Physical Chemistry , 1st ed.

Consider, then, the two-step process illustrated in Figure 3.3, where an irre-

versible step takes a system from a set 1 of conditions to a set 2 of conditions,

and then a reversible step takes it back to the original conditions. As a state

function, the sum of the steps equals the overall change for the entire process.

But from equation 3.15, the overall integral’s value must be less than zero.

Separating the integral into two parts:



2

1



dq

T

irrev+ 

1

2



dq

T

rev 0

The expression inside the second integral is, by the definition in equation 3.12,

dS. If we reverse the limits on the second integral (so both terms refer to the

same process going in the same, not opposite, directions), it becomes dS.We

therefore have



2

1

dq

T

irrev+ 

2

1

(dS) 0

or



2

1

dq

T

irrev

2

1

dS 0

The integral ofdSis S, so for this step we have



2

1



dq

T

irrevS 0



2

1



dq

T

irrevS

Reversing and generalizing for any step, we simply remove the specific limits:

S 

dq

T

irrev (3.16)

If we want to keep this in terms of infinitesimals (that is, without integral signs)

as well as include the original definition ofdSfrom equation 3.12, this becomes

dS 

d

T

q

 (3.17)

where again the equality is applicable to reversible processes, and the inequal-

ity is applicable to irreversible processes.

But consider that a spontaneous process isan irreversible process.

Spontaneous processes will occur if they can. With that in mind, we have the

following generalizations:

dS 

d

T

q

 for irreversible, spontaneous processes

dSd

T

q for reversible processes

74 CHAPTER 3 The Second and Third Laws of Thermodynamics

System with initial

set of conditions

(p 1 , V 1 , T 1 )

System with initial

set of conditions

(p 2 , V 2 , T 2 )

Step 1: IRREVERSIBLE

Step 2: REVERSIBLE

Figure 3.3 A representation of a process that has an irreversible step. See text for discussion.

Most real processes can be described like this, giving entropy a meaningful place in the under-

standing of real processes.