Chapter 19: Problem 56
A cylinder with a piston contains 0.150 mol of nitrogen at 1.80 \(\times\) 10\(^5\) Pa and 300 K. The nitrogen may be treated as an ideal gas. The gas is first compressed isobarically to half its original volume. It then expands adiabatically back to its original volume, and finally it is heated isochorically to its original pressure. (a) Show the series of processes in a \(pV\)-diagram. (b) Compute the temperatures at the beginning and end of the adiabatic expansion. (c) Compute the minimum pressure.
Short Answer
Step by step solution
Understand the Initial Conditions
Calculate the Initial Volume
Isobaric Compression
Adiabatic Expansion to Original Volume
Temperature After Adiabatic Expansion
Isochoric Heating Back to Original Pressure
Compute Minimum Pressure
Draw the \(pV\)-Diagram
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Ideal Gas Law
- \(PV = nRT\)
- \(P\) stands for pressure, \(V\) for volume, \(n\) for the number of moles, \(R\) for the ideal gas constant (8.314 J/mol·K), and \(T\) is the temperature in Kelvin.
- \[V_1 = \frac{nRT_1}{P_1}\]
Adiabatic Process
- The key equation for an adiabatic process in terms of temperature and volume is: \(TV^{\gamma-1} = \text{constant}\), where \(\gamma\) is the adiabatic index (the ratio of specific heats).
- \[T_3 = T_2 \left(\frac{V_2}{V_1}\right)^{\gamma-1}\]
Isobaric Process
- The formula to describe the relationship between volume and temperature in an isobaric process is: \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \).
- It talks about calculating the final temperature \(T_2\) based on the compressional changes at constant pressure, where in one step, it was incorrectly calculated, reminding us of the importance of accurate calculations in thermodynamic processes. In reality, \(T_2\) should be carefully evaluated, respecting the state changes in terms of volume first.
Isochoric Process
- The Ideal Gas Law can be rearranged for an isochoric process as \(\frac{P_1}{T_1} = \frac{P_2}{T_2}\).
- The final pressure \(P_3\) equals the initial pressure, indicating the system has returned to its starting conditions in terms of pressure through this constant volume heating.