JEE Main & Advanced Physics Thermodynamical Processes Adiabatic Process

When a thermodynamic system undergoes a change in such a way that no exchange of heat takes place between System and surroundings, the process is known as adiabatic process.

In this process P, V and T changes but \[\Delta Q=0\].

(1) Essential conditions for adiabatic process

(i) There should not be any exchange of heat between the system and its surroundings. All walls of the container and the piston must be perfectly insulating.

(ii) The system should be compressed or allowed to expand suddenly so that there is no time for the exchange of heat between the system and its surroundings.

Since, these two conditions are not fully realised in practice, so no process is perfectly adiabatic.

(2) Some examples of adiabatic process

(i) Sudden compression or expansion of a gas in a container with perfectly non-conducting walls.

(ii) Sudden bursting of the tube of bicycle tyre.

(iii) Propagation of sound waves in air and other gases.

(iv) Expansion of steam in the cylinder of steam engine.

(3) FLOT in adiabatic process : From \[\Delta Q=\Delta U+\Delta W\]

For adiabatic process \[\Delta Q=0\] \[\Rightarrow \] \[\Delta U=-\,\Delta W\]

If \[\Delta W=\] positive then \[\Delta U=\] negative so temperature decreases i.e. adiabatic expansion produce cooling.

If \[\Delta W=\] negative then \[\Delta U=\] positive so temperature increases i.e. adiabatic compression produce heating.

(4) Equation of state : In adiabatic change ideal gases do not obeys Boyle's law but obeys Poisson's law. According to it

\[P{{V}^{\gamma }}=\] constant; where \[\gamma =\frac{{{C}_{P}}}{{{C}_{V}}}\]

(i) For temperature and volume 

\[T{{V}^{\gamma -1}}=\]constant \[\Rightarrow \] \[{{T}_{1}}{{V}_{1}}^{\gamma -1}={{T}_{2}}{{V}_{2}}^{\gamma -1}\] or \[T\propto {{V}^{1-\gamma }}\]

(ii) For temperature and pressure 

\[\frac{{{T}^{\gamma }}}{{{P}^{\gamma -1}}}\]= const. \[\Rightarrow \]\[{{T}_{1}}^{\gamma }{{P}_{1}}^{1-\gamma }={{T}_{2}}^{\gamma }{{P}_{2}}^{1-\gamma }\]or\[T\propto {{P}^{\frac{\gamma -1}{\gamma }}}\]or \[P\propto {{T}^{\frac{\gamma }{\gamma -1}}}\]  

Special cases of adiabatic process

(5) Indicator diagram

(i) Curve obtained on PV graph are called adiabatic curve.

(ii) Slope of adiabatic curve : From \[P{{V}^{\gamma }}=\text{constant}\]

By differentiating, we get         

\[dP\,{{V}^{\gamma }}+P\gamma {{V}^{\gamma -1}}\,dV=0\]

\[\frac{dP}{dV}=-\gamma \frac{P{{V}^{\gamma -1}}}{{{V}^{\gamma }}}=-\gamma \left( \frac{P}{V} \right)\]

\[\therefore \] Slope of adiabatic curve\[\tan \varphi =-\gamma \left( \frac{P}{V} \right)\]

(iii) But we also know that slope of isothermal curve \[\tan \theta =\frac{-P}{V}\]

Hence \[{{\text{(Slope)}}_{\text{Adi}}}=\gamma \times {{\text{(Slope)}}_{\text{Iso}}}\] or \[\frac{{{\text{(Slope)}}_{\text{Adi}}}}{{{\text{(Slope)}}_{\text{Iso}}}}>1\]

(6) Specific heat : Specific heat of a gas during adiabatic change is zero As \[C=\frac{Q}{m\Delta T}=\frac{0}{m\Delta T}=0\] [As Q = 0]

(7) Adiabatic elasticity \[({{E}_{\phi }}):\] \[P{{V}^{\gamma }}=\text{constant}\]

Differentiating both sides \[d\,P{{V}^{\gamma }}+P\gamma {{V}^{\gamma -1}}dV=0\]

\[\gamma P=\frac{dP}{-dV/V}=\frac{\text{Stress}}{\text{Strain}}={{E}_{\varphi }}\]\[\Rightarrow \]\[{{E}_{\varphi }}=\gamma P\]

i.e. adiabatic elasticity is g times that of pressure

Also isothermal elasticity \[{{E}_{\theta }}=P\] \[\Rightarrow \] \[\frac{{{E}_{\varphi }}}{{{E}_{\theta }}}=\gamma =\frac{{{C}_{P}}}{{{C}_{V}}}\]

i.e. the ratio of two elasticity of gases is equal to the ratio of two specific heats.

(8) Work done in adiabatic process

\[W=\int_{{{V}_{i}}}^{{{V}_{f}}}{P\,dV}\]\[=\int_{{{V}_{i}}}^{{{V}_{f}}}{\frac{K}{{{V}^{\gamma }}}dV}\]\[\Rightarrow \]\[W=\frac{[{{P}_{i}}{{V}_{i}}-{{P}_{f}}{{V}_{f}}]}{(\gamma -1)}=\frac{\mu R({{T}_{i}}-{{T}_{f}})}{(\gamma -1)}\]

(As \[P{{V}^{\gamma }}=K,{{P}_{f}}{{V}_{f}}=\mu R{{T}_{f}}\] and \[{{P}_{i}}{{V}_{i}}=\mu R{{T}_{i}}\])

(i) \[W\propto \] quantity of gas (either M or \[\mu \])

(ii) \[W\propto \] temperature difference \[({{T}_{i}}-{{T}_{f}})\]

(iii)\[W\propto \frac{1}{\gamma -1}\] \[\because \] \[{{\gamma }_{mono}}>{{\gamma }_{di}}>{{\gamma }_{tri}}\]\[\Rightarrow \]\[{{W}_{mono}}<{{W}_{dia}}<{{W}_{tri}}\]

(9) Comparison between isothermal and adiabatic indicator diagrams : Always remember that adiabatic curves are more steeper than isothermal curves  

(i) Equal expansion : Graph 1 represent isothermal process and 2 represent adiabatic process

\[{{W}_{\text{isothermal}}}>{{W}_{\text{adiabatic}}}\]

\[{{P}_{\text{isothermal}}}>{{P}_{\text{adiabatic}}}\]

\[{{T}_{\text{isothermal}}}>{{T}_{\text{adiabatic}}}\]

\[{{(Slope)}_{\text{isothermal}}}<{{(Slope)}_{\text{Adiabatic}}}\]

(ii) Compression : Graph 1 represent adiabatic process and 2 represent isothermal process

\[{{W}_{Adiabatic}}>{{W}_{Isothermal}}\]

\[{{P}_{Adiabatic}}>{{P}_{Isothermal}}\]

\[{{T}_{Adiabatic}}>{{T}_{Isothermal}}\]

\[{{(Slope)}_{\text{Isothermal}}}<{{(Slope)}_{\text{Adiabatic}}}\]

(10) Free expansion : Free expansion is adiabatic process in which no work is performed on or by the system. Consider two vessels placed in a system which is enclosed with thermal insulation (asbestos-covered). One vessel contains a gas and the other is evacuated. When suddenly the stopcock is opened, the gas rushes into the evacuated vessel and expands freely.

\[\Delta W=0\] (Because walls are rigid)

\[\Delta Q=0\] (Because walls are insulated)

\[\Delta U={{U}_{f}}-{{U}_{i}}=0\] (Because \[\Delta Q\] and \[\Delta W\] are zero. Thus the final and initial energies are equal in free expansion.