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  4. Polytropic Process - Practice Questions & MCQ

Polytropic Process - Practice Questions & MCQ

Updated on Sep 18, 2023 18:34 AM

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 An ideal gas undergoes a quasi-static, reversible process in which its molar heat capacity C remains constant.  If during  this process the relation of pressure P and volume V is given by PVn=constant,  then n is given by (Here CP and CV are molar specific heat at constant pressure and constant volume, respectively)

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Concepts Covered - 1

Polytropic Process

A process PV^{N} = C  is called polytropic process. So, any process in this world related to thermodynamics can be explained by polytropic process. 

For example - 1. If N = 1 , then the process become isothermal.

                       2. If N=0, then the process become isobaric.

                       3. If N = \gamma, then the process become adiabatic

Work done by polytropic process - 

                                                       W_{1-2}=\int P d V

                                     

                                                       \begin{array}{l}{\text { For a polytropic process, }} \\ {\qquad \begin{aligned} P V^{N} &=P_{1} V_{1}^{N}=P_{2} V_{2}^{N}=C \\ & P=\frac{C}{V^{N}} \end{aligned}}\end{array} 

                                                       \begin{array}{l}{\text { Subsiting in Equation } \text { , we get, }} \\ \\ {\qquad \begin{aligned} \int P d V &=\int \frac{C d V}{V^{N}}=C \int V^{-N} d v \\ \\ &=\left[V^{1-N}\right]_{1}^{2}=\left(V_{2}^{1-N}-V_{1}^{1-N}\right) \\ \\ W_{1-2} &=\frac{P_{2} V_{2}-P_1 V_{1}}{1-N} \text { or } \frac{P_1 V_{1}-P_{2} V_{2}}{N-1}. . . . . . (1) \\ \\ P_{1} V_{1} &=n R T_{1} \\ \\ P_{2} V_{2} &=n R T_{2} \end{aligned}}\end{array}

So, equation (1) can be written as - 

                                                        \begin{aligned} W_{1-2} &=\frac{n R\left(T_{2}-T_{1}\right)}{1-N} \\ \text { And for one mole, } & W_{1-2}=\frac{R\left(T_{2}-T_{1}\right)}{1-N} \end{aligned}

Specific heat for polytropic process - 

  We can write equation of heat as - Q=C \Delta T

Here C = Molar specific heat - 

                                                      \begin{array}{l}{\text { From the first law of thermodynamics }} \\ \\ {\qquad Q=\Delta U+W} \\ \\ {\text { or } C \Delta T=C_{v} \Delta T-\frac{R \Delta T}{(N-1)}} \\ \\ {\therefore \quad C=C_{v}-\frac{R}{(N-1)}=\frac{R}{(\gamma-1)}-\frac{R}{(N-1)}}\end{array}

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Polytropic Process

Study other Related Concepts

Polytropic Process Current Topic

Introduction To Thermodynamics
Thermodynamic State Variables And Equation Of State
Thermodynamic Equilibrium
Heat, Internal Energy And Work - Thermodynamics
First Law Of Thermodynamics
Isobaric Process
Isochoric Process
Isothermal Process
Adiabatic Process
Polytropic Process

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