# Raising temperature with no external work applied

• BOYLANATOR
In summary, the conversation discusses a system of three bodies with the same heat capacity, initially at different temperatures. It is shown that without any external energy or heat, the highest temperature that can be reached by operating heat engines between the bodies is approximately 488 K. The solution involves finding a conserved quantity in the process, which is important in thermodynamics.
BOYLANATOR

## Homework Statement

A system consists of three bodies with the same heat capacity. These are
initially at 200 K,400 K and 400 K respectively. Show that, without the
supply of any mechanical energy or heat from outside the system, the highest
temperature to which any of the bodies can be raised by operating heat
engines between them is 488 K.

## Homework Equations

efficiency = 1- (Tc/Th)
dU=dQ-dW

## The Attempt at a Solution

I believe the solution may have something to do with the work lost in one heat engine being used as useful work in another.

With A=200K, B=400K, C=400K, you could heat A and cool B - this gives work to heat C and cool B. And if you find a quantity which is conserved in such a process, you can show that no other process can be better.

I would add that the OP can assume TCfinal = 488K and work back to show that the "conserved quantity" is indeed conserved, rather than solve for TCfinal = 488 K.
.

Last edited:
488 is not the exact answer, but it is possible to show that it has to be close to 488 with that approach.

Ok I'm guessing the conserved quantity is energy and the rounding error may be from 487.5 which is like 7/8 but I'm not sure how to prove this.

BOYLANATOR said:
Ok I'm guessing the conserved quantity is energy and the rounding error may be from 487.5 which is like 7/8 but I'm not sure how to prove this.
Since it's a closed system, energy will be conserved. E.g. you could just let two bodies equalise in temperature. What thermodynamic quantity is conserved in 100% efficient processes?

Energy conserved, but there is another conserved quantity, which is very important in thermodynamics.
Energy conservation alone would allow to cool two bodies to 0K and to heat the third one to 1000K.

## 1. How is it possible to raise temperature without applying external work?

It is possible to raise the temperature of a system without applying external work by introducing heat energy into the system through various processes, such as chemical reactions, nuclear reactions, or electromagnetic radiation.

## 2. Can temperature be raised indefinitely without external work?

No, temperature cannot be raised indefinitely without external work. The laws of thermodynamics dictate that energy cannot be created or destroyed, only transferred. Therefore, eventually the energy source that is raising the temperature will run out and the temperature will reach an equilibrium.

## 3. What are some examples of raising temperature without external work?

Examples of raising temperature without external work include burning wood to create a fire, generating heat through friction, and using a microwave to heat up food. In all of these cases, energy is being transferred into the system without the need for external work to be applied.

## 4. Can raising temperature without external work lead to a perpetual motion machine?

No, raising temperature without external work cannot lead to a perpetual motion machine. The laws of thermodynamics state that energy cannot be continuously created without an external source, therefore a perpetual motion machine is impossible.

## 5. Is raising temperature without external work a sustainable method of heating?

It depends on the energy source being used to raise the temperature. If the energy source is renewable, such as solar or wind power, then it can be considered a sustainable method of heating. However, if the energy source is fossil fuels, then it is not sustainable as these resources are finite and contribute to climate change.

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