expansion does not have to involve a fluid. One technique used to reach very low temperatures (thousandths and even millionths of a degree above absolute zero) is via adiabatic demagnetisation, where the change in magnetic field on a magnetic material is used to provide adiabatic expansion. Also, the contents of an expanding universe can be described (to first order) as an adiabatically expanding fluid. (See heat death of the universe.) Rising magma also undergoes adiabatic expansion before eruption, particularly significant in the case of magmas that rise quickly from great depths such as kimberlites. In the Earth's convecting mantle (the asthenosphere) beneath the lithosphere, the mantle temperature is approximately an adiabat. The slight decrease in temperature with shallowing depth is due to the decrease in pressure the shallower the material is in the Earth. Such temperature changes can be quantified using the ideal gas law, or the hydrostatic equation for atmospheric processes. In practice, no process is truly adiabatic. Many processes rely on a large difference in time scales of the process of interest and the rate of heat dissipation across a system boundary, and thus are approximated by using an adiabatic assumption. There is always some heat loss, as no perfect insulators exist.
== Ideal gas (reversible process) ==
The mathematical equation for an ideal gas undergoing a reversible (i.e., no entropy generation) adiabatic process can be represented by the polytropic process equation
P
V
γ
=
c
o
n
s
t
a
n
t
,
{\displaystyle P\ V^{\gamma }={\mathsf {constant}}\ ,}
where P is pressure, V is volume, and γ is the adiabatic index or heat capacity ratio defined as
γ
=
C
P
C
V
=
f
+
2
f
.
{\displaystyle \gamma ={\frac {C_{P}}{C_{V}}}={\frac {f+2}{f}}~.}
Here CP is the specific heat for constant pressure, CV is the specific heat for constant volume, and f is the number of degrees of freedom (3 for a monatomic gas, 5 for a diatomic gas or a gas of linear molecules such as carbon dioxide). For a monatomic ideal gas, γ = 5/3, and for a diatomic gas (such as nitrogen and oxygen, the main components of air), γ = 7/5. Note that the above formula is only applicable to classical ideal gases (that is, gases far above absolute zero temperature) and not Bose–Einstein or Fermi gases. One can also use the ideal gas law to rewrite the above relationship between P and V as
P
1
−
γ
T
γ
=
c
o
n
s
t
a
n
t
,
T
V
γ
−
1
=
c
o
n
s
t
a
n
t
.
{\displaystyle {\begin{aligned}P^{1-\gamma }\ T^{\gamma }&={\mathsf {constant}}\ ,\\T\ V^{\gamma -1}&={\mathsf {constant}}~.\end{aligned}}}
where T is the absolute or thermodynamic temperature.
=== Example of adiabatic compression === The compression stroke in a gasoline engine can be used as an example of adiabatic compression. The model assumptions are: the uncompressed volume of the cylinder is one litre (1 L = 1000 cm3 = 0.001 m3 ); the gas within is the air consisting of molecular nitrogen and oxygen only (thus a diatomic gas with 5 degrees of freedom, and so γ = 7/5 ); the compression ratio of the engine is 10:1 (that is, the 1 L volume of uncompressed gas is reduced to 0.1 L by the piston); and the uncompressed gas is at approximately room temperature and pressure (a warm room temperature of ~27 °C, or 300 K, and a pressure of 1 bar = 100 kPa, i.e. typical sea-level atmospheric pressure).
P
1
V
1
γ
=
c
o
n
s
t
a
n
t
1
=
100
000
Pa
×
(
0.001
m
3
)
7
5
=
10
5
×
6.31
×
10
−
5
P
a
m
21
/
5
=
6.31
Pa
m
21
/
5
,
{\displaystyle {\begin{aligned}P_{1}V_{1}^{\gamma }&={\mathsf {constant}}_{1}\\&=100\,000~{\text{Pa}}\times (0.001~{\mathsf {m}}^{3})^{\frac {7}{5}}\\&=10^{5}\times 6.31\times 10^{-5}~{\mathsf {Pa}}\,{\mathsf {m}}^{21/5}\\&=6.31~{\text{Pa}}\,{\mathsf {m}}^{21/5}\ ,\end{aligned}}}
so the adiabatic constant for this example is about 6.31 Pa·m4.2 . The gas is now compressed to a 0.1 L (0.0001 m3) volume, which we assume happens quickly enough that no heat enters or leaves the gas through the walls. The adiabatic constant remains the same, but with the resulting pressure unknown
P
2
V
2
γ
=
c
o
n
s
t
a
n
t
1
=
6.31
P
a
m
21
/
5
=
P
×
(
0.0001
m
3
)
7
5
,
{\displaystyle {\begin{aligned}P_{2}V_{2}^{\gamma }&={\mathsf {constant}}_{1}\\&=6.31~{\mathsf {Pa}}\,{\mathsf {m}}^{21/5}\\&=P\times (0.0001~{\mathsf {m}}^{3})^{\frac {7}{5}},\end{aligned}}}
We can now solve for the final pressure
P
2
=
P
1
(
V
1
V
2
)
γ
=
100
000
P
a
×
10
7
/
5
=
2.51
×
10
6
P
a
,
{\displaystyle {\begin{aligned}P_{2}&=P_{1}\left({\frac {V_{1}}{V_{2}}}\right)^{\gamma }\\&=100\ 000~{\mathsf {Pa}}\times {10}^{7/5}\\&=2.51\times 10^{6}~{\mathsf {Pa}}\ ,\end{aligned}}}