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## Homework Statement

Could someone look through my work? The parts where I wrote (??) are steps I am especially unsure about.

Many thanks in advance.

A large reservoir at temperature ##T_r## is placed in thermal contact with a small system at temperature ##T##. They end up at temperature ##T_r##. Given that ##C## is the heat capacity of the system, find the total change in entropy in the universe.

## Homework Equations

## The Attempt at a Solution

I started by looking for the entropy of the system. Since the volumes don't change, I can write

$$ dU_{s} = TdS_{s}$$

Since ##U_{s}## is a state function that depends only on the initial and final states of the system, I take some reversible pathway (??), which means

$$dU_{s} = TdS_{s} = \delta Q_{rev}$$

$$dS_{s} = \frac{1}{T} C dT$$

Integrating gives me $$\Delta S_{sys} = C \ln{\frac{T_r}{T}}$$

Then for the entropy of the reservoir: Conservation of energy, and the fact that the reservoir doesn't change its volume allows me to write (again, using some reversible path [?])

$$dU_{r} = T_{r}dS_{r} = -dU_{s} = -CdT$$

$$dS_{r} = -\frac{1}{T_{r}}CdT$$

Integrating gives me ##\Delta S_{r} = \frac{C(T - T_r)}{T_r}##

The total change is then the sum of these two changes in entropy.