Ideal Gas Expansion: Calculating Work And Heat Transfer

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Ideal Gas Expansion: Calculating Work and Heat Transfer

Hey guys! Let's dive into a classic thermodynamics problem involving the expansion of an ideal gas (specifically, air) undergoing a polytropic process. We're going to calculate the work done and the heat transferred during this expansion. This kind of problem is super common in engineering and physics, so understanding the concepts and the calculations is a total win. We'll break it down step-by-step to make sure it's crystal clear. Ready to roll?

Understanding the Scenario: Polytropic Process of Air

Alright, so here's the deal. We have an ideal gas – think of it as a bunch of air molecules – that's initially at a pressure of 600 kPa (kilopascals) and occupies a volume of 0.1 mΒ³ (cubic meters). Now, this gas expands, meaning it takes up more space, until its pressure drops to 150 kPa. This expansion happens in a polytropic process. What's a polytropic process, you ask? Basically, it's a thermodynamic process that follows the relationship: PVn = constant, where:

  • P is the pressure.
  • V is the volume.
  • n is the polytropic index (in our case, n = 1.3).

This 'n' value is super important because it dictates how the pressure and volume change relative to each other during the expansion. Different values of 'n' describe different types of processes. For example, if n = 1, we have an isothermal process (constant temperature). If n equals the specific heat ratio (Ξ³), we get an adiabatic process (no heat transfer). But in our case, we have a polytropic process where n = 1.3. This means that both heat transfer and work are involved.

So, our goal is to figure out two things: (1) how much work the gas does during this expansion (because it's pushing against its surroundings), and (2) how much heat is either added to or removed from the gas during the process. This is where our thermodynamics equations come into play. It's like a puzzle, and we have to find all the missing pieces. This problem is not that difficult when you follow the steps. Let's see how this works! Get ready for some equations and calculations, but don’t worry, we'll keep it as simple as possible.

Polytropic Process Explained

The polytropic process is a fundamental concept in thermodynamics, describing a wide range of real-world processes. It's a generalization that encompasses several special cases like isothermal, adiabatic, isobaric, and isochoric processes. Understanding the polytropic process is crucial for analyzing the behavior of gases, particularly in engines, compressors, and other thermodynamic systems. Unlike an adiabatic process (where there's no heat exchange with the surroundings) or an isothermal process (where temperature is constant), a polytropic process allows for both heat transfer and changes in temperature. The key to analyzing a polytropic process lies in the polytropic index, 'n'. This index determines the specific relationship between pressure and volume during the process. The equation PVn = constant is the defining characteristic of a polytropic process. The value of 'n' influences how the gas behaves during expansion or compression, affecting the work done and the heat transfer involved. In the context of an ideal gas, the polytropic process is a practical model for many real-world scenarios. It allows engineers and scientists to estimate the work and heat transfer involved, vital for designing efficient and reliable thermodynamic systems. Therefore, the polytropic process is a versatile tool for analyzing and optimizing the performance of various engineering systems, making it a critical topic in thermodynamics.

Calculating the Work Done During Expansion

Alright, let's get down to business and calculate the work done during this expansion. The work done (W) during a polytropic process can be calculated using the following formula:

W = (Pβ‚‚Vβ‚‚ - P₁V₁) / (1 - n)

Where:

  • P₁ is the initial pressure (600 kPa).
  • V₁ is the initial volume (0.1 mΒ³).
  • Pβ‚‚ is the final pressure (150 kPa).
  • Vβ‚‚ is the final volume.
  • n is the polytropic index (1.3).

But wait, we don't have Vβ‚‚! We need to figure that out first. Since we know that PVn = constant during a polytropic process, we can use the following relationship:

P₁V₁n = Pβ‚‚Vβ‚‚n

Let's rearrange this to solve for Vβ‚‚:

Vβ‚‚ = (P₁ / Pβ‚‚)1/n V₁

Now we can plug in the values:

Vβ‚‚ = (600 kPa / 150 kPa)(1/1.3) * 0.1 mΒ³ Vβ‚‚ β‰ˆ 0.303 mΒ³

Excellent! We have our final volume. Now we can go back to our work formula and plug in the values:

W = (150 kPa * 0.303 mΒ³ - 600 kPa * 0.1 mΒ³) / (1 - 1.3) W β‰ˆ (-15.0 - 60.0) / -0.3 W β‰ˆ -45 / -0.3 W β‰ˆ 15 kJ

Therefore, the work done by the air during the expansion is approximately 15 kJ. This means the air did 15 kilojoules of work on its surroundings. That is awesome, right?

Work Done: Step-by-Step Breakdown

The calculation of work done in a polytropic process, as shown, involves a series of logical steps that reflect fundamental principles of thermodynamics. Here's a breakdown to make sure you've got it:

  1. Understand the Formula: Start with the work equation for a polytropic process: W = (Pβ‚‚Vβ‚‚ - P₁V₁) / (1 - n). This equation is a direct application of the work done by a gas during expansion or compression, accounting for the changing pressure and volume. It's essentially a modification of the general work equation, accounting for the polytropic index. Remember that this formula assumes that the process is quasi-static, meaning it happens slowly enough that the system remains close to equilibrium.
  2. Determine Vβ‚‚: Since the final volume Vβ‚‚ isn’t always immediately known, you first have to find it using the polytropic relationship P₁V₁n = Pβ‚‚Vβ‚‚n. This equation is derived from the definition of a polytropic process (PVn = constant). It states that the product of pressure and volume raised to the power of the polytropic index remains constant throughout the process. Solving for Vβ‚‚ gives you Vβ‚‚ = (P₁ / Pβ‚‚)(1/n) V₁.
  3. Plug in the Values: Substitute the known values of P₁, Pβ‚‚, V₁, and n into the equation to calculate Vβ‚‚. The accuracy of your calculation relies on correct units. Ensure that all pressure values are in the same unit (e.g., kPa) and that all volumes are in the same unit (e.g., mΒ³). Also, make sure that you are using the correct polytropic index.
  4. Calculate the Work: With all the necessary variables known, go back to the original work equation and plug in P₁, V₁, Pβ‚‚, Vβ‚‚, and n to find the work done W. Double-check your calculations to prevent errors. Ensure you maintain correct units throughout the calculation.
  5. Interpret the Result: A positive value of W means the gas did work on its surroundings (expansion), while a negative value means work was done on the gas (compression). The magnitude of W represents the energy transferred in the form of work during the process.

Calculating the Heat Transfer

Now, let's figure out the heat transferred during this process. We can use the first law of thermodynamics, which states:

Ξ”U = Q - W

Where:

  • Ξ”U is the change in internal energy.
  • Q is the heat transferred.
  • W is the work done.

We already calculated W, so we need to find Ξ”U. For an ideal gas, the change in internal energy depends only on the change in temperature. The formula for Ξ”U is:

Ξ”U = m cv Ξ”T

Where:

  • m is the mass of the gas.
  • cv is the specific heat at constant volume.
  • Ξ”T is the change in temperature.

Since we don't know the mass (m) or the specific heat at constant volume (cv), we can't directly calculate Ξ”U. However, we can use the following approach.

First, we need to know the initial and final temperatures. We can use the ideal gas law: PV = mRT, where:

  • R is the specific gas constant for air (approximately 0.287 kJ/kgΒ·K).
  • T is the temperature in Kelvin.

Let's calculate the initial and final temperatures (T₁ and Tβ‚‚).

For the initial state: T₁ = P₁V₁ / (mR)

For the final state: Tβ‚‚ = Pβ‚‚Vβ‚‚ / (mR)

Instead of calculating the exact temperatures, we can find the ratio of Tβ‚‚ / T₁

Tβ‚‚ / T₁ = (Pβ‚‚Vβ‚‚ / mR) / (P₁V₁ / mR) = (Pβ‚‚Vβ‚‚) / (P₁V₁)

Tβ‚‚ / T₁ = (150 kPa * 0.303 mΒ³) / (600 kPa * 0.1 mΒ³) = 0.7575

Let's assume the mass and specific heat are constant. Therefore, we can find a relationship between the heat and work done, because Q = W + Ξ”U. With the specific heat at constant volume we have cv = 0.718 (kJ/kgΒ·K)

Q = W + m cv Ξ”T

Using the polytropic relation:

Q = W + m cv (Tβ‚‚ - T₁)

We know W = 15 kJ, now we need to calculate m cv (Tβ‚‚ - T₁)

W = (Pβ‚‚Vβ‚‚ - P₁V₁) / (1 - n) = 15 kJ

First, from the initial conditions, we get m = P₁V₁ / (R T₁) , we get T₁

T₁ = P₁V₁ / (m R)

Then:

Tβ‚‚ = T₁ (Pβ‚‚Vβ‚‚) / (P₁V₁) = T₁ (0.7575)

Therefore:

Q = 15 + m cv (T₁ (0.7575) - T₁)

Q = 15 + m cv T₁ (0.7575 - 1)

Q = 15 - m cv T₁ (0.2425)

Remember that m cv = cv P₁V₁ / (R T₁) = 0.718 * (600 * 0.1) / (0.287 * T₁) = 150 / T₁.

So finally:

Q = 15 - 150 / T₁ T₁ * (0.2425)

Q = 15 - 36.375

Q β‰ˆ -21.375 kJ

The heat transferred during the process is approximately -21.375 kJ. The negative sign indicates that heat was removed from the system. This makes sense because, during expansion, the gas is doing work, and in a polytropic process, we can have a heat transfer as well.

Heat Transfer: Detailed Calculations

To determine heat transfer in a polytropic process, a systematic approach is essential. The first law of thermodynamics, which states Ξ”U = Q - W, is the foundation of the calculation. With the work done (W) already computed, the focus shifts to internal energy change (Ξ”U). For an ideal gas, internal energy is solely dependent on temperature change, which is reflected in the equation: Ξ”U = m cv Ξ”T, where:

  • m is the mass of the gas.
  • cv is the specific heat at constant volume.
  • Ξ”T is the change in temperature (Tβ‚‚ - T₁).

The challenge lies in the fact that we don't know the mass (m) or the specific heat at constant volume (cv) directly. Therefore, calculating Ξ”U requires us to find these unknown values. You can calculate the initial and final temperatures using the ideal gas law: PV = mRT. We can calculate a ratio Tβ‚‚ / T₁ = (Pβ‚‚Vβ‚‚) / (P₁V₁). This allows us to use an alternative route to find a solution. By knowing W and calculating Ξ”U we can obtain Q and thus calculate the heat transferred during the process.

Conclusion: Work, Heat, and Polytropic Magic

So, there you have it, folks! We've successfully calculated the work done and the heat transferred during the polytropic expansion of an ideal gas. We found that the air did approximately 15 kJ of work, and approximately -21.375 kJ of heat was transferred out of the system. This type of analysis is crucial for understanding how energy behaves in thermodynamic systems. Remember, the polytropic index is key to understanding the process.

Keep practicing these problems, guys! The more you work through them, the more comfortable you'll become with the concepts and equations. Good luck, and keep those thermodynamic wheels turning!

I hope this helps! If you have any questions, feel free to ask. Let me know if you want to try another example or have any questions about any of the steps.