[eschewthecashew]

Why is it that the change in entropy depends upon heat supplied reversibly (qrev)? why must the change in entropy be calculated along a reversible path?

Also, I'm not undersanding how the efficiency of a heat engine correlates to entropy. any input is helpful.thanks

[FeLiXe]

that's just the definition of entropy: heat supplied reversibly divided by temperature
because of this definition, it makes sense to look at reversible paths when entropy is calculated

[eschewthecashew]

that's just the definition of entropy: heat supplied reversibly divided by temperature
because of this definition, it makes sense to look at reversible paths when entropy is calculated

Right... but there must be some qualitative reason for that definition, right?   I have a vague idea why,  but I'm still not entirely sure if my reasoning is okay. 

[Yggdrasil]

Recall that the total differential for the internal energy of a thermodynamic system is given by:

dE = TdS - PdV

Rearranging this to solve for dS, we get:

dS = (1/T)(dE + PdV)

Also recall that the amount of work done to expand a gas by an infinitessimal volume (dV) is given by:

dw = -P_extdV

From the first law of thermodynamics that dE = dw + dq

Combining this with our previous expression, we see that:

dE = dq - P_extdV
or
dq = dE + P_extdV

Now, here is the crucial answer to your question.  For a reversible process, P_ext = P and dw_rev = -PdV

Therefore, we now see that dE + PdV = dq_rev and therefore dS = dq_rev/T.

[0000000]

Why is it that the change in entropy depends upon heat supplied reversibly (qrev)? why must the change in entropy be calculated along a reversible path?

Also, I'm not undersanding how the efficiency of a heat engine correlates to entropy. any input is helpful.thanks

One update: it have to be a process with out a hysteresis also!!!

About the Carlo engine, the efficiency of it is smaller then 1, and this is not expected because this engine is imagined like idealized process, here is the need of defining the second thermodynamical principle!