# Polytropic Process – Derivation Of Polytropic Process Equation and Work Done

Polytropic Process can be defined as the process in which heat absorbed by the gas due to unit rise in temperature is constant.

A polytropic process is a thermodynamic process that can be expressed using the following equation.**PV**^{n}**=C,** It is a**Polytropic process equation.**

Where,**P** is the pressure**V** is the volume**n** is the polytropic index**C** is a constant

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The exponential n can have any value minus infinity to plus infinity.**Poly** means many and this process is called the polytropic process because many processes are represented by this process based on the value of **n**.

Polytropic process equation can be used to describe various compression and expansion processes which include heat transfer.

In the polytropic processes, the specific heat is constant and in the non-polytropic process the specific heat is variable. Specific heat is constant means per degree rise in temperature same amount of heat is supplied

All the standard processes like **isobaric**, **isochoric**, **adiabatic**, and **isothermal** processes belong to the category of **polytropic **process.

Polytropic process equation can be used to describe various compression and expansion processes which include heat transfer.

Some of the most common values of exponential **n** of the polytropic process equation which corresponds to a particular process are:**Case 1: If n=0 Then pV**

Then, PV^{o} = Constant

=> P = constant

It corresponds to isobaric i.e constant pressure process.

Case 2: If n=1

Then, PV = Constant

It corresponds to isothermal i.e constant temperature process**Case 3: If n=**

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**∞**

it corresponds to isochoric i.e constant volume process

Case 4: If n=Gm

It corresponds to isentropic i.e constant entropy process

**Derivation Of Polytropic process equation:**

**Prequisites:****1)** First Law Of Thermodynamics:

dQ = dU + dW**2)** Ideal Gas Equation:

PV = nRT**3)** C_{p }=C_{v}+R**Starting From 1**^{st}** Law Of Thermodynamics, We have**

dQ = dU + dW

=> nCdT = nC_{v}dT + PdV

=> n (C – C_{v}) dT = PdV

( Now using ideal gas equation, PV = nRt

In differential form, PdV + Vdp = nRdT

=> C PdV + CVdP – C_{V}PdV-C_{V}VdP = 0

=> (C –C_{v}-R)PdV + (C-C_{V})VdP = 0

(Using C_{P}=C_{v}+R)

=> (C-C_{P})PdV + (C-C_{v}) Vdp=0

Now integrating both side we get,

=> PV^{n}= Constant**Hence the polytropic process equation is derived.**

=> n (C-C_{v})= C – C_{p}

=> nC – nC_{v}= C – C_{p}=> C (1-n) = C_{p}-nC_{v}=> C (1-n) = C_{v} + R – xC_{v}=> C (1-n) = C_{v}(1-n) + R

**Now we will see the polytropic process equation for different major processes:**

**i) Isobaric Process:**

In Isobaric process, C=C_{p}

**PV**^{0}** = Constt**, It is a polytropic process equation for an isobaric process.**ii) Isothermal Process:**In isobaric process, C=∞.

If c=∞

So polytropic process equation for isothermal process becomes:

=> PV^{1} = Constant

=> PV = Constant**iii) Adiabatic Process:**

In the Adiabatic process, C=0

So polytropic process equation for adiabatic process becomes:

**iv) Isohoric Process:**

In the isochoric process, C=C_{v}

So polytropic process equation for isochoric process becomes:

**Derivation Of Work Done For Polytropic Process:**

We know that in the polytropic process, PV^{n} = Constant = C

So , we have