How do temperature differences affect the work done in an ideal Carnot process?

[Pouyan]

Homework Statement

An ideal Carnot process operates between a heating bath with the temperature of 20 C and a mole of hydrogen is in a container with constant volume. During the process, the work is done to remove heat from the hydrogen gas and emit heat to the heat bath. Calculate the work necessary to cool the hydrogen gas to temperature -10 C, if the original's temperature is 0 C.

Homework Equations

With heat pump and Carnot:
Qc/W = Tc/Th -Tc

and in this case W= Qc (Th-Tc)/Tc

Qc = ν*2.5*R*dTc

The Attempt at a Solution

Well I figured I can solve Qc = 1 * 2.5* 8.31 * (10) J = 207.75 J
and W = 207.75 (30)/263 J= 23.7 J
But I see in my solution:

Qc =ν*2.5*R*dTc and W= Qc (Th-Tc)/Tc ⇒ dW =ν*2.5*R*((Th-Tc)/Tc)* dTc

and W = ∫dW=∫ν*2.5*R*((Th-Tc)/Tc)* dTc and this is between 273 K and 263 K. And W = 19.4 J.

I know with these engines we will do work with temperature differences (Th-Tc) but should we always use integrals if Th or Tc is changing like this exercise ?!
Think if for example we would find the value of work to change the heat reservoir from 20 C to 30 C, should we use again integrals to do this ?!

 

[DrClaude]

Yes, if the temperature of one of the reservoirs changes, you must integrate.