Adiabatic process

In thermodynamics, an adiabatic process is designated as one in which the thermodynamic system (generally, a fluid that performs work) does not exchange heat with its surroundings. An adiabatic process that is also reversible is known as an isentropic process. The opposite extreme, where maximum heat transfer takes place, causing the temperature to remain constant, is called an isothermal process.

The term adiabatic refers to volumes that prevent heat transfer with the surroundings. An isolated wall is very close to an adiabatic limit. Another example is the adiabatic flame temperature, which is the temperature that a flame could reach if there was no heat loss to the surroundings. In air conditioning, the humidification processes (supply of water vapor) are adiabatic, since there is no heat transfer, despite the fact that the air temperature and relative humidity can be varied.

Adiabatic heating and cooling are processes that commonly occur due to a change in the pressure of a gas, which entails variations in volume and temperature. The new values of the state variables can be quantified using the ideal gas law.

According to the first law of thermodynamics,

Δ Δ U+W=0(1){displaystyle Delta U+W=0qquad qquad qquad (1)}

where U is the internal energy of the system and W is the work done by the system. Any work (W) done must be done at the expense of energy U, as long as no heat Q has been supplied from outside. The work W done by the system is defined as

W=PΔ Δ V(2){displaystyle W=PDelta Vqquad qquad qquad (2)}

If the issue of the adiabatic process is related to waves, it must be taken into account that the adiabatic process or character only occurs in longitudinal waves

Relations P, V, T in the adiabatic expansion of the ideal gas

Joule, in his famous experiment on free expansion, showed that the internal energy of a perfect gas was independent of the volume (V), or the pressure (P), only a function of temperature.

This conclusion leads to the fact that, for an ideal gas:

(a)dU=nCvdT=δ δ Q− − δ δ W{displaystyle {text{(a)}qquad dU=n C_{v} dT=delta Q-delta W}

But in the adiabatic expansion:

(b)δ δ Q=0;δ δ W=Pδ δ V{displaystyle {text{(b)}qquad delta Q=0;qquad delta W=P delta V}

With which the following relationship is obtained:

(c)δ δ U=nCvδ δ T=− − Pδ δ V{displaystyle {text{(c)}qquad delta U=n C_{v} delta T=-Pdelta V}

In the ideal gas it is fulfilled:

PV=nRT{displaystyle P V=n R T}

Cp− − Cv=R{displaystyle C_{p}-C_{v}=R}

γ γ =Cp/Cv{displaystyle gamma =C_{p}/C_{v}}}

Values Cp{displaystyle C_{p}} and Cv{displaystyle C_{v}} are function of the number of atoms in the molecule.

Clearing P{displaystyle P} and replace P{displaystyle P} and R{displaystyle R} in the Ec.(c) remains, the differential relationship:

(d)dTT=− − (γ γ − − 1)dVV{displaystyle {text{(d)}qquad {frac {dT}{T}=}-{frac {left(gamma -1right) dV}{V}}}}}}}}

E integrating between the initial and final states:

(e)TfTi=(ViVf)(γ γ − − 1){displaystyle {text{(e)}}qquad {frac {T_{f}{T_{i}}=left({frac {V_{i}}}{V_{f}}}}}{right){left(gamma -1right)}}}}}

Considering that working with perfect gases is met T=PV/nR{displaystyle T=PV/nR}the Ec.(e) can get:

PfVfPiVi=(ViVf)(γ γ − − 1)→ → PfPi=(ViVf)(γ γ ){displaystyle qquad {frac {P_{f}{V_{f}}}{P_{i}}}}}{left({frac {V_{i}}{V_{f}}}{ right}{left(gamma)}{qquad to qquad {

Finally:

(f)PfVfγ γ =PiViγ γ =Cornstante{displaystyle {text{(f)}}qquad P_{f}V_{f}{f}{gamma} =P_{i}V_{i}{gamma }=constant}

operating on Eq.(e) and Eq.(f):

(g)TfTi=(PfPi)(γ γ − − 1γ γ ){displaystyle {text{(g)}}qquad {frac {T_{f}{T_{i}}}=left({frac {P_{f}}}{P_{i}}}}}{right)}{left({frac {gamma-1}{gamma }}}}}}}}}}{right

Calculation example

We will assume that the system is a monoatomic gas, so:CV=32R{displaystyle C_{V}={3 over 2}R} where R is the universal constant of gases.

Done Δ Δ P{displaystyle Delta P} and Δ Δ V{displaystyle Delta V} then. W=PΔ Δ V{displaystyle W=PDelta V} and Δ Δ U=32nRΔ Δ T=32Δ Δ (PV)=32(PΔ Δ V+VΔ Δ P)(3){displaystyle Delta U={3 over 2}nRDelta T={3 over 2}Delta (PV)={3 over 2}(PDelta V+VDelta P)qquad (3)}

Now substituting equations (2) and (3) into equation (1) we get

− − PΔ Δ V=32PΔ Δ V+32VΔ Δ P{displaystyle -PDelta V={3 over 2}PDelta V+{3 over 2}VDelta P}

simplifying

− − 52PΔ Δ V=32VΔ Δ P{displaystyle -{5 over 2}PDelta V={3 over 2}VDelta P}

dividing both sides of the equality by PV

− − 5Δ Δ VV=3Δ Δ PP{displaystyle -5{Delta V over V}=3{Delta P over P}}}

Applying the rules of differential calculus (that is, integrating both sides) we obtain that

− − 5Δ Δ (ln V)=3Δ Δ (ln P){displaystyle -5Delta (operatorname {ln} V)=3Delta (operatorname {ln} P)}

to take:Δ Δ lnV=(lnV− − lnV0);Δ Δ lnP=(lnP− − lnP0){displaystyle Delta lnV=(lnV-lnV_{0});Delta lnP=(lnP-lnP_{0})} can be expressed as

ln P− − ln P0ln V− − ln V0=− − 53{displaystyle {operatorname {ln} P-operatorname {ln} P_{0} over operatorname {ln} V-operatorname {ln} V_{0}}}}}=-{5 over 3}}}

For certain constants P0{displaystyle P_{0}} and V0{displaystyle V_{0}} from the initial state. Then

ln (P/P0)ln (V/V0)=− − 53,{displaystyle {operatorname {ln} (P/P_{0}) over operatorname {ln} (V/V_{0})}=-{5 over 3},}

ln (PP0)=− − 5/3ln (VV0){displaystyle operatorname {ln} left({P over P_{0}}right)=-5/3operatorname {ln} left({V over V_{0}}}}{right)}}

raising to the exponent both sides of the equality

(PP0)=(VV0)− − 5/3{displaystyle left({P over P_{0}}}right)=left({V over V_{0}right)^{-5/3}}}

removing the minus sign

(PP0)=(V0V)5/3{displaystyle left({P over P_{0}}}right)=left({V_{0} over Vright)^{5/3}}}

therefore

(PP0)(VV0)5/3=1{displaystyle left({P over P_{0}}}right)left({V over V_{0}right)^{5/3}=1}

and

PV5/3=P0V05/3=P0V0γ γ =constant.{displaystyle PV^{5/3}=P_{0}V_{0}^{5/3}=P_{0}V_{0}{gamma }=operatorname {constant}}. !

Graphical representation of adiabatic curves

During an adiabatic process, the internal energy of the fluid that performs the work must necessarily decrease.

Scheme of an adiabatic expansion.

The properties of adiabatic curves in a P-V diagram are as follows:

1. Each adiabatic approaches both axis of the P-V diagram as well as isothermics.
2. Each adiabatic gets intersected with each isoterma at one point.
3. An adiabatic curve looks like an isoterma, except that during an expansion, an adiabatic loses more pressure than an isoterma, so inclination is greater (it is more vertical).
4. If the isoterms are concaves towards the direction "northeast" (45°), then the adiabatics are concaves towards the direction "northeast" (31°).
5. If adiabatics and isothermics are drawn separately with regular changes in entropy and temperature, then as we move away from axis (in the northeast direction), it seems that the density of isothermics remains constant, but the density of adiabatics decreases. The exception is very close to absolute zero, where the density of adiabatics falls strongly and become very rare (See also: Nernst Theorem).

Calculation of the work involved

As described above, the equation that describes an adiabatic process of ideal gas, in a reversible process: PVγ γ =constant{displaystyle PV^{gamma }=operatorname {constant} } where P It's gas pressure, V volume and γ γ =CPCV{displaystyle gamma ={C_{P} over C_{V}}}}} the adiabatic coefficient, being CP{displaystyle C_{P}} the specific heat grind to constant pressure and CV{displaystyle C_{V}} the specific heat grind to constant volume.

For an ideal monoatomic gas, γ γ =5/3{displaystyle gamma =5/3}. For a diatomic gas (such as nitrogen or oxygen, the main components of the air) γ γ =7/5=1,4{displaystyle gamma =7/5=1.4}

As there is no external heat supply, any work (W) done will be at the expense of energy U, In the formula:

PVγ γ =K{displaystyle PV^{gamma }=operatorname {K} }

we make a small change, so it would look like this:

P=K/Vγ γ (4){displaystyle P=K/V^{gamma }qquad qquad qquad (4)}

Now deriving the work formula and integrating it at the same time we have:

∫ ∫ 12dW{displaystyle int _{1}^{2},dW} =∫ ∫ 12PdV(5){displaystyle int _{1}^{2},PdVqquad qquad qquad (5)}

Now we replace (4) in (5):

∫ ∫ 12K/Vγ γ dV{displaystyle int _{1}^{2},K/{V^{gamma }dV}

Now we know that "K" is a constant, therefore, it comes out of the integral:

K∫ ∫ 121/Vγ γ dV{displaystyle Kint _{1}^{2},1/{V^{gamma }d}V}

then we see that we have everything depending on the volume, so we integrate it:

(KV1− − γ γ )/(1− − γ γ ){displaystyle (KV^{1-{gamma }})/(1-{gamma }}}

as we know that:

PVγ γ =K{displaystyle PV^{gamma }=operatorname {K} }

then we substitute in the equation:

PVγ γ (V1− − γ γ )12/(1− − γ γ ){displaystyle PV^{gamma }(V^{1-{gamma }}})_{1}^{2}/(1-{gamma })}}

and multiply:

(P2V2γ γ V21− − γ γ − − P1V1γ γ V11− − γ γ )/(1− − γ γ ){displaystyle (P_{2}V_{2}{2}^{gamma }V_{2}{2}{1-{gamma }}}{1}V_{1}{gamma }V_{1⁄2}{1-{gamma }}}}{1-{gamma }}}}}}}}

after solving the equation we will have this form:

[chuckles]P2V2− − P1V1]/(1− − γ γ ){displaystyle [P_{2}V_{2}-P_{1}V_{1}]/(1-{gamma })}

and by definition we would be left with:

Δ Δ (PV)/(1− − γ γ ){displaystyle Delta (PV)/(1-{gamma })}

which in the end will give us:

nRΔ Δ T/(1− − γ γ ){displaystyle nRDelta T/(1-{gamma })}

and this will be equal to work:

W=(P2V2− − P1V1)/(1− − γ γ )=nR(T2− − T1)/(1− − γ γ ){displaystyle W=(P_{2}V_{2}-P_{1}V_{1}V_{1}})/(1-{gamma })=nR(T_{2}-T_{1})/(1-{gamma })}}

Adiabatic cooling of air

There are three relationships in the adiabatic cooling of air:

1. The ambient relationship of the atmosphere, which is the proportion to which the air cools as it gains altitude.
2. The adiabatic dry rate is about -1° per 100 meters high.
3. The adiabatic wet rate is about -0.6° - 0.3o per 100 meters of rise.

The first relation is used to describe the temperature of the surrounding air through which the rising air is passing. The second and third proportions are the references for an air mass that is rising in the atmosphere. The adiabatic dry rate applies to air that is below the dew point, for example if it is not saturated with water vapor, while the adiabatic wet rate applies to air that has reached its dew point. Adiabatic cooling is a common cause of cloud formation.

Adiabatic cooling doesn't have to involve a fluid. A technique used to achieve very low temperatures (thousandths or millionths of a degree above absolute zero) is adiabatic demagnetization, where the change in a magnetic field in a magnetic material is used to achieve adiabatic cooling.

Adiabatic processes in quantum mechanics

In quantum mechanics an adiabatic transformation is a slow change in quantum Hamiltonian H^ ^ {displaystyle {hat {H},} which describes the system and which results in a change in Hamiltonian's own values but not his own states, which is known as a avoided cross. For example, if a system starts in its fundamental state it will remain in the fundamental state even though the properties of this state can change. If in such a process a qualitative change occurs in the properties of the fundamental state, such as a change of spin the transformation is called quantum phase transition. The transitions of this type are phase transitions prohibited by classical mechanics.