Polytropic Process.pdf 6q4326

4-1 BOUNDARY WORK FOR POLYTROPIC PROCESS OF GASES

P

closed system, m=const

n,C are constants

n PV 1 1 = C

1

ln P

PV n = C P=

C Vn

ln P = −n ⋅ lnV + C

P2V2n = C

2

n 1 1

n 2 2

PV = P V

n=0

⇒

P = const isobaric

n=1

⇒

T = const isothermal ideal gas

n=k

⇒

s = const

isentropic

n=∞

⇒

v = const

isochoric

lnV

W V

V2

V1

PV n = C polytropic process

n≠1

W

2

2

1

1

= ∫ PdV = C ∫ V − n dV 1 = ( CV21− n − CV11− n ) 1− n

W

=

PV = mRT

C = P1V 1n = P2V 2n

P2V2 − PV 1 1 1− n

ideal gas

PV PV 1 1 = 2 2 T1 T2

⇐

P  T2 = 2  T1  P1 

1−

1 n

( 4.9 )

V  = 1   V2 

n −1

PV 1 1 = mRT1 mR (T2 − T1 ) 1− n

P2V2 = mRT2

W

=

PV = C process

W

=∫

W

= PV 1 1 ln

( 4.10 )

2

n=1

P PV 1 1 = C 1

P=

C V

V C dV = C ln 2 V V1 1 V2 V1

= P2V2 ln

V2 V1

= PV 2 2 ln

P1 P2

For ideal gas: PV = mRT = C ⇒ T = const

( 4.7 ) (isothermal process)

2 P2V2 = C

W

W

V

V1

n=0

= mRT ln

V2 V1

= mRT ln

P1 P2

PV 1 1 = P2V2

V2

P=C

isobaric process

(constant pressure process)

P

1

W

2 P = const W V

V1

V2

= P ⋅ (V2 − V1 ) = P ⋅ m ⋅ ( v2 − v1 )

( 4.6 )