**Principle**

To
understand the concept of adiabatic phenomena, first Boyle’s law is applied to
the simple set-up of Fig. 1. It is assumed that both compartments have constant
volume and temperature.

Fig.
1

The
amount of gas in both compartments together is proportional p_{a}·V_{a}
+ p_{b}·V_{b }which holds with the closed valve (at time zero)
as well as after equilibration of the pressures (at time t) when the valve has
been opened. So, it holds that :

p_{a, 0}·V_{a} + p_{b, 0}·V_{b
= }p_{t}·(V_{a} + V_{b}). (0)

With
given p_{a, 0}, p_{b, 0}, V_{a} and V_{b =, }p_{t}·
can be calculated. However, this is only true when T in the whole system
remains constant, i.e. isothermic compression and expansion holds. In daily
life, a bicycle pump becomes warm when a tire is inflated. This is due to heat
transfer from the heated, compressed air in the pump. When the heat produced by
compression is not given off to the environment but “absorbed” by the gas it
self, or when the energy needed for expansion is not provided by the
environment but provided by the expanding gas itself, than Boyle’s law does not
hold. So, a compressing gas heats and an expanding gas cools down. These
processes are called adiabatic compression and expansion. The deviation from
Boyle’s law can be very substantial (Fig. 2), as will be
explained conceptually with the example of Fig. 1.

When
the valve has a very narrow opening, the process of equilibration takes so much
time (tens of minutes) that the whole process is isothermic. There is enough
time for heat release and uptake between the system and the environment. When
the equilibration is very fast (about a second; pipe and valve diameter large)
the process is practically adiabatic and the changes of temperature can rather
well be calculated.

For
ideal gases under pure adiabatic conditions the p-V relation is:

p·V^{γ
}= constant, (1)

with
γ a numerical value greater than 1 and depending on the type of gas.
γ is the so-called c_{p}/c_{V} ratio, the ratio of heat
capacity of the gas (c_{p}) with constant p and the specific heat
capacity of the gas (c_{V}) with constant V.* * It depends on the number of
atoms in the gas molecule. For mono-atomic gases, e.g. the noble gases, γ
is 5/3. For diatomic gases, for instance air, γ is 7/5 (= 1.4). More
precisely, c_{p}/c_{V} =is :

c_{p}/c_{V}
= c_{p}/(c_{p}-R), (2)

where R the universal gas constant (see __Gas laws__).

Fig.2
Isothermic and adiabatic p-V relation. The dashed curve gives the p-V
relation according to Boyle’s law and the dashed straight horizontal line the
temperature belonging to it. The solid curve represents the adiabatic p-V
relation and the thick solid curve gives the adiabatic temperature. *n* = 0.1 kmol.

As
follows from Eq (1), reducing a volume by a factor k^{1} gives a
pressure increase much higher than k (being k^{1.4}). Conceptually, one
can say that a factor k is due to the volume decrease according to Boyle and
that the remaining factor, k^{0.4}, is caused by the temperature
increase.

An
isothermal p-V curve according to Boyle and an adiabatic p-V curve are depicted
in Fig. 2. Numerically, the resulting temperature effect can be very
substantial. A further description of adiabatic processes can be found in More info. Since textbooks seldom give
examples how to calculate the adiabatic effect, an example is given in
More Info.

**Application**

Adiabatic
effects are seldom applied and in contrary, are generally unwanted. So,
prevention is required. (Semi)-artificial breathing apparatus (home care
pulmonary diseases, pulmonology, IC, surgery theatre) should have valves, tubes
etc. designed in such a way that valves are not blocked by freezing and the
breathing gas is not cooled down by adiabatic effects. Adiabatic effects also
play a role in the technology of (clinical) hyperbaric chambers, high pressure
gas tanks and diving regulators.

In
the human (and mammalian) body), compression and expansion of gas filed
cavities, adiabatic effects play now role. Even when one dives in the water
from some tens of meters altitude, the temperature rise in the cavities is
physiologically irrelevant since the process is sufficiently isotherm.

A
medical imaging method, __Optoacoustic Imaging__ is based on
adiabatic processes.

**More
Info**

The
exponent γ is the c_{p}/c_{V} ratio, the ratio of
heat capacity of the gas (c_{p}) with constant p and the specific heat
capacity of the gas (c_{V}) with constant V. The complicated derivation
can be found in physical textbooks. Here, a conceptual explanation is given.

The
process of compression always has the same energetic cost, so independent of
the velocity of the compression. This external energy is added to the intrinsic
energy of the gas, which is actually the kinetic energy of the particles. The
kinetic energy of the particles is proportional with the temperature. When the
external energy is supplied instantaneously, it is completely added to the
kinetic energy of the particles. When the compression is slow, then during the
time of compression, the particles have the time to transfer a part of their
kinetic energy to the vessel wall. The result is that the temperature of the
gas raises less and the temperature of the wall increases. When the compression
is extremely slow, the gas remains its temperature since all external energy is
(indirectly) transferred to the environment. Now, the compression is isothermic
and Boyle's law holds.

For
instantaneous compression, the increase can be calculated as follows. Suppose
that volume V with pressure p is compressed to V’ yielding a new pressure p’.
Applying Eq. (1) means that:

p·V^{γ}
^{ }= p’(V’)^{γ }. (3).

Before
compression and after compression the universal gas law (see __Gas laws__) holds, so
that pV = nRT and p’V’ = nRT’ (T’ is the new temperature), and some more
calculation yields:

ΔT = T’ – T = T((V/V’)^{(}^{γ}^{ -1)} - 1) (4)

This
relation also holds for volume expansion.

In
a similar way, for a given change of pressure it can be derived that:

ΔT
= T((p/p’)^{(γ-1)/ γ} - 1). (5)

__Example
of calculation__

Suppose,
everything ideal, that a gas volume at 300 K (27 ^{o}C) is
instantaneously reduced in volume by a factor 2.5 and that there is no heat
transport. According to Eq. (4) this yields ΔT = 300 x (2.5^{0.4}
- 1) = 133 K. However, due to the temperature increase, a much higher factor of
pressure increase is obtained (being 2.5^{1.4} = 3.6) rather than the
factor of 2.5 according to Boyle’s law. Actually, a compression-factor of 1.92
(= 2.5^{1/1.4}) is required to obtain p = 2.5 bar. Then, the
temperature increase becomes 300 x (1.92^{0.4} – 1) = 90 K.

The
equation p·V^{γ }= constant works also the other way around.
Suppose that the pressure p is increased instantaneously to k·p by a factor of
k. Then, according to Boyle, the volume V reduces with a factor k, but now, the
reduction is only a factor k ^{1/1.4}. Again, in the pressure increase,
a “Boyle factor” k^{1/1.4} and a “temperature factor” k^{0.4/1.4}
are comprised. Of course, these two factors together have to yield k (= k^{1/1.4}·k^{0.4/1.4}).
Now, with k = 2.5 and T = 300 K the temperature increase is again 90 K (2.5^{0.4/1.4}x300
K).

In
practice, it appears that the processes of compression and expansion are seldom
purely adiabatic or purely isothermic. When the process is in between the
calculation is complicated and, moreover in practice valves may block by
freezing, for instance the valve of Fig.1. This may result in oscillatory
behavior between blocking and unblocking by freezing and unfreezing.