Anadiabatic process is athermodynamic process where a fluid becomes warmer or cooler without getting heat from, or giving it to, something else. Usually thetemperature instead changes because of changes inpressure .
Adiabatic cooling is the usual cause ofcloud convection or other cause, water condenses to make clouds, and in some casesprecipitation . Convection also causes cold air to sink. This warms it adiabatically, often destroying a cloud and sometimes causing any precipitation to evaporate before hitting the ground.
Adiabatic process If we take some fluids in an insulated system and expand or compress the system very fast, the system won't be able to gain or loose any heat (from outside of the system). (In reality, you can't get a perfectly insulated system. But in some cases, theheat transfer is so low that we can call them adiabatic process.)
From thefirst law of thermodynamics , we know that,[ 1]
Δ U = Δ Q − Δ W {\displaystyle \Delta U=\Delta Q-\Delta W} Here,Δ U {\displaystyle \Delta U} internal energy ;Δ Q {\displaystyle \Delta Q} heat andΔ W {\displaystyle \Delta W} work .
Since there occurs no heat transfer in adiabatic process, theΔ Q {\displaystyle \Delta Q} equation is,
Δ W = − Δ U {\displaystyle \Delta W=-\Delta U} Again, we know thatΔ W = p Δ V {\displaystyle \Delta W=p\Delta V} (W = F s {\displaystyle W=Fs} F = p A {\displaystyle F=pA} V = A s {\displaystyle V=As} so,Δ W = p Δ V {\displaystyle \Delta W=p\Delta V}
Here,V {\displaystyle V} volume ;F {\displaystyle F} force ;p {\displaystyle p} pressure ;A {\displaystyle A} area ands {\displaystyle s} distance .
So, the equation stands as,
Δ U = − p Δ V {\displaystyle \Delta U=-p\Delta V} Which means, by doing work (changing volume), one can change the internal energy (and so the temperature) in a adiabatic process.
In adiabatic process, increasing the volume will result in decrease of the internal energy In the case of Adiabatic process (see the upper section),
(1) p Δ V = − Δ U {\displaystyle {\text{(1)}}\qquad p\Delta V=-\Delta U} Again, in constant volume, theheat capacity ofonemole gas is,
C v = Δ Q Δ T , {\displaystyle C_{v}={\frac {\Delta Q}{\Delta T}},} Here,Δ T {\displaystyle \Delta T} temperature andT {\displaystyle T}
But, in constant volume (as no work is done),
Δ Q = Δ U {\displaystyle \Delta Q=\Delta U} So,
C v = Δ U Δ T , {\displaystyle C_{v}={\frac {\Delta U}{\Delta T}},} From the first (1) equation,
(2) p Δ V = − C v Δ T {\displaystyle {\text{(2)}}\qquad p\Delta V=-C_{v}\Delta T} Now, theIdeal gas law for one mole gas is,
p V = R T , {\displaystyle pV=RT,} Here,R {\displaystyle R} gas constant .
Bydifferentiating the last equation,
p Δ V + V Δ p = R Δ T {\displaystyle p\Delta V+V\Delta p=R\Delta T} Or,Δ T = p Δ V + V Δ p R {\displaystyle \Delta T={\frac {p\Delta V+V\Delta p}{R}}}
Now from equation (2), we get,
C v ( p Δ V + V Δ p R ) + p Δ V = 0 {\displaystyle C_{v}\left({\frac {p\Delta V+V\Delta p}{R}}\right)+p\Delta V=0} Since,R = C p − C v {\displaystyle R=C_{p}-C_{v}} Mayer's relation ),
(3) C v ( p Δ V + V Δ p C p − C v ) + p Δ V = 0 {\displaystyle {\text{(3)}}\qquad C_{v}\left({\frac {p\Delta V+V\Delta p}{C_{p}-C_{v}}}\right)+p\Delta V=0} Here,C p {\displaystyle C_{p}} heat capacity of gas at constant pressure.
Simplifying the equation (3),
V Δ p + p Δ V ( C p C v ) = 0 {\displaystyle V\Delta p+p\Delta V\left({\frac {C_{p}}{C_{v}}}\right)=0} If,
C p C v = γ {\displaystyle {\frac {C_{p}}{C_{v}}}=\gamma } Then,
(4) V Δ p + γ p Δ V = 0 {\displaystyle {\text{(4)}}\qquad V\Delta p+\gamma p\Delta V=0} By dividing both sides of the equation (4) withp V {\displaystyle pV}
(5) Δ p p + γ Δ V V = 0 {\displaystyle {\text{(5)}}\qquad {\frac {\Delta p}{p}}+\gamma {\frac {\Delta V}{V}}=0} Integrating (Integrationleaves a constant) the equation (5),
ln p + γ ln V = (constant) {\displaystyle \ln p+\gamma \ln V={\text{(constant)}}} If we assume that,
(constant) = ln k {\displaystyle {\text{(constant)}}=\ln k} We get,
(6) ln p + γ ln V = ln k {\displaystyle {\text{(6)}}\qquad \ln p+\gamma \ln V=\ln k} Simplifying the equation (6),
p V γ = (constant) = k {\displaystyle pV^{\gamma }={\text{(constant)}}=k} This is the relation between pressure and volume in adiabatic process.
