Problem 64
Question
CALC Acylinder with a piston contains 0.250 mol of \(0 x y-\) gen at \(2.40 \times 10^{5}\) Pa and 355 \(\mathrm{K}\) . The oxygen may be treated as an ideal gas. The gas first expands isobarically to twice its original volume. It is then compressed isothermally back to its original volume, and finally it is cooled isochorically to its original pressure. (a) Show the series of processes on a \(p V\) -diagram. (b) Compute the temperature during the isothermal compression. (c) Compute the maximum pressure. (d) Compute the total work done by the piston on the gas during the series of processes.
Step-by-Step Solution
Verified Answer
(a) The processes form a rectangular path on PV-diagram. (b) Temperature during isothermal compression is 710 K. (c) Maximum pressure is 4.80 x10^5 Pa. (d) Total work involves adding work from expansion and compression.
1Step 1: Understand the Process
We have three processes in sequence: an isobaric expansion, an isothermal compression, and an isochoric cooling. We need to analyze them separately, noting that it starts at point (original pressure, volume) in the system.
2Step 2: Show Processes on PV-Diagram
On a PV diagram, start at point A (original pressure and volume). During isobaric expansion, the volume doubles, so move horizontally to point B. For isothermal compression, move vertically back to the original volume, reaching point C. Finally, cool isochorically back to the original pressure, returning to point A.
3Step 3: Calculate Temperature During Isothermal Compression
Use the ideal gas law. Initially, the gas had a temperature of 355 K. After isobaric expansion, calculate the temperature at point B using \(PV = nRT\) leading to a final temperature at point C during isothermal compression. Since volume returns to initial, the new temperature \(T_i\) is: \[ T_i = T_f = \frac{V_f}{V_i} \times T_i = 2 \times 355 \] Thus, the temperature \(T_i = 710\, \text{K}\).
4Step 4: Calculate Maximum Pressure
The maximum pressure occurs at point C during isothermal compression when the gas returns to the initial volume but seeks higher pressure. Use \(PV = nRT\) with constant temperature calculated previously. \[ P_{max} = P_i \times \frac{V_f}{V_i} = 2.40 \times 10^5 \times 2 = 4.80 \times 10^5 \, \text{Pa}\]
5Step 5: Compute the Total Work Done
Calculate work done for each process:1. Isobaric expansion: \(W_1 = P \Delta V = P(V_f - V_i) \)}2. Isothermal compression: \(W_2 = - nRT \ln(\frac{V_f}{V_i})\)Sum the work of two processes,Noting that during isochoric process, no work is done.Total Work done:\[ W = W_1 + W_2 + 0 \]
Key Concepts
Ideal Gas LawPV DiagramIsothermal ProcessesIsobaric Processes
Ideal Gas Law
The ideal gas law is one of the most fundamental equations in thermodynamics. It connects the four critical properties of gases: pressure (P), volume (V), number of moles (n), and temperature (T). The equation is written as \( PV = nRT \), where \( R \) is the universal gas constant.
Understanding this equation helps us predict how a gas will behave under different conditions. For instance, if you increase the temperature while keeping the volume constant, the pressure will go up. Conversely, if you increase the volume while keeping the temperature constant, the pressure will decrease.
In practical scenarios, like the one described in the exercise, the ideal gas law helps us calculate unknown values when we know the others. It is crucial for tracking the changes in a gas's state as it undergoes different processes like isothermal, isobaric, and even isochoric changes.
Understanding this equation helps us predict how a gas will behave under different conditions. For instance, if you increase the temperature while keeping the volume constant, the pressure will go up. Conversely, if you increase the volume while keeping the temperature constant, the pressure will decrease.
In practical scenarios, like the one described in the exercise, the ideal gas law helps us calculate unknown values when we know the others. It is crucial for tracking the changes in a gas's state as it undergoes different processes like isothermal, isobaric, and even isochoric changes.
PV Diagram
A PV diagram, or pressure-volume diagram, is a graphical representation of the changes in pressure and volume for a gas as it undergoes different thermodynamic processes.
This diagram is incredibly useful because it visually reflects the steps taken in a thermodynamic cycle. Each line or curve on the diagram represents a different process of change, such as expansion or compression of the gas.
In the context of this exercise, the PV diagram helps in visualizing the isobaric expansion, isothermal compression, and isochoric cooling processes. It starts from point A, moving horizontally to B during isobaric expansion, then vertically back to the original volume at C during isothermal compression, and finally, it returns to A as we cool isochorically. This cycle of movements allows us to assess the gas's state and energy changes effectively.
This diagram is incredibly useful because it visually reflects the steps taken in a thermodynamic cycle. Each line or curve on the diagram represents a different process of change, such as expansion or compression of the gas.
In the context of this exercise, the PV diagram helps in visualizing the isobaric expansion, isothermal compression, and isochoric cooling processes. It starts from point A, moving horizontally to B during isobaric expansion, then vertically back to the original volume at C during isothermal compression, and finally, it returns to A as we cool isochorically. This cycle of movements allows us to assess the gas's state and energy changes effectively.
Isothermal Processes
Isothermal processes are those where the temperature remains constant. For an ideal gas, this occurs when the system is adjusted slowly, allowing heat exchange with the surroundings to maintain a consistent temperature.
The ideal gas law simplifies in this case, as temperature doesn't change, meaning \( PV = nRT \) could be rewritten as \( P \propto \frac{1}{V} \). The result is a hyperbolic curve on a PV diagram.
During the isothermal compression in the exercise, the volume shrinks while the temperature remains constant, causing the pressure to rise. This inverse relationship between pressure and volume is pivotal in understanding how pressure increases when volume decreases in an isothermal manner. Calculating the temperature, like we did by recognizing that the volume ratio defined the final temperature, was a direct application of isothermal process understanding.
The ideal gas law simplifies in this case, as temperature doesn't change, meaning \( PV = nRT \) could be rewritten as \( P \propto \frac{1}{V} \). The result is a hyperbolic curve on a PV diagram.
During the isothermal compression in the exercise, the volume shrinks while the temperature remains constant, causing the pressure to rise. This inverse relationship between pressure and volume is pivotal in understanding how pressure increases when volume decreases in an isothermal manner. Calculating the temperature, like we did by recognizing that the volume ratio defined the final temperature, was a direct application of isothermal process understanding.
Isobaric Processes
Isobaric processes take place at constant pressure. When a gas expands or contracts at a fixed pressure, the relationship between the volume and temperature changes directly.
In the context of our exercise, the gas in the cylinder expanded isobarically. The volume of the gas doubled while the pressure remained steady, which, according to the ideal gas law, caused the temperature to rise. We used the formula \( PV = nRT \) to evaluate these changes, which makes isobaric processes a model for understanding everyday phenomena like inflating a tire or heating a gas in an open container.
On a PV diagram, an isobaric process is represented by a horizontal line. This visual cue indicates that as volume increases, we move directly to the right on the V axis, while maintaining the same pressure level throughout the process.
In the context of our exercise, the gas in the cylinder expanded isobarically. The volume of the gas doubled while the pressure remained steady, which, according to the ideal gas law, caused the temperature to rise. We used the formula \( PV = nRT \) to evaluate these changes, which makes isobaric processes a model for understanding everyday phenomena like inflating a tire or heating a gas in an open container.
On a PV diagram, an isobaric process is represented by a horizontal line. This visual cue indicates that as volume increases, we move directly to the right on the V axis, while maintaining the same pressure level throughout the process.
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