91Ó°ÊÓ

The drawing shows an ideal gas confined to a cylinder by a massless piston that is attached to an ideal spring. Outside the cylinder is a vacuum. The crosssectional area of the piston is \(A=2.50 \times 10^{-3} \mathrm{m}^{2}\) The initial pressure, volume, and temperature of the gas are, respectively, \(P_{0}, V_{0}=6.00 \times 10^{-4} \mathrm{m}^{3},\) and \(T_{0}=273 \mathrm{K},\) and the spring is initially stretched by an amount \(x_{0}=0.0800 \mathrm{m}\) with respect to its unstrained length. The gas is heated, so that its final pressure, volume, and temperature are \(P_{\mathrm{f}}, V_{\mathrm{f}},\) and \(T_{\mathrm{f}},\) and the spring is stretched by an amount \(x_{\mathrm{f}}=0.1000 \mathrm{m}\) with respect to its unstrained length. What is the final temperature of the gas?

Step by step solution

01

Understand the Relationship Between Force and Pressure

The force exerted by the spring, when it is stretched by an amount \( x \), is given by Hooke's Law, \( F = kx \), where \( k \) is the spring constant. The pressure exerted by the gas \( P \) on the piston can be related to the force exerted by the spring through \( P = \frac{F}{A} = \frac{kx}{A} \). Both the initial and final conditions will use this relationship.

02

Determine the Change in Spring Force

Initially, the spring is stretched by \( x_0 = 0.0800 \) m. In its final state, it's stretched by \( x_f = 0.1000 \) m. The change in the force exerted by the spring, \( \Delta F \), relates to the change in the spring's elongation \( \Delta x = x_f - x_0 = 0.1000 - 0.0800 \) m = 0.0200 m.

03

Set Up the Relation Using Ideal Gas Law

The initial and final states of the gas are related by the Ideal Gas Law, which can be expressed as \( P_0 V_0 = nRT_0 \) and \( P_f V_f = nRT_f \). We're solving for the final temperature \( T_f \). Assuming \( n \) and \( R \) are constants, the ratios of pressure, volume, and temperature between the initial and final states can be compared as \( \frac{P_0 V_0}{T_0} = \frac{P_f V_f}{T_f} \).

04

Calculate Final Pressure Using Spring Force Equation

The pressure exerted by the spring on the piston can be calculated for initial and final positions. Since \( P = \frac{k \cdot x}{A} \), and knowing that \( \Delta P = \frac{k \cdot \Delta x}{A} \), the final pressure \( P_f \) is \( P_0 + \Delta P \). Since the spring force is only affected by \( x \), and knowing \( \Delta x = 0.0200 \) m, compute \( \Delta P \).

05

Solve for Final Temperature

Substitute \( V_f \) as \( V_0 + A \cdot \Delta x \) because the volume change due to piston movement relates to change in spring elongation. Using \( P_f V_f = P_0 V_0 \frac{T_f}{T_0} \), solve for \( T_f \) with the calculated \( P_f \) and \( V_f \). Input values \( V_0 = 6.00 \times 10^{-4} \) m³ and \( A = 2.50 \times 10^{-3} \) m².

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hooke's Law

Hooke's Law is a fundamental principle that describes the behavior of springs when they are deformed. It states that the force exerted by a spring is directly proportional to the amount it is stretched or compressed from its original length. Mathematically, this is expressed as:

• \( F = kx \)

Here, \( F \) is the force exerted by the spring, \( k \) is the spring constant (a measure of the spring's stiffness), and \( x \) is the displacement from its equilibrium position. This relationship is crucial because it helps us calculate the force exerted by the spring on the piston in the given exercise, allowing us to connect the mechanical behavior of the spring to the thermodynamic properties of the gas.

Pressure and Volume

Pressure and volume are two interrelated physical properties of a gas that are essential for understanding how gases behave under different conditions. According to Boyle's Law, for a given amount of gas at a constant temperature, the pressure is inversely proportional to its volume:

• \( P \propto \frac{1}{V} \)

However, in combination with other variables, pressure and volume can be better understood through the Ideal Gas Law:

• \( PV = nRT \)

Where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles of gas, \( R \) is the ideal gas constant, and \( T \) is the temperature. Understanding how the change in volume due to the piston's movement affects the system pressure is key to solving the given problem and predicting the behavior of the gas during heating.

Spring Mechanics

Spring mechanics delve into the behavior and characteristics of springs, particularly how they store and exert energy. In our problem, the spring inside the cylinder influences the piston's motion and consequently the gas's volume and pressure. The mechanics of the spring can be examined through:

• Spring Constant \( k \): Determines the rigidity or softness of a spring.
• Displacement \( x \): The change in spring length when the force is applied.
• Force exertion: Calculating the exerted force via Hooke’s Law as \( F = kx \).

This exerted force results in changes to the gas pressure, which alters the thermodynamic state. By understanding these mechanics, you can manipulate how the piston moves and predict how these changes impact gas behavior.

Thermodynamics

Thermodynamics is the branch of physics that deals with heat and its relation to energy and work. The Ideal Gas Law is a cornerstone in thermodynamics, especially when working with gases confined in a system. In our scenario, the gas's initial temperature, pressure, and volume change with heating, meaning that the thermodynamic equilibrium shifts:

• Initial and final states relate mathematically as \( \frac{P_0 V_0}{T_0} = \frac{P_f V_f}{T_f} \).
• Conservation of mass allows \( n \) and \( R \) to remain constant during the process.
• Change in energy influences the gas's thermodynamic state and affects temperature \( T_f \).

Applying these thermodynamic principles enables us to calculate the final temperature when the system reaches equilibrium after being heated, using the adjustments in pressure and volume.