Various close system processes

A close system may undergoes change of state by following methods.

1. Constant Volume process ( isochoric process)

2. Constant Pressure process ( isobaric process )

3. Constant temperature process ( isothermal process )

4. Hyperbolic process

5. Adiabatic process

6. Polytropic process

1. Constant Volume process ( isochoric process)

A process is said to be isochoric process if during the change of state, volume of the system remains constant.

Fig. P-V diagram of Isochoric Process

We know that

P₁V₁/ T₁ = P₂V₂ /T₂ { General Gas Equation }

V₁ = V₂ { constant volume process }

Therefore P₁/T₁ = P₂ /T₂

dU = mC𝗏dT

dH = mC𝗉dT

Work Transfer

W = ʃ PdV

In this case change in volume = 0

⇒ W = 0

No work is done in isochoric process.

Heat Transfer

From 1st Law of Thermodynamics, during any process

Q₁₋₂ = dU + W₁₋₂

In constant volume process

dU = mC𝗏dT

since W₁₋₂ = 0

Q₁₋₂ = mC𝗏dT

Or Q₁₋₂ = mC𝗏( T₂ – T₁)

2. Constant Pressure process ( isobaric process )

A process is said to be isobaric process if during any change of state, pressure of the system remains constant.

Fig. P-V diagram of Isobaric Process

We know that

P₁V₁/ T₁ = P₂V₂ /T₂ { General Gas Equation }

P₁ = P₂ { constant pressure process }

Therefore V₁/T₁ = V₂ /T₂

dU = mC𝗏dT

dH = mC𝗉dT

Work Transfer

v₂

W = ʃ PdV

v₁

v₂ v₂

W = P ʃdV = P│V│

v₁ v₁

W = P ( V₂ – V₁)

P = P₁ = P₂ { constant pressure process }

W = P₂V₂ – P₁V₁ -----(1)

Or PV = mRT

Or P₁V₁ = mRT₁

Or P₂V₂ = mRT₂

Inserting above values in eq. (1)

W = mR ( T₂ – T₁) -----(2)

Heat Transfer

Q₁₋₂ = dU + W₁₋₂ -----(3)

dU = mC𝗏 ( T₂ – T₁­)

W₁₋₂ = mR ( T₂ – T₁­)

By inserting the values of dU and W1-2 in eq. (3)

Q₁₋₂ = mC𝗏 ( T₂ – T₁) + mR ( T₂ – T₁)

Q₁₋₂ = m ( T₂ – T₁)( C𝗏 + R)

Q₁₋₂ = mC𝗉 ( T₂ – T₁) -----(4) { C𝗉 = C𝗏 + R }

3. Constant Temperature process ( isothermal process )

A process is said to be isothermal process in which during any change of state, the temperature of the system remains constant.

Fig. P-V diagram of Isothermal Process

We know that

P₁V₁/ T₁ = P₂V₂ /T₂ { General Gas Equation }

T₁ = T₂ { constant temperature process }

Therefore P₁V₁ = P₂V₂

When we draw this process on the P-V diagram it comes in the form of hyperbola.

dU = mC𝗏 ( T₂ – T₁)

T₁ = T₂ { Constant temperature process }

⇒ dU = 0

dH = mC𝗉 ( T₂ – T₁)

T₁ = T₂ { Constant temperature process }

⇒ dH = 0

Work Transfer

W = ʃ PdV -----(1)

During this process both P and V changes and hence the integration of equation (1) become difficult. So to simplify the process one term should be converted in the form of other terms.

We know for isothermal process

PV = P₁V₁ = P₂V₂ = C

Therefore P = C/V

Inserting this value in eq. (1)

W₁₋₂ = ʃ CdV/V

v₂ v₂

⇒ W₁₋₂ = C ʃ dV/V = C│log V │

v₁ v₁

W₁₋₂ = C [ log V₂ – log V₁ ]

W₁₋₂ = C log (V₂ /V₁) -----(2)

We know that the value of C in eq. (2) for isothermal process

C = P₁V₁ = P₂V₂

⇒ W₁₋₂ = P₁V₁ ln (V₂ /V₁)

We know that

PV = mRT

Therefore W₁₋₂ = mRT1 ln( V₂ /V₁) -----(3)

Heat Transfer

Q₁₋₂ = dU + W₁₋₂ -----(4)

dU = mC𝗏 ( T₂ – T₁­)

T₁ = T₂ { constant temperature process }

Therefore dU = 0

W₁₋₂ = P₁V₁ ln( V₂ /V₁)

Inserting above values in eq. (4)

Q₁₋₂ = 0 + W₁₋₂

Q₁₋₂ = W₁₋₂ = P₁V₁ ln( V₂ /V₁) -----(5)

Therefore during constant temperature process, total heat interaction is equal to total work interaction.

4. Hyperbolic Process

A process is said to be hyperbolic process when during any change of state, system follows the law PV=C

When we draw this process on P-V diagram, then it form the shape of hyperbola, so this process is called hyperbolic process.

Fig. P-V diagram of Hyperbolic Process

There is no physical difference between hyperbolic process and isothermal process. So all the formulae of isothermal process are applicable for hyperbolic process.

5. Reversible Adiabatic Process

A process is said to be reversible adiabatic process if during change of state there is no heat interaction with the surrounding takes place. Although the work interaction may takes place.

Fig. P-V diagram of reversible adiabatic process

Let us consider a process which is going from state 1-2 reversibly and adiabatically.

Let P₁ V₁ T₁ are the properties at initial state.

Let P₂ V₂ T₂ are the properties at final state.

𝛾

PV = C

If the process is going from state 1-2, then from 1st Law of Thermodynamics

Q₁₋₂ = dU + W₁₋₂

For reversible adiabatic process

Q₁₋₂ = 0; dU = mC𝗏dT; W₁₋₂ = ʃ PdV

Therefore 0 = mC𝗏dT + ʃ PdV

⇒ dT = -PdV/mCv -----(1)

PV = mRT

d ( PV ) = d ( mRT )

PdV + VdP = mRdT

dT = ( PdV +VdP )/mR

dT = ( PdV +VdP )/m(C𝗉 – C𝗏 ) ` -----(2) { R = C𝗉 – C𝗏 }

equating (1) and (2)

-PdV/mC𝗏 = ( PdV +VdP )/m(C𝗉 – C𝗏 )

- mC𝗉PdV + mC𝗏PdV = mC𝗏PdV + mC𝗏VdP

- mC𝗉PdV - mC𝗏VdP = 0

mC𝗉PdV + mC𝗏VdP = 0

Dividing both sides by mC𝗏VP

( C𝗉/C𝗏 ) × ( dV/V ) + ( dP/P) = 0

C𝗉/C𝗏 = 𝛾 {for adiabatic process}

𝛾 × ( dV/V ) + ( dP/P) = 0

integrating both sides

𝛾 ln ( V ) + ln ( P ) = C

𝛾

ln ( V ) + ln ( P ) = C

𝛾

ln ( PV ) = C

𝛾 ᴄ

PV = 𝘦

𝛾

⇒ PV = K -----(3)

𝛾 𝛾

P₁V₁ = P₂V₂

𝛾

⇒ P₁/P₂ = ( V₂ /V₁) -----(4)

We know that

P₁V₁/ T₁ = P₂V₂ / T₂ { General Gas Equation }

P₁/P₂ = ( V₂ /V₁) × ( T₁/ T₂) -----(5)

From eq. (4)

𝛾

P₁/P₂ = ( V₂ /V₁)

Inserting this value in eq. (5)

𝛾

( V₂ /V₁) = ( V₂ /V₁) × ( T₁/T₂)

𝛾

( V₂ /V₁) × ( V₁/V₂) = ( T₁/T₂)

𝛾-1

⇒ ( V₂ /V₁) = T₁/T₂ -----(6)

We know that

P₁V₁/ T₁ = P₂V₂ / T₂ { General Gas Equation }

P₁/P₂ = ( V₂ /V₁) × ( T₁/T₂) -----(7)

From eq. (4)

𝛾 1/𝛾

P₁/P₂ = ( V₂ /V₁) ⇒ V₂ /V₁ = ( P₁/P₂)

1/𝛾

P₁/P₂ = ( P₁/P₂) × ( T₁/T₂)

1/𝛾

( P₁/P₂) × ( P₂ /P₁) = T₁/T₂

1-(1/𝛾)

( P₁/P₂) = T₁/T₂

(𝛾-1)/𝛾

T₁/T₂ ­= ( P₁/P₂) -----(8)

Change in internal energy

dU = mC𝗏dT

change in enthalpy

dH = mC𝗉dT

Work done

W₁₋₂ = ʃ PdV

𝛾

PV = C

𝛾

⇒ P = C/V

𝛾

W₁₋₂ = ʃ CdV/V

v₂ 𝛾

W₁₋₂ = C ʃ dV/V

v₁

-𝛾+1 v₂

W₁₋₂ = C │V /(-𝛾+1)│

v₁

-𝛾+1 -𝛾+1

W₁₋₂ = C [ (V₂ /(-𝛾+1)) - (V₁ /(-𝛾+1)) ]

1-𝛾 1-𝛾

W₁₋₂ = C [(V₂ - V₁ ) /(1-𝛾)]

We know that

𝛾 𝛾

C = P₁V₁­ = P₂V₂

W₁₋₂ = [( P₂V₂ – P₁V₁)/(1-𝛾)]

Multiplying both numerator and denominator by -1

W₁₋₂ = [( P₁V₁ – P₂V₂)/(𝛾-1)]

Or W₁₋₂ = mR ( T₁ – T₂)/(𝛾-1) -----(9)

Heat Transfer

We know that

Q₁₋₂ = dU + W1-2

Q₁₋₂ = mC𝗏 ( T₂ – T₁) + mR ( T₁ – T₂)/(𝛾-1)

Q₁₋₂ = mC𝗏 ( T₂ – T₁) + mCv ( T₁ – T₂) { R/(𝛾-1) = Cv }

Q₁₋₂ = 0 -----(10)

6. Polytropic Process

It is a general process which is used to decide the behavior of various processes and follows

n

the general law PV = C.

n

PV = C -----(1)

Where n→0 - ∞

Expression for P-V-T ( Pressure – Volume – Temperature ) relation of various processes

Fig. P-V diagram of various processes

0

When n = 0 ⇒ PV = C ⇒ P = C ⇒ isobaric process

∞ 1/∞ 0

When n = ∞ ⇒ PV = C ⇒ P ×V = C ⇒ P × V = C ⇒ 1×V=C

⇒ V = C ⇒ isochoric process

When n = 1 ⇒ PV = C ⇒ isothermal process

𝛾

When n = 𝛾 ⇒ PV = C ⇒ reversible adiabatic process