# Volumes at C and D in a Carnot cycle

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In summary, the Carnot cycle, with conditions of Va = 7.5 L, Vb = 15 L, TH = 470°C, and TL = 260°C, uses a diatomic gas with y = 1.4. The pressures at a and b are Pa = 4.1 x 105 Pa and Pb = 2.1 x 105 Pa, respectively. To find the volumes at c and d, we use the adiabatic process equation PVy = constant, where points b and c are on the same adiabat, giving Pb Vbγ = Pc Vcγ. This can be further understood by considering the analogous situation for an isothermal process, where PV
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## Homework Statement

A Carnot cycle, shown in Fig. 20-7, has the following conditions: Va = 7.5 L, Vb = 15 L, TH = 470°C, and TL = 260°C. The gas used in the cycle is 0.50 mil of a diatomic gas, y = 1.4. Calculate (a) the pressures at a and b; (b) the volumes at c and d. (The rest of the problem...)

PVy = Constant

## The Attempt at a Solution

I already solved part (a), with Pa = 4.1 x 105 Pa and Pb = 2.1 x 105 Pa.

For part (b), I tried
PVy = constant
nRT⋅Vy-1 = constant.

Annnnd then I get lost. I don't understand what constant represents or if I am even on the right track. The book says Vc = 34 L and Vd = 17 L.

What do the points a, b, c, and d represent in the Carnot cycle? You didn't post the figure.

My apologies, here is the figure:

Points a, b, c, and d represent stages of the Carnot engine as it works.

For an adiabatic process, you have ##PV^\gamma = \text{constant}##. Since points b and c are on the same adiabat, you can say that ##P_b V_b^\gamma = P_c V_c^\gamma##.

If it still seems confusing, consider the analogous situation for an isothermal process. The righthand side of the ideal gas law, ##PV = nRT##, is constant for an isothermal process. In other words, on an isotherm, we have that ##PV = \text{constant}##. Points a and b are on the same isotherm, so you can say ##P_a V_a = P_b V_b##, which should look very familiar to you.

## 1. What is the purpose of calculating volumes at C and D in a Carnot cycle?

The volumes at C and D are important in understanding the efficiency of a Carnot cycle, as they represent the maximum and minimum volumes in the cycle. This information can be used to calculate the Carnot efficiency, which is a measure of how efficiently the engine converts heat into work.

## 2. How are the volumes at C and D determined in a Carnot cycle?

The volumes at C and D can be determined by using the ideal gas law, which states that the product of pressure and volume is directly proportional to the temperature. In a Carnot cycle, the volumes at C and D are the points where the temperature is at its highest and lowest, respectively.

## 3. Why is the Carnot efficiency highest at the volumes at C and D?

The Carnot efficiency is highest at the volumes at C and D because these points represent the maximum and minimum temperatures in the cycle. In a perfect Carnot cycle, all of the heat energy is converted into work at these temperatures, resulting in the highest possible efficiency.

## 4. How do the volumes at C and D affect the overall efficiency of a Carnot cycle?

The volumes at C and D play a crucial role in determining the overall efficiency of a Carnot cycle. The larger the difference in volume between these points, the higher the Carnot efficiency will be. This is because a larger volume difference means a greater temperature difference, allowing for more efficient conversion of heat into work.

## 5. Are the volumes at C and D the same for all Carnot cycles?

No, the volumes at C and D can vary depending on the specific conditions of the Carnot cycle, such as the type of gas used and the temperature at which the cycle operates. However, the efficiency of a Carnot cycle will always be highest when the volumes at C and D are at their maximum and minimum, respectively.

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