September 5, 2001 Reading: Chapter Three Homework: 3.1,3.2,3.4,3.5,3.6 An isothermal reversible expansion of idea gas: Consider the expansion of an idea gas from (P1,V1,T1) to (P2,V2,T1) by an isothermal reversible process, what’s change of entropy? Isothermal: T = constant, dU = 0. PdVwqrev == maxδδ

dVVRdV

TP

TqdS rev

gas ===δ

1

2ln2

1

2

1VVRdV

VRdSS

V

V

S

Sgasgas ===∆ ��

As V2 > V1, the entropy is increased in the gas. However, since it is a reversible process, no degradation occurs, so overall no entropy created. Entropy is simply transferred from the surrounding (heat reservoir) to the gas, i.e., the entropy change in the heat reservoir must be

2

1

1

2 lnlnVVR

VVRSreser =−=∆

An adiabatic reversible expansion of idea gas: Consider the expansion of an idea gas from (P1,V1,T1) to (P’2,V2,T2) by an adiabatic reversible process, what’s change of entropy? (Here, we choose the same final volume as in the isothermal process) Idiabatic: δqrev = 0.

0==TqdS rev

gasδ

02

1

==∆ �S

Sgasgas dSS

In an adiabatic reversible process, PVγ = constant. And there is no change in entropy. So, during the adiabatic reversible expansion, all the states of the idea gas lies on a PVγ = constant line of equal entropy. In comparison, during the isothermal reversible expansion, all the states of the idea gas lies on a PV = RT line of equal internal enrgy. Thus, an adiabatic reversible process is an isentropic process. The work done by the gas, wmax, equals the decrease in the internal energy, which can be calculated as the following:

PA F=PA vacuum

diaphragm

P1V1T

Heat reservoir

reversible irreversible

γγ2

'211 VPVP = , γ

γ

2

11'2 V

VPP =

112

11

12

111

12

112'

22

)(11 TVV

VVRT

RVVP

RRVPT −

−

−

−

− ==== γ

γ

γ

γ

γ

γ

So, )(23)(

23

112

11

112 TVVTRTTRUw −

−

−=−−=∆−= γ

γ

We note that T2 < T1, as V1 < V2. So, let’s now heat the gas at constant volume (V2) to T1, to arrive at the same final state (P2,V2,T1) as arrived by an isothermal reversible process. Then during this constant volume process, 0=wδ dTCdUq Vrev ==δ

dTT

CTqdS Vrev

gas ==δ

2

1ln2

1

2

1TTCdT

TCdSS V

T

T

VS

Sgasgas ===∆ ��

1

21

1

12 ln)1(ln

VVC

VVCS VVgas −==∆ −

−

γγ

γ

For idea gas, 3/213/51 =−=−γ and 2/3RCV = ,

So, 1

2lnVVRS gas =∆

This is the same as we obtained for the isothermal reversible process. Therefore, once again, we show that the change of entropy is a unique function of states, pass independent; i.e., S is a state function. An reversible vs. irreversible isothermal expansion of idea gas: Expansion from V1 to V2 at constant temperature (T):

In a reversible process:

1

2lnVVRS gas =∆

1

2lnVVRSreser −=∆

0=∆+∆=∆ resergastotal SSS In a fully irreversible process (free expansion of gas): Removing the diaphragm, no work is done, w = 0; no heat exchange, q = 0. As the state of gas changes from (P1,V1,T) to (P2,V2,T), the change of entropy in the gas is

1

2lnVVRS gas =∆

i.e, regardless whether the process is carried out reversibly or irreversibly, as long as the change of state is the same, the change of entropy is the same, because S is a state function. However, there is no heat exchange between gas and heat reservoir, so

0=∆ reserS .

Then, 0ln1

2 >=∆+∆=∆VVRSSS resergastotal ,

which represents the maximum degradation of all “potential” work into heat. Therefore, for the same change of state of the gas, the change in S is the same regardless of whether the process is reversible or irreversible; the change of entropy in the surrounding (heat reservoir) is different. If we consider the total combined system of gas and reservoir, the total change of entropy is zero in the reversible process and is positive (increase) in the irreversible process. Second Law of Thermodynamics:

1. The entropy S, defined as TqdS rev /δ= , is a state function. 2. The entropy of a system in an adiabatic enclosure can never decrease; it

increases during an irreversible process and remains constant during a reversible process.

Math form: 0≥=� iirr dSdS

Maximum Work: The maximum work is done when the process is reversible.

irrsysirrsys dSTwdUdSTqdS ++=+= /)(/ δδ

syssysirrsyssys dUTdSTdSdUTdSw −≤−−=δ

syssys USTw ∆−∆≤