Thermodynamic polytropic compression

[Vathral]

Homework Statement

I'm reading through my textbook and doing examples on polytropic process. I see this step and cannot figure out how the textbook gets V₃ = 0.0169 m³

P₂V₂ⁿ = P₃V₃ⁿ
(500 kPa)(0.05364 m³)^1.2 = (2000 kPa)V₃^1.2
V₃ = 0.01690 m³

Homework Equations

Not sure what to add other than it is a polytropic compression process

The Attempt at a Solution

I've tried multiplying the left side so its 26.82^1.2 then diving it it by 2000 to get rid of it from the right side to leave V₃ alone. Tried subtracting 1.2 by 1 and tried again...no dice.

 

[stewartcs]

Homework Statement

I'm reading through my textbook and doing examples on polytropic process. I see this step and cannot figure out how the textbook gets V₃ = 0.0169 m³

P₂V₂ⁿ = P₃V₃ⁿ
(500 kPa)(0.05364 m³)^1.2 = (2000 kPa)V₃^1.2
V₃ = 0.01690 m³

Homework Equations

Not sure what to add other than it is a polytropic compression process

The Attempt at a Solution

I've tried multiplying the left side so its 26.82^1.2 then diving it it by 2000 to get rid of it from the right side to leave V₃ alone. Tried subtracting 1.2 by 1 and tried again...no dice.

Solve for V3:

$$V_3 = \left(\frac{V_2^nP_2}{P_3}\right)^{1/n}$$

CS

 

[piercas]

I would like to clarify that the given problem is a specific case of the more general polytropic process, which is described by the equation P1V1n = P2V2n. This equation relates the initial pressure (P1) and volume (V1) to the final pressure (P2) and volume (V2) under a polytropic process, where n is the polytropic index. In this case, n = 1.2, indicating a non-ideal, non-isothermal compression process.

To solve for V3, we can rearrange the equation to be V3 = (P1V1n/P2)n. Plugging in the given values, we get V3 = (500 kPa)(0.05364 m3)1.2/(2000 kPa)1.2 = 0.0169 m3. Therefore, the textbook's solution is correct.

In general, solving for the final volume in a polytropic process involves using the equation V3 = (P1V1n/P2)n, where n is the polytropic index. It may also be helpful to use logarithms to simplify the calculations. Additionally, it is important to pay attention to the units and make sure they are consistent throughout the calculation.

 

[Mene]

I can understand your confusion and frustration with this step. It is important to carefully follow the equations and units to ensure accurate calculations. In this case, it seems that you are on the right track but may have made a small error in your calculations.

The equation for a polytropic process is P1V1^n = P2V2^n, where n is the polytropic index. In this case, n = 1.2. So, let's start by plugging in the given values:

(500 kPa)(0.05364 m^3)^1.2 = (2000 kPa)V3^1.2

Next, we can solve for V3 by dividing both sides by (2000 kPa):

(500 kPa)(0.05364 m^3)^1.2 / (2000 kPa) = V3^1.2

(500 kPa)(0.05364 m^3)^1.2 / (2000 kPa) = V3

V3 = 0.0169 m^3

So, it seems that the textbook's answer is correct. It is possible that you made a small error in your calculations or units, which resulted in a different answer. I would recommend double checking your work and units to see if you can identify where the discrepancy may have occurred. Alternatively, you can also ask your instructor or a classmate for help in understanding this step.

Overall, it is important to carefully follow the equations and units in thermodynamics to ensure accurate and meaningful results. Keep practicing and seeking help when needed, and you will become more comfortable with these concepts. Good luck with the rest of your homework!