How Does Changing Temperatures Affect Carnot Engine Efficiency?

[JSGhost]

Homework Statement

A Carnot heat engine operates between temperatures Th and Tc.
(a) If Th = 515 K and Tc = 363 K, what is the efficiency of the engine?
0.30 Your answer is correct.

(b) What is the change in its efficiency for each degree of increase in Th above 515 K?
? K-1

(c) What is the change in its efficiency for each degree of change in Tc?
? K-1

(d) Does the answer to part (c) depend on Tc?
Yes or No?

Explain.

Homework Equations

e_c = 1 - T_c/T_h

The Attempt at a Solution

(a)
T_h = 515k
T_c = 363k
e_c = 1 - T_c/T_h = 1 -363k/515k = 0.2951456311 = 0.30

I'm don't know how to start part (b). Do I use the same equation or a different one?

 

[kuruman]

You want to find the rate of change of the efficiency e with respect to T_h. Rate of change is also known as ...? (Hint: It's the "D" word.)

 

[JSGhost]

So do the derivative of e = 1 - T_c/T_h ?

 

[kuruman]

Yes.

 

[JSGhost]

e = 1 - T_c/T_h

How do you do the derivative of that equation?

dE = -ln(T_h/T_c) ?

 

[kuruman]

Would it help if you said let x = T_C and wrote

e = 1 - x/T_H?

Can you find de/dx ?

 

[JSGhost]

Actually I did not have to do the derivative. It was so simple roflol.

b) Increase T_h by one and calculate the efficiency

1 - 363/516= 0.2965

calculate difference from the original efficiency

0.2965-0.2951=.0014

c) Decrease T_c by one and calculate

1 - 362/515= 0.2971

Now find the difference

.2951 - .2971= 0.002

d) No