Adiabatic process

This article covers adiabatic processes in thermodynamics. For adiabatic processes in quantum mechanics, see adiabatic process (quantum mechanics). For atmospheric adiabatic processes, see lapse rate.

In thermodynamics, an adiabatic process or an isocaloric process is a thermodynamic process in which no heat is transferred to or from the working fluid. The term "adiabatic" literally means impassable (from Greek ἀ-διὰ-βαῖνειν, not-through-to pass), corresponding here to an absence of heat transfer. Conversely, a process that involves heat transfer (addition or loss of heat to the surroundings) is generally called diabatic.

For example, an adiabatic boundary is a boundary that is impermeable to heat transfer and the system is said to be adiabatically (or thermally) insulated; an insulated wall approximates an adiabatic boundary. Another example is the adiabatic flame temperature, which is the temperature that would be achieved by a flame in the absence of heat loss to the surroundings. An adiabatic process that is reversible is also called an isentropic process. Additionally, an adiabatic process that is irreversible and extracts no work is in an isenthalpic process, such as viscous drag, progressing towards a nonnegative change in entropy.

One opposite extreme—allowing heat transfer with the surroundings, causing the temperature to remain constant—is known as an isothermal process. Since temperature is thermodynamically 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. At the opposite extreme, 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.

1 Adiabatic heating and cooling
2 Ideal gas (reversible case only)
- 2.1 Derivation of continuous formula
- 2.2 Derivation of discrete formula
3 Graphing adiabats
4 See also
5 External links

Adiabatic heating and cooling

Adiabatic changes in temperature occur due to changes in pressure of a gas while not adding or subtracting any heat.

Adiabatic heating occurs when the pressure of a gas is increased from work done on it by its surroundings, ie a piston. Diesel engines rely on adiabatic heating during their compression stroke to elevate the temperature sufficiently to ignite the fuel. Similarly, jet engines rely upon adiabatic heating to create the correct compression of the air to enable fuel to be injected and ignition to then occur.

Adiabatic heating also occurs 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 as it does work on its surroundings. 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 Earth's atmosphere with orographic lifting and lee waves, and this can form pileus or lenticular clouds if the air is cooled below the dew point.

Rising magma also undergoes adiabatic cooling before eruption.

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 (reversible case only)

For a simple substance, during an adiabatic process in which the volume increases, the internal energy of the working substance must decrease

The mathematical equation for an ideal fluid undergoing a reversible (i.e., no entropy generation) 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. $\alpha$ 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, $\gamma = 5/3$ , and for a diatomic gas (such as nitrogen and oxygen, the main components of air) $\gamma = 7/5$ . Note that the above formula is only applicable to classical ideal gases and not Bose-Einstein or Fermi gases.

For reversible 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, $\delta Q=0$ . 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, $P V = n R T$ , 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 $C_{V} = \alpha R$ .

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

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

simplify:

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

and divide both sides by PV:

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

After integrating the left and right sides from $V_0$ to V and from $P_0$ 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,

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

and eliminate the negative sign to obtain

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

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

$\text{(2)} \qquad \Delta W = \int_{V_1}^{V_2}P\, dV$

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

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

By the previous derivation,

$\text{(4)} \qquad P V^\gamma = \text{constant} = P_1 V_1^\gamma$

Rearranging (4) gives

$P = P_1 \left(\frac{V_1}{V} \right)^\gamma$

Substituting this into (2) gives

$\Delta W = \int_{V_1}^{V_2}P_1 \left(\frac{V_1}{V} \right)^\gamma\, dV$

Integrating,

$\Delta W = P_1 V_1^\gamma \frac{V_2^{1-\gamma}-V_1^{1-\gamma}}{1-\gamma}$

Substituting $\lambda = \frac{\alpha+1}{\alpha}$ ,

$\Delta W = \alpha P_1 V_1^{\gamma} \left( V_2^{1-\gamma} - V_1^{1-\gamma} \right)$

Rearranging,

$\Delta W = \alpha P_1 V_1 \left( \left( \frac{V_2}{V_1} \right)^{1-\gamma} - 1 \right)$

Using the ideal gas law and assuming a constant molar quantity (as often happens in practical cases),

$\Delta W = \alpha n R T_1 \left( \left( \frac{V_2}{V_1} \right)^{1-\gamma} - 1 \right)$

By the continuous formula,

$\frac{P_2}{P_1}=\left(\frac{V_2}{V_1}\right)^{-\gamma}$

Or,

$\left(\frac{P_2}{P_1}\right)^{-1 \over \gamma}=\frac{V_2}{V_1}$

Substituting into the previous expression for $\Delta W$ ,

$\Delta W = \alpha n R T_1 \left( \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} - 1 \right)$

Substituting this expression and (1) in (3) gives

$\alpha n R (T_2 - T_1) = \alpha n R T_1 \left( \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} - 1 \right)$

Simplifying,

$T_2 - T_1 = T_1 \left( \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} - 1 \right)$

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 horizontal axis and pressure is the vertical axis.

External links

Article in HyperPhysics Encyclopaedia