**Title: Adiabatic Compression of Ethane in a Piston-Cylinder Assembly****Overview:**This problem involves two moles of ethane undergoing a reversible adiabatic compression in a piston-cylinder assembly. The initial volume of ethane is 0.1 m³, and it is compressed to a final volume of 0.002 m³. The initial pressure is 0.5 bar.**Heat Capacity of Ethane:**For ethane as an ideal gas, the heat capacity (\( \frac{c_p}{R} \)) is expressed as:\[ \frac{c_p}{R} = 1.131 + 1.9225 \times 10^{-2} T - 5.561 \times 10^{-6} T^2 \]where \( T \) is the temperature in Kelvin (K).**Tasks:**Use the van der Waals equation of state to determine:1. **Initial Temperature:** Calculate the initial temperature in Kelvin using both the van der Waals equation and compare it with the result obtained using the ideal gas law.2. **Final Pressure:** Determine the final pressure in bar once the compression is completed.3. **Final Temperature:** Calculate the final temperature in Kelvin after the compression.4. **Work Done:** Compute the work done during this adiabatic process, expressed in Joules (J).This exercise provides insights into the behavior of real gases under non-ideal conditions, demonstrating the application of both the ideal gas law and the van der Waals equation for realistic approximations.

Initial volume= 0.1 m3Final volume= 0.002 m3Initial pressure= 0.5 barStep 1: Analyze the van der Waals equation given: Step 2: Identify the given values:For ethane;a=5.57 L2 bar/mol2b=0.06499 L/molP=0.5 barStep 3: Calculate the solution:To solve this problem, we follow the steps outlined below:Start by analyzing the given problem and the equations provided.Step 1 involves finding the final temperature, Tf, for adiabatic process by using the equationSubstitute the given valuesEvaluate the value of γ (gamma) usingγ = cp/cv,where,Calculate Tf using the values obtained from step 3 and step 4.Simplify the equation and calculate the value of Tf.To find Pf, we use the equationgiven P1 = 0.5 and use the valuesto substitute them into the equation.Rearrange the equation to solve for Pf.Finally, we have the solution:Result:1. Initial temperature: 301.644 K2. Final temperature: 615.725 K3. Final pressure= 51.03 bar4. Work done = 44.114 x 103 J

Engineering

Mechanical Engineering

Two moles of ethane in a piston-cylinder assembly undergo a reversible adiabatic compression from the initial volume of 0.1 m² to the final volume of 0.002 m³. The initial pressure is 0.5 bar. Note that if ideal gas, the heat capacity of ethane is given by Cp = 1.131 + 1.9225 x 10-² T-5.561 x 10-6 T², where T is in K. Use the van der Waals equation of state to determine (1) the initial temperature in K (also compares the result with that calculated using the ideal gas law), (2) the final pressure in bar, (3) the final temperature in K, and (4) the work done during this process in J.

Two moles of ethane in a piston-cylinder assembly undergo a reversible adiabatic compression from the initial volume of 0.1 m² to the final volume of 0.002 m³. The initial pressure is 0.5 bar. Note that if ideal gas, the heat capacity of ethane is given by Cp = 1.131 + 1.9225 x 10-² T-5.561 x 10-6 T², where T is in K. Use the van der Waals equation of state to determine (1) the initial temperature in K (also compares the result with that calculated using the ideal gas law), (2) the final pressure in bar, (3) the final temperature in K, and (4) the work done during this process in J.

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Concept explainers

Work and Heat Transfer

It is generally related to thermal engineering which tells about the use, creation, conversion and transfer of heat energy between the systems. It is also considered that to transfer heat from one source to another, transfer of masses is also required. There are various mechanisms by which, transfer of heat takes place i.e. thermal conduction, convection, radiation or by change of phase. All these mechanisms have their own specific features which sometimes occur simultaneously within the same system.

Question

**Title: Adiabatic Compression of Ethane in a Piston-Cylinder Assembly**

**Overview:**
This problem involves two moles of ethane undergoing a reversible adiabatic compression in a piston-cylinder assembly. The initial volume of ethane is 0.1 m³, and it is compressed to a final volume of 0.002 m³. The initial pressure is 0.5 bar.

**Heat Capacity of Ethane:**
For ethane as an ideal gas, the heat capacity (\( \frac{c_p}{R} \)) is expressed as:
\[ \frac{c_p}{R} = 1.131 + 1.9225 \times 10^{-2} T - 5.561 \times 10^{-6} T^2 \]
where \( T \) is the temperature in Kelvin (K).

**Tasks:**
Use the van der Waals equation of state to determine:

1. **Initial Temperature:**
Calculate the initial temperature in Kelvin using both the van der Waals equation and compare it with the result obtained using the ideal gas law.

2. **Final Pressure:**
Determine the final pressure in bar once the compression is completed.

3. **Final Temperature:**
Calculate the final temperature in Kelvin after the compression.

4. **Work Done:**
Compute the work done during this adiabatic process, expressed in Joules (J).

This exercise provides insights into the behavior of real gases under non-ideal conditions, demonstrating the application of both the ideal gas law and the van der Waals equation for realistic approximations.

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