Heat Engine and temperature Question

[joker_900]

Homework Statement

Three identical bodies of constant thermal capacity are at temperatures 300 K, 300 K
and 100 K. If no work or heat is supplied from outside, what is the highest temperature
to which anyone of these bodies can be raised by the operation of heat engines?

Homework Equations

The Attempt at a Solution

I don't think I'm on the right lines at all. Apparently I am supposed to get a cubic equation to solve, so I think I have approached this all wrong. But here's what I did:

So I thought the highest temperature would be achieved by using one of the 300K and the 100K as a heat engine, and using the work from that to pump heat from that same 100K to the other 300K.

I set up expressions of the form dQ = CdT for net heat energy entering each reservoir, to find the total heat energy that enters each body throughout the whole process. I also did conservation of energy and related work out of engine = work into pump.

I said that the whole process stops when the first 300K body and the 100K body reach the same temperature, Tf, as then the engine stops and there is no more energy into pump heat.

I got: TH = T3i - 2Tf + T2i + T1i

where TH is the final temperature of the heated 300K body, Tji initial values 300K, 100K, 300K.

?




Thanks

 

[anathema]

for your question! Your approach is on the right track, but there are a few things that need to be clarified and corrected.

First, when calculating the net heat energy entering each reservoir, you should use the formula dQ = CdT, where C is the specific heat capacity of the body and dT is the change in temperature. This will give you the total heat energy absorbed by each body throughout the process.

Next, you correctly applied the conservation of energy principle, but the expression you wrote is incorrect. The correct expression is:

Work out of engine = Work into pump + Heat energy absorbed by heated body

You also correctly identified that the process will stop when the first 300K body and the 100K body reach the same temperature, but this temperature is not Tf. It is actually the average temperature between the two bodies, which can be calculated as (300K + 100K)/2 = 200K.

Putting all of this together, your final equation should be:

TH = T3i - 2*200K + T2i + T1i

where TH is the final temperature of the heated 300K body, Tji initial values 300K, 100K, 300K.

Now, to get a cubic equation, you can substitute in values for T3i, T2i, and T1i (which are known) and solve for TH. This will give you a cubic equation that can be solved to find the final temperature of the heated body.

I hope this helps! Let me know if you have any further questions.

 

[thomate1]

for your question. Your approach is somewhat on the right track, but you are missing some key concepts and equations. Let's break it down step by step.

First, let's define some terms:

- Heat engine: A device that converts heat energy into mechanical work.
- Thermal capacity: The amount of heat energy required to raise the temperature of a substance by 1 degree.
- Temperature: A measure of the average kinetic energy of the particles in a substance.

Now, let's look at the given scenario. We have three identical bodies with constant thermal capacity at temperatures of 300 K, 300 K, and 100 K. We want to know the highest temperature that any one of these bodies can be raised to by the operation of heat engines.

To solve this problem, we need to apply the first law of thermodynamics, which states that the total change in energy of a system is equal to the heat added to the system minus the work done by the system. In this case, we can assume that there is no work done by the heat engines, so we only need to consider the heat added to the system.

Next, we need to consider the efficiency of the heat engines. The efficiency of a heat engine is defined as the ratio of the work output to the heat input. In other words, it is the percentage of heat energy that is converted into work. The maximum efficiency of a heat engine is given by the Carnot efficiency, which is equal to 1 - (Tc/Th), where Tc is the temperature of the cold reservoir and Th is the temperature of the hot reservoir.

In this scenario, we can use one of the 300 K bodies as the hot reservoir and the 100 K body as the cold reservoir. Using the Carnot efficiency equation, we can calculate the maximum possible efficiency of the heat engine as 1 - (100/300) = 2/3.

Now, we can use the first law of thermodynamics to calculate the maximum temperature that one of the 300 K bodies can be raised to by the heat engine. Let's call this temperature Tf. The first law of thermodynamics tells us that the total change in energy of the system is equal to the heat added to the system, which is equal to the heat added to the hot reservoir (300 K body) minus the heat removed from the cold reservoir (100 K body). In equation form:

dU = Qh - Qc

where dU is