# Work Check: Heat transfer between reservoir and small system

• WWCY
In summary, the conversation discusses the calculation of the total change in entropy in the universe when a large reservoir at temperature ##T_r## is placed in thermal contact with a small system at temperature ##T##. The solution involves finding the entropy of the system and the reservoir, and then taking the sum of these two changes.
WWCY

## Homework Statement

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

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.

## 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.

WWCY said:

## Homework Statement

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

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.

## 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.
Looks good. Nice job.

## What is heat transfer?

Heat transfer is the process of exchanging thermal energy between two systems due to a temperature difference.

## What is a reservoir in the context of heat transfer?

In the context of heat transfer, a reservoir is a large system with a constant temperature that can supply or absorb heat without changing its own temperature significantly.

## What is a small system in the context of heat transfer?

A small system in the context of heat transfer is a system that can exchange heat with a reservoir, but its temperature can change significantly due to the heat transfer.

## What factors affect the rate of heat transfer between a reservoir and a small system?

The rate of heat transfer between a reservoir and a small system is affected by the temperature difference between the two systems, the thermal conductivity of the materials, and the surface area of contact between the two systems.

## How is heat transfer quantified between a reservoir and a small system?

Heat transfer between a reservoir and a small system is quantified using the heat transfer equation, which takes into account the temperature difference, thermal conductivity, and surface area of contact between the two systems.

• Biology and Chemistry Homework Help
Replies
13
Views
421
• Biology and Chemistry Homework Help
Replies
2
Views
296
• Thermodynamics
Replies
11
Views
687
Replies
1
Views
1K
• Biology and Chemistry Homework Help
Replies
4
Views
308
• Introductory Physics Homework Help
Replies
2
Views
803
• Thermodynamics
Replies
19
Views
2K
• Thermodynamics
Replies
15
Views
2K
• Biology and Chemistry Homework Help
Replies
1
Views
191
• Introductory Physics Homework Help
Replies
2
Views
1K