Adiabatic process

This article covers adiabatic processes in lapse rate.

The opposite extreme -- of maximum heat transfer with the surroundings, causing the temperature to remain constant -- is known as an conjugate to entropy, the isothermal process is conjugate to the adiabatic process for reversible transformations.

A transformation of a thermodynamic system can be considered adiabatic when it is quick enough that no significant heat is transferred between the system and the outside. The adiabatic process can also be called quasi-static. At the opposite, a transformation of a thermodynamic system can be considered isothermal if it is slow enough so that the system's temperature remains constant by heat exchange with the outside.

Adiabatic heating and cooling

Adiabatic heating and cooling are processes that commonly occur due to a change in the Earth's atmosphere when an air mass descends, for example, in a katabatic wind or Foehn wind flowing downhill.

Adiabatic cooling occurs when the pressure of a substance is decreased, such as when it expands into a larger volume. An example of this is when the air is released from a pneumatic tire; the outlet air will be noticeably cooler than the tire. Adiabatic cooling 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 adiabatic demagnetisation, where the change in magnetic field on a magnetic material is used to provide adiabatic cooling. Adiabatic cooling also occurs in the dew point.

Such temperature changes can be quantified using the ideal gas law, or the hydrostatic equation for atmospheric processes.

It should be noted that no process is truly adiabatic. Many processes are close to adiabatic and can be easily approximated by using an adiabatic assumption, but there is always some heat loss. There is no such thing as a perfect insulator.

Ideal gas

The mathematical equation for an ideal fluid undergoing an adiabatic process is

$P V^{\gamma} = \operatorname{constant} \qquad$

where P is pressure, V is volume, and

$\gamma = {C_{P} \over C_{V}} = \frac{\alpha + 1}{\alpha},$

$C P$ being the specific heat for constant pressure and $C V$ being the specific heat for constant volume. $α$ comes from the number of degrees of freedom divided by 2 (3/2 for monatomic gas, 5/2 for diatomic gas). For a monatomic ideal gas, $γ = 5 / 3$ , and for a diatomic gas (such as air) $γ = 7 / 5$ . Note that the above formula is only applicable to classical ideal gases and not Bose-Einstein or Fermi gases.

For adiabatic processes, it is also true that

$P^{\gamma-1}T^{-\gamma}= \operatorname{constant}$

$VT^\alpha = \operatorname{constant}$

where T is an absolute temperature.

This can also be written as

$TV^{\gamma - 1} = \operatorname{constant}$

Derivation of continuous formula

The definition of an adiabatic process is that heat transfer to the system is zero, $δ Q = 0$ . Then, according to the first law of thermodynamics,

$\text{(1)} \qquad d U + \delta W = \delta Q = 0$

where dU is the change in the internal energy of the system and δW is work done by the system. Any work (δW) done must be done at the expense of internal energy U, since no heat δQ is being supplied from the surroundings. Pressure-volume work δW done by the system is defined as

$\text{(2)} \qquad \delta W = -P \, dV.$

However, P does not remain constant during an adiabatic process but instead changes along with V.

It is desired to know how the values of dP and dV relate to each other as the adiabatic process proceeds. For an ideal gas the internal energy is given by

$\text{(3)} \qquad U = \alpha n R T$

where R is the universal gas constant and n is the number of moles in the system (a constant).

Differentiating Equation (3) and use of the ideal gas law yields

$\text{(4)} \qquad d U = \alpha n R \, dT = \alpha \, d (P V) = \alpha (P \, dV + V \, dP).$

Equation (4) is often expressed as $d U = n C_{V} \, d T$ because $C V = α R$ .

Now substitute equations (2), (3), and (4) into equation (1) to obtain

$-P \, dV = \alpha P \, dV + \alpha V \, dP \,$

simplify,

$- (\alpha + 1) P \, dV = \alpha V \, dP \,$

divide both sides by PV,

$-(\alpha + 1) {d V \over V} = \alpha {d P \over P}.$

After integrating the left and right sides from Vo to V and from Po to P and changing the sides respectively,

$\ln \left( {P \over P_0} \right) = {-{\alpha + 1 \over \alpha}} \ln \left( {V \over V_0} \right).$

Exponentiate both sides,

$\left( {P \over P_0} \right) = \left( {V \over V_0} \right)^{-{\alpha + 1 \over \alpha}},$

eliminate the negative sign,

$\left( {P \over P_0} \right) = \left( {V_0 \over V} \right)^{\alpha + 1 \over \alpha}.$

Therefore

$\left( {P \over P_0} \right) \left( {V \over V_0} \right)^{\alpha+1 \over \alpha} = 1$

and

$P V^{\alpha+1 \over \alpha} = P_0 V_0^{\alpha+1 \over \alpha} = P V^\gamma = \operatorname{constant}.$

Derivation of discrete formula

The change in internal energy of a system, measured from state 1 to state 2, is equal to

$\text{(1)} \qquad \delta U = \alpha R n_2T_2 - \alpha R n_1T_1 = \alpha R (n_2T_2 - n_1T_1)$

At the same time, the work done by the pressure-volume changes as a result from this process, is equal to

$\text{(2)} \qquad \delta W = P_2V_2 - P_1V_1$

Since we require the process to be adiabatic, the following equation needs to be true

$\text{(3)} \qquad \delta U + \delta W = 0$

Substituting (1) and (2) in (3) leads to

$\alpha R (n_2T_2 - n_1T_1) + (P_2V_2 - P_1V_1) = 0 \qquad \qquad \qquad$

$\frac {(P_2V_2 - P_1V_1)} {-(n_2T_2 - n_1T_1)} = \alpha R \qquad \qquad \qquad$

If it's further assumed that there are no changes in molar quantity (as often in practical cases), the formula is simplified to this one:

$\frac {(P_2V_2 - P_1V_1)} {-(T_2 - T_1)} = \alpha n R \qquad \qquad \qquad$

Graphing adiabats

An adiabat is a curve of constant entropy on the P-V diagram. Properties of adiabats on a P-V diagram are:

Every adiabat asymptotically approaches both the V axis and the P axis (just like isotherms).
Each adiabat intersects each isotherm exactly once.
An adiabat looks similar to an isotherm, except that during an expansion, an adiabat loses more pressure than an isotherm, so it has a steeper inclination (more vertical).
If isotherms are concave towards the "north-east" direction (45 °), then adiabats are concave towards the "east north-east" (31 °).
If adiabats and isotherms are graphed severally at regular changes of entropy and temperature, respectively (like altitude on a contour map), then as the eye moves towards the axes (towards the south-west), it sees the density of isotherms stay constant, but it sees the density of adiabats grow. The exception is very near absolute zero, where the density of adiabats drops sharply and they become rare (see Nernst's theorem).

The following diagram is a P-V diagram with a superposition of adiabats and isotherms:

The isotherms are the red curves and the adiabats are the black curves. The adiabats are isentropic. Volume is the abscissa (horizontal axis) and pressure is the ordinate (vertical axis).