Ideal Gas Carnot engine refrigeration

[supermanii]

Homework Statement

What is the relationship between the heat absorbed and rejected by the reservoirs and their temperature. If a heat pump is used to transfer heat what is the rate at which heat is added to the hot reservoir in terms of the temperatures of the reservoir and the work done by the pump?

Homework Equations

Answer to the first part is $$\frac{Q_{h}}{Q_{c}}=\frac{\theta_{h}}{\theta_{c}}$$

The Attempt at a Solution

I got a solution of $$Q_{h}=\frac{W\theta_{h}}{\theta_{h}-\theta_{c}}$$ however I am very unsure of this and think I went about it the wrong way. Not even sure I have answered the question. Without a correct answer I can not do the rest of the question.

 

[collinsmark]

I got a solution of $$Q_{h}=\frac{W\theta_{h}}{\theta_{h}-\theta_{c}}$$ however I am very unsure of this and think I went about it the wrong way. Not even sure I have answered the question. Without a correct answer I can not do the rest of the question.

Looks fine to me. (I'm assuming you are using the symbol θ to represent temperature.)

But if you'd like a more formal discussion on the subject, look in your textbook or coursework for coefficient of performance (COP). It will explain the concept probably better than what I can do here.

 

[Andrew Mason]

I got a solution of $$Q_{h}=\frac{W\theta_{h}}{\theta_{h}-\theta_{c}}$$ however I am very unsure of this and think I went about it the wrong way.

Your answer is correct. Here's why: Since the system returns to its initial state after a complete cycle, there is no change internal energy so $\Delta Q = |Q_h|-|Q_c| = \Delta U + W = 0 + W = W$.

If you are using a Carnot heat pump, Qh/Qc = Th/Tc. So Qh/W = Qh/(|Qh|-|Qc|) = Th/(Th-Tc)

AM

 

[supermanii]

Thanks :)

 

[rwisz]

I would like to clarify that the question is asking about the relationship between heat absorbed and rejected by the reservoirs and their temperatures in an Ideal Gas Carnot engine refrigeration system. This system follows the Carnot cycle, which is a reversible thermodynamic cycle consisting of two isothermal and two adiabatic processes. The relationship between heat and temperature in this system is described by the Carnot efficiency, which is given by the equation:

η = (T_hot - T_cold) / T_hot

where T_hot and T_cold are the temperatures of the hot and cold reservoirs, respectively. This equation shows that the efficiency of the Carnot cycle is dependent on the temperature difference between the two reservoirs.

In terms of a heat pump, the rate at which heat is added to the hot reservoir can be calculated using the equation:

Q_hot = W * (T_hot / (T_hot - T_cold))

where W is the work done by the pump. This equation shows that the rate at which heat is added to the hot reservoir is directly proportional to the temperature of the hot reservoir and inversely proportional to the temperature difference between the two reservoirs. Therefore, the higher the temperature of the hot reservoir and the smaller the temperature difference, the more heat can be added to the hot reservoir by the heat pump.

I hope this clarifies the relationship between heat absorbed and rejected by the reservoirs and their temperatures in an Ideal Gas Carnot engine refrigeration system.