Chapter 19: Problem 41
Two moles of an ideal monatomic gas go through the cycle \(abc\). For the complete cycle, 800 J of heat flows out of the gas. Process \(ab\) is at constant pressure, and process \(bc\) is at constant volume. States \(a\) and \(b\) have temperatures \(T_a\) = 200 K and \(T_b\) = 300 K. (a) Sketch the \(pV\)-diagram for the cycle. (b) What is the work \(W\) for the process \(ca\)?
Short Answer
Step by step solution
Understand the Cycle
Draw the pV Diagram
Calculate Work in Process CA
Relate Heat and Work
Solve for W_ca
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Ideal Gas Processes
In the context of the exercise, the gas undergoes several types of processes, including constant pressure (isobaric) and constant volume (isochoric). Each of these processes results in different changes to the internal energy and work done by the system.
These processes are part of a cycle, which is pivotal in thermodynamics as it allows us to understand how heat and work are exchanged in a closed system. In any complete cycle, the change in internal energy (\(\Delta U\)) is zero, leading to a clear relationship between heat exchanged (\(Q\)) and work done (\(W\)). This relationship is critical for solving problems related to cycles and comprehending the efficiency of thermodynamic systems.
Isothermal Process
In an isothermal process involving an ideal gas, the internal energy stays constant since it is a function of temperature, which does not change. Therefore, the work done (\(W\)) by or on the gas is equal to the heat exchanged (\(Q\)) with the surroundings:
\[ Q = W \]
The formula to calculate work done during an isothermal expansion or compression is:\
\[ W = nRT \ln\left(\frac{V_f}{V_i}\right) \]
where \(V_f\) and \(V_i\) are the final and initial volumes, respectively. This formula derives from the integral of pressure with respect to volume under constant temperature conditions. This understanding is crucial when analyzing processes like \(ca\) in the exercise, where such conditions apply.
pV Diagram
In this exercise, the cycle involves moving from one state to another under constant pressure and volume, forming a series of straight lines and thus creating a rectangle or loop on the diagram.
- **Isobaric Process (ab):** Represented by a horizontal line, showing constant pressure with changing volume.
- **Isochoric Process (bc):** Depicted as a vertical line, illustrating constant volume while pressure changes.
The final part, process \(ca\), typically draws a curve (often an isothermal line), indicating simultaneous changes in pressure and volume at a constant temperature.
Using such diagrams makes it easier to calculate the work done during each process, as the area under the curve (or within the loop) invariably relates to the work performed by or on the system. Understanding the pV diagram equips you with deeper insights into various thermodynamic processes and how systems evolve from one state to another.