# The General Entropy Inequality

">" for irreversible processes
"=" for internally reversible processes
Path A is irreversible and
path B is internally reversible.
The general

#### entropy inequality

:
Rearrange:
Put into
differential form:
Definition of

#### entropy

:
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### Ch 7, Lesson C, Page 2 - The General Entropy Inequality

• Let’s use the definition of entropy to eliminate the 2nd integral in the Clausius Inequality.
• Next, let’s bring S2 and S1 over to the right-hand side of the equation.
• We get S2 minus S1 is greater than the integral from 1 to 2 of dQ over T along irreversible path A.
• This looks very similar to the integral from of the definition of entropy, doesn’t it ?
• The key difference is that this equation has a “greater than” symbol instead of an equal sign.
• In differential form, we see that dS is greater than dQ over T along an irreversible path.
• I just thought I would remind you that the differential form of the definition of entropy is dS is equal to dQ over T along an internally reversible path.
• We can combine these last two equations to obtain the general form of the entropy inequality.
• We find that dS is greater than or equal to dQ over T, where the greater than applies for irreversible processes and the equal sign applies for internally reversible processes.
• Another way to look at this is to observe what happens when we evaluate dQ over T along different paths connecting the same two states.
• The integral of dQ over T is always greater along a reversible path than it is along any real, irreversible path.
• So, what does all this have to do with entropy generation ?  Flip the page and let’s see.