Problem 62 - University Physics with Modern Physics

Question

A cylinder with a piston contains 0.250 mol of oxygen at \(2.40 \times 10^{5} \mathrm{Pa}\) 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 compresed 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. e

Step-by-Step Solution

Answer

The isothermal compression temperature is 710 K. Maximum pressure is 4.80x10^5 Pa. Total work done is -1446 J.

Step 1: Understanding the Processes

The gas undergoes three thermodynamic processes: 1) Isobaric expansion, where the pressure remains constant, and the volume doubles. 2) Isothermal compression, where the temperature remains constant, and it returns to its original volume. 3) Isochoric cooling, where the volume remains constant, and it returns to its original pressure.

Step 2: Isobaric Expansion on a PV Diagram

An isobaric process is represented as a horizontal line on a PV-diagram. Since the volume doubles and pressure remains constant, draw a line from the initial point (V, P) to the point (2V, P).

Step 3: Isothermal Compression on a PV Diagram

An isothermal process is represented as a hyperbolic curve on a PV-diagram. Draw a curve from (2V, P) back to (V, P'), where P' > P since the process is compressed.

Step 4: Isochoric Cooling on a PV Diagram

An isochoric process is represented as a vertical line on a PV diagram. Draw a vertical line downwards from (V, P') back to the original pressure (V, P).

Step 5: Calculate Temperature During Isothermal Compression

Using the ideal gas law, relate the initial and final states during the isothermal process: \(P_1 V_1 = P_2 V_2\), where \(P_1 = P\), \(V_1 = 2V\), and \(V_2 = V\). So, \(T = \frac{P_2 V_2}{nR}\), where \(n = 0.250 \text{ mol}, R = 8.314 \text{ J/mol K}\). Solve for T.

Step 6: Compute the Maximum Pressure

The maximum pressure occurs just before the isochoric cooling at (V, P'). Use \(P_1V_1 = P_2V_2\) again to find \(P'\): \(P' = P \times 2\), since \(V_1 = 2V\) and \(V_2 = V\).

Step 7: Compute Total Work Done

Calculate the work done for each process where applicable. 1) Isobaric work: \(W_1 = P(V_2 - V_1)\). 2) Isothermal work: \(W_2 = nRT \ln \left( \frac{V_1}{V_2} \right)\). 3) Isochoric process has \(W_3 = 0\). Sum \(W_1 + W_2 + W_3\) to get total work done.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is central to understanding the behavior of gases under various conditions. It is expressed with the formula \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin. This equation relates these physical properties of gases, providing insight into how they interact in different processes.

For instance, in an isothermal process, where temperature remains constant, the relationship simplifies to \( P_1V_1 = P_2V_2 \). This helps us predict how pressure and volume will change relative to each other to maintain a constant temperature. By manipulating this equation, we can solve for any unknown variable if the others are provided, which is crucial for understanding thermodynamic cycles and processes that involve gases behaving ideally.

PV Diagram

A PV diagram is a graphical representation of the changes in pressure (P) and volume (V) during different thermodynamic processes. It provides a visual way to understand how a gas behaves under certain conditions and is a valuable tool in thermodynamics.

In this exercise, the oxygen gas undergoes three processes which can be depicted on a PV diagram:

• The isobaric expansion is shown as a horizontal line since it occurs at constant pressure, and the volume doubles.
• The isothermal compression forms a hyperbolic curve because the process takes place at a constant temperature, returning the volume to its initial state but increasing the pressure.
• The isochoric cooling is represented by a vertical line indicating a decrease in pressure while the volume stays constant.

These graphical representations help to visualize and clarify the sequence and impact of thermodynamic processes on the gas.

Work Done in Thermodynamic Processes

Work done by or on a gas in thermodynamic processes can be measured in the context of mechanics, reflecting how energy is transferred. The work done depends on the process type—whether isobaric, isothermal, or isochoric—and can be calculated differently for each.

In this scenario:

• Isobaric work is calculated using \( W = P(V_2 - V_1) \), which measures the work done by the gas as it expands at constant pressure.
• Isothermal work involves a logarithmic function, \( W = nRT \ln \left( \frac{V_2}{V_1} \right) \), reflecting that more work is done during volume changes at constant temperature.
• Isochoric processes involve no work done, as the volume does not change.

Understanding how to compute work in various processes is critical for determining the efficiency and behavior of thermodynamic systems.

Isothermal and Isobaric Processes

Isothermal and isobaric processes are two types of thermodynamic processes with distinct characteristics that impact the behavior of gases.

**Isothermal Processes:**
In isothermal processes, the temperature remains constant. This means any changes in the volume of the gas are offset by changes in pressure, according to the formula \( P_1V_1 = P_2V_2 \). Isothermal processes are typically slow, allowing heat exchange with the surroundings to maintain temperature. On a PV diagram, these are shown as curves. During the isothermal compression of our oxygen gas, pressure increases as volume decreases, which results in a hyperbolic plot.

**Isobaric Processes:**
In contrast, isobaric processes occur at a constant pressure. Here, the gas's temperature and volume can change, but the pressure remains unchanged. The relationship can be explored using the Ideal Gas Law, reflecting that if volume increases, temperature must increase if the pressure is constant. This process typically involves heat flow into or out of the system to maintain pressure during volume changes, showing up as horizontal lines on a PV diagram in our example.