- #1

Pouyan

- 103

- 8

- Homework Statement
- Derive the Clausius' theorem

- Relevant Equations
- Qh/Ql= Th/Tl

ΔW= ΣdQi

I see this in my book but there is something I don't get!

If we consider a Carnot cycle where heat Qh enters and heat Ql leaves,

We know Qh/Ql=Th/Tl

And we define ΔQ_rev then :

∑(ΔQ_rev/T) = (Qh/Th) - (Ql/Tl) =0

I insert an image:

Which shows the heat dQi entering the reservoir at Ti from a reservoir at temperature T via a Carnot (Ci).

We know:

heat to reservoir at Ti / Ti = heat from the reservoir at T/ T

So : dQi/Ti = (dQi+dWi)/T

and rearranging:

dWi=dQi((T/Ti) -1)

The system in the image seems to convert heat to work but it cannot convert 100% of heat to work according to Kelvin's statement of the second law and hence we must insist that this is not the case. Hence:

Total work produced per cycle = ΔW + ∑(dWi) =< 0

But there is something I don't understand!

ΔW = ∑(dQi)

and dWi=dQi((T/Ti) -1)

So the total work produced per cycle = T ∑(dQi/Ti)

But why we say that this is less or equal than zero ?

T ∑(dQi/Ti) =< 0

T > 0.

Ti must be more than zero of course (Ti > 0)

How can we say that dQi =< 0 ?

Is that because it leaves from the reservoir with lower temperature ?

If we consider a Carnot cycle where heat Qh enters and heat Ql leaves,

We know Qh/Ql=Th/Tl

And we define ΔQ_rev then :

∑(ΔQ_rev/T) = (Qh/Th) - (Ql/Tl) =0

I insert an image:

Which shows the heat dQi entering the reservoir at Ti from a reservoir at temperature T via a Carnot (Ci).

We know:

heat to reservoir at Ti / Ti = heat from the reservoir at T/ T

So : dQi/Ti = (dQi+dWi)/T

and rearranging:

dWi=dQi((T/Ti) -1)

The system in the image seems to convert heat to work but it cannot convert 100% of heat to work according to Kelvin's statement of the second law and hence we must insist that this is not the case. Hence:

Total work produced per cycle = ΔW + ∑(dWi) =< 0

But there is something I don't understand!

ΔW = ∑(dQi)

and dWi=dQi((T/Ti) -1)

So the total work produced per cycle = T ∑(dQi/Ti)

But why we say that this is less or equal than zero ?

T ∑(dQi/Ti) =< 0

T > 0.

Ti must be more than zero of course (Ti > 0)

How can we say that dQi =< 0 ?

Is that because it leaves from the reservoir with lower temperature ?