- #1

- 150

- 0

## Homework Statement

You want to boil a pot of water at 20C by heating it to 100C. I suggest a way of heating the pot in a reversible manner: simply inserting a Carnot engine between the reservoir (at 100C) and the pot. The Carnot engine operates between two temperatures, absorbing heat [tex]dQ_1[/tex] from the 100C reservoir and rejecting heat [tex]dQ_2[/tex] to the pot at temperature T (heat capacity C). At the same time, work [tex]dW[/tex] is generated in this process (n.b.: I don't know if this is relevant; it was used in the second part of the question). Starting from 20C water temperature, receiving heat in a reversible manner, the pot of water reached 100C.

Show that the total entropy of the system (reservoir + pot) remains the same in the process. Ignore the heat capacity of the pot and the temperature dependence of the water heat capacity.

## Homework Equations

For a Carnot engine:

[tex]\frac{Q_H}{Q_L} = \frac{T_H}{T_L}[/tex]

[tex]dS = \frac{dQ}{T}[/tex]

## The Attempt at a Solution

Since all Carnot cycles imply zero change in entropy, rather than use numbers, I just use [tex]T_R,T_P[/tex] for the temperature of the reservoir and the pot, respectively. When I got stuck, I tried inserting the actual numbers but to no avail. For the reservoir, since its temperature remains constant, and we know how much work it does, its change in entropy is just [tex]\frac{-dQ_1}{T_R}[/tex]. Given the way that the problem is worded, I'm unsure if [tex]dQ_1[/tex] is already negative or not, so I put it here to be safe.

For the pot, its change in heat is [tex]dQ_2[/tex], so its change in entropy becomes:

[tex]\Delta S_P = dQ_2\int_{T_P}^{T_R} \frac{dT}{T} = dQ_2 \ln \frac{T_R}{T_P}[/tex]

We want [tex]\Delta S_P = -\Delta S_R[/tex], so we have

[tex]\frac{dQ_1}{T_R} = dQ_2 \ln \frac{T_R}{T_P}[/tex]

We know that

[tex]\frac{dQ_1}{dQ_2} = \frac{T_R}{T_P}[/tex], so dividing both sides by [tex]dQ_2[/tex], we get

[tex]\frac{1}{T_P} = \ln \frac{T_R}{T_P}[/tex]

But clearly this can't be true, and in fact, substituting in the given numbers show this. Where did I go wrong?

Thanks in advance! :)