Problem 3 - University Physics with Modern Physics

Question

Two moles of an ideal gas are compressed in a cylinder at a constant temperature of \(85.0^{\circ} \mathrm{C}\) until the original pressure has tripled. (a) Sketch a \(p V\) -diagram for this process. (b) Calculate the amount of work done.

Step-by-Step Solution

Answer

Work done is about 6552.6 J.

Step 1: Understand the Process

This is an isothermal process, meaning the temperature remains constant. As the gas is compressed, its pressure increases while its volume decreases. The initial pressure is denoted as \( P_1 \) and the initial volume as \( V_1 \), with the final pressure being \( P_2 = 3P_1 \).

Step 2: Sketch the pV Diagram

For an isothermal compression, the pV diagram will display a hyperbolic curve. Since the volume decreases as pressure triples, you will sketch a downward curve starting at the initial point \((V_1, P_1)\) and ending at the point \((V_2, 3P_1)\) on the \(pV\) diagram.

Step 3: Use the Ideal Gas Law

Apply the ideal gas law equation \( PV = nRT \). Here, \(n = 2 \text{ moles}, T = 85^{\circ} \text{C} = 358.15 \text{K}, \text{and } R = 8.314 \text{ J/mol K}\). Initially, \( P_1V_1 = nRT \). Then \( P_2V_2 = nRT \) at the final state. Referring to these, realize \( V_2 = \frac{V_1}{3} \) since \( P_2 = 3P_1 \).

Step 4: Calculate Work Done

For an isothermal process, the work done on the gas is given by: \[ W = nRT \ln\left(\frac{V_1}{V_2}\right) \]Since \( V_2 = \frac{V_1}{3} \), we have:\[ W = 2 \times 8.314 \times 358.15 \ln(3) \]Compute the numerical value: \[ W = 2 \times 8.314 \times 358.15 \times 1.0986 \approx 6552.6 \text{ J} \]

Step 5: Final Verification

Recheck each step for consistency: constants used, proper natural logarithm calculation, and ensure the temperature is correctly converted into Kelvin. Confirm that the calculated work is reasonable given the scenario.

Key Concepts

Isothermal Process

In the realm of thermodynamics, an isothermal process is one where the temperature remains constant throughout the procedure. This means that any change in the pressure or volume of the gas occurs without a change in its temperature.
This is typically observed in ideal gases where the heat exchanged compensates for the energy requirement of the process. An isothermal process is significant because:

• It helps maintain equilibrium as the system adjusts pressure and volume.
• It ensures that the temperature of the system stays the same, despite the alterations.
• When ideal gases undergo isothermal changes, they follow the equation \(PV = nRT\), because temperature \(T\) remains constant.

In the example discussed, the compression of the gas at \(85.0^{\circ}C\) while keeping it isothermal implies that the heat exchange is managed so that the system neither cools down nor heats up.

Work Done by Gas

The concept of work done by gas in an isothermal process is quite intriguing. In physics, 'work' refers to the energy required to displace something, in this case, the molecules in an ideal gas. During an isothermal compression, the work done is calculated using the formula:
\[ W = nRT \ln\left(\frac{V_1}{V_2}\right) \] where \(W\) is the work done, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the consistent temperature which is held constant. Key points include:

• The natural logarithm of the volume ratio reflects how much compression or expansion occurs.
• The work done can be understood as the area under the pressure-volume curve in the PV diagram.
• The work done is positive when the gas is compressed, suggesting energy input to the gas.

In the solution, we computed this, obtaining approximately \(6552.6 \text{ J}\) as the work required to compress the gas until the pressure triples.

pV Diagram

A pressure-volume (pV) diagram visually represents the relationship between the pressure and volume of a gas in any thermodynamic process. In an isothermal process, the pV diagram showcases a hyperbolic curve. This is because, according to Boyle's Law (a subset of the ideal gas law), if the temperature is constant, as pressure increases, the volume decreases, and vice versa.

• An isothermal curve in a pV diagram is known as an "isotherm." This curve slopes downward and to the right as we transition from higher pressure and lower volume to lower pressure and higher volume and vice versa.
• The area under the curve represents the work done during the process.
• In the solution, this involved sketching a curve beginning at \((V_1, P_1)\) towards \((V_2, 3P_1)\), illustrating constant temperature compression.

Using a pV diagram in thermodynamics allows one to easily visualize and understand changes occurring in a system concerning volume and pressure.

Thermodynamics

Thermodynamics is a branch of physics that deals with the relationships between heat, work, temperature, and energy in systems. It is fundamental to understanding the behavior of gases in processes like the isothermal compression discussed. Key concepts in thermodynamics related to this problem include:

• The first law of thermodynamics, which relates to energy conservation: \( \Delta U = Q - W \), where \( \Delta U \) is the change in internal energy, \( Q \) is heat added to the system, and \( W \) is work done by the system.
• Isothermal processes illustrate how heat is conducted to maintain a constant temperature, thus not altering internal energy post-process.
• Ideal gases provide a simplified model to explain real gas behaviors under many conditions, enabling easier calculation and understanding of thermodynamic processes within a controlled environment.

Understanding thermodynamics helps us predict how energy transactional systems evolve over time and how other types of energy, like work and heat, interact within gas systems.

Compression of Gases

Compression involves reducing the volume of a gas, subsequently increasing its pressure as observed through Boyle's Law, which maintains that, at constant temperature, the pressure of a gas is inversely proportional to its volume. Important aspects of gas compression include:

• The temperature must be controlled to ensure a smooth, isothermal process. In the problem discussed, the temperature remained solid at \(85.0^{\circ}C\).
• The process of compression requires external work to be performed on the gas, often involving mechanical manipulation like piston movement within a cylinder.
• Thermodynamic processes, like those governed by the ideal gas law, provide insightful details on what happens internally to the gases during compression.

In an isothermal compression scenario, the pressure will rise as volume decreases, which involves careful regulation of heat to keep the temperature constant, demonstrating a perfect equilibrium of thermal dynamics in practice.