Universe entropy variation of one body and a reservoir

[Sabian]

Homework Statement

One body of constant pressure heat capacity $C_P$ at temperature $T_i$ it's placed in contact with a thermal reservoir at a higher temperature $Tf$. Pressure is kept constant until the body achieves equilibrium with the reservoir.
a) Show that the variation in the entropy of the universe equals:
$C_P [x − ln(1 + x )]$,
onde $x = − \frac {(Tf− Ti )}{Tf}$

(There might be some translation issues from Portuguese to English).

Homework Equations

All I can think is
$C_P= T \left ( \frac {\partial S}{\partial T} \right )_P$
$dU = T dS$
$dQ = C_P dT$
$dS = \frac {dQ}{T}$

The Attempt at a Solution

I haven't been able to understand the underlying physics properly. I attempted a "no idea what I'm doing but here I go" with the first equation, which led to nothing. I assume that the entropy variation of the universe should be the addition of the variation in the body and in the reservoir but can't seem to calculate them correctly.Thank you for your time people.

 

[Sabian]

I know it's rude to bump threads, but does anybody have the time and knowledge to help me?

 

[Chestermiller]

I know it's rude to bump threads, but does anybody have the time and knowledge to help me?

To get the total entropy change for the combination of system and surroundings, you have to take each of them separately along reversible paths from the initial to the final state. Can you think of a reversible path between the initial and final state for the system? Can you think of a reversible path between the initial and final state for the reservoir? How much heat is transferred between the system and the reservoir when it goes from the initial to the final state? All this heat is transferred to the reservoir at its (constant) temperature T_f.

That's all the hints I can give you without revealing every last detail.

 

[Sabian]

I could figure it out, thanks man, really.

 

[paranormal]

As a scientist, my response would be that the variation in entropy of the universe is a fundamental concept in thermodynamics and is related to the concept of irreversibility. In this particular scenario, the entropy variation can be calculated using the equation provided in the homework statement, which takes into account the constant pressure heat capacity, initial and final temperatures, and the temperature of the reservoir. This equation is derived from the fundamental equations of thermodynamics, such as the heat capacity, internal energy, and entropy equations.

To understand the physics behind this equation, we can think of the body as a closed system that is in contact with a thermal reservoir. When the body is placed in contact with the reservoir, heat will flow from the reservoir to the body until they reach thermal equilibrium. During this process, there is an exchange of energy and entropy between the body and the reservoir. The body will absorb heat and increase its internal energy, while the reservoir will lose heat and decrease its internal energy. However, the entropy of the universe, which is the sum of the entropy of the body and the reservoir, will increase due to the transfer of heat and the irreversible nature of the process.

The equation provided in the homework statement shows that the entropy variation of the universe is directly proportional to the heat capacity of the body and the temperature difference between the reservoir and the initial temperature of the body. It also takes into account the logarithmic relationship between the temperature difference and the temperature of the reservoir. This equation can be used to calculate the entropy variation in various thermodynamic processes, and it highlights the importance of considering the universe as a whole when analyzing thermodynamic systems.

In conclusion, the equation provided in the homework statement is a fundamental tool in understanding the concept of entropy variation and its relation to thermodynamic processes. It shows that the entropy of the universe always increases in irreversible processes, and it can be used to calculate the entropy variation in various scenarios, such as the one described in the homework statement.