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Chapter 14: Problem 64

A gas fills the right portion of a horizontal cylinder whose radius is \(5.00 \mathrm{cm} .\) The initial pressure of the gas is \(1.01 \times 10^{5} \mathrm{Pa}\) A frictionless movable piston separates the gas from the left portion of the cylinder, which is evacuated and contains an ideal spring, as the drawing shows. The piston is initially held in place by a pin. The spring is initially unstrained, and the length of the gas-filled portion is \(20.0 \mathrm{cm} .\) When the pin is removed and the gas is allowed to expand, the length of the gas-filled chamber doubles. The initial and final temperatures are equal. Determine the spring constant of the spring.

Short Answer

Expert verified
The spring constant is 1965 N/m.

Step by step solution

01

Identify Known Values and Conditions

Firstly, identify the known values and conditions:- Initial pressure of the gas, \( P_1 = 1.01 \times 10^5 \, \text{Pa} \).- Radius of the cylinder, \( r = 5.00 \, \text{cm} = 0.05 \, \text{m} \).- Initial length of the gas-filled portion, \( L_1 = 20.0 \, \text{cm} = 0.2 \, \text{m} \).- Final length of the gas-filled portion, \( L_2 = 2L_1 = 0.4 \, \text{m} \).- The initial and final temperatures are equal, implying it is an isothermal expansion.
02

Calculate Initial and Final Volumes

Calculate the initial volume \( V_1 \) and the final volume \( V_2 \) of the gas using the formula for the volume of a cylinder, \( V = \pi r^2 L \):\[V_1 = \pi (0.05)^2 (0.2) = 1.57 \times 10^{-3} \, \text{m}^3\]\[V_2 = \pi (0.05)^2 (0.4) = 3.14 \times 10^{-3} \, \text{m}^3\]
03

Use the Ideal Gas Law for Isothermal Process

Since temperature is constant, use the ideal gas law in the form for isothermal processes \( P_1 V_1 = P_2 V_2 \) to find \( P_2 \).\[1.01 \times 10^5 \cdot 1.57 \times 10^{-3} = P_2 \cdot 3.14 \times 10^{-3}\]Solving for \( P_2 \):\[P_2 = \frac{1.01 \times 10^5 \cdot 1.57 \times 10^{-3}}{3.14 \times 10^{-3}} = 5.05 \times 10^4 \, \text{Pa}\]
04

Determine Force Exerted by the Spring

The force exerted by the spring \( F_s \) is equal to the difference in force due to the gas pressure initially and finally, which can be expressed as:\[F = P_1 A - P_2 A \]where \( A = \pi r^2 \) is the area of the piston. Calculating \( A \):\[A = \pi (0.05)^2 = 7.85 \times 10^{-3} \, \text{m}^2\]Thus, \( F \) becomes:\[F = (1.01 \times 10^5 - 5.05 \times 10^4) \times 7.85 \times 10^{-3} = 393 \, \text{N}\]
05

Calculate the Spring Constant

The spring exerts a force \( F_s = k \cdot x \), where \( x \) is the change in length of the spring (0.2m since the gas-filled chamber doubles in length, causing the spring to be compressed by 0.2m). Thus, solve for \( k \):\[393 = k \cdot 0.2\]\[k = \frac{393}{0.2} = 1965 \, \text{N/m}\]
06

Final Calculation

The spring constant \( k \) of the spring is \( 1965 \, \text{N/m} \). This is calculated by understanding the pressure and volume relationships via the ideal gas law and translating that into the force exerted by the spring.

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Key Concepts

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

Isothermal Expansion
Isothermal expansion refers to a process where gas expands at a constant temperature. Since temperature remains the same, this implies no net change in the internal energy of the gas.
The ideal gas law, expressed as \( PV = nRT \), plays a crucial role here since \( n \), and \( R \) are constants, and here \( T \) is constant too. This results in the relationship \( P_1V_1 = P_2V_2 \) during isothermal processes.
This equation helps us understand how pressure and volume interact when a gas expands without changing temperature. In the problem, the expansion of the gas while holding the temperature steady means it performs work on the surroundings, specifically by moving a piston against a spring.
Cylinder Volume
When solving problems involving gases, determining the correct volume is key. Here, the volume of the gas is contained within a cylindrical portion. The formula for the volume of a cylinder is \( V = \pi r^2 L \), where \( r \) is the radius, and \( L \) is the length of the cylinder.
In the given exercise, the volume is essential to understand how the gas behaves when allowed to expand.
  • The initial volume \( V_1 \) is computed based on the initial length of 20.0 cm, or 0.2 meters.
  • The final volume \( V_2 \) is double this initial length since the gas expands to twice its original volume, highlighting how changes in the cylinder鈥檚 dimensions impact overall gas behavior.
This change is necessary to calculate pressure at different stages and determine how the spring constant is derived.
Spring Constant
The spring constant \( k \) is a measure of a spring's stiffness, defined in Hooke's Law as \( F = kx \). Here, \( F \) is the force exerted by the spring, and \( x \) is the displacement or compression of the spring.
In the scenario, as the gas expands, it moves the piston and compresses the spring. Calculating the spring constant involves knowing the change in force exerted on the piston and relating it to the displacement.
  • The force on the piston is obtained from pressure differences before and after the expansion.
  • The displacement \( x \) in this case is equivalent to the extension of the gas-filled area (0.2 meters).
Using these values, the spring constant \( k \) is calculated as 1965 N/m.
Pressure and Volume Relationships
In gas laws, pressure and volume are inversely related under isothermal (constant temperature) conditions. This relationship is critical in understanding gas behavior during expansions or compressions.
By the ideal gas law, the relationship \( P_1V_1 = P_2V_2 \) emerges during an isothermal process. Here:
  • \( P_1 \) and \( V_1 \) are the initial pressure and volume.
  • \( P_2 \) and \( V_2 \) are the final pressure and volume after expansion.
This aids in calculating the pressure after expansion when the volume doubles, essential for finding the spring force. This exercise effectively shows how manipulating one quantity affects the other, giving insight into both mathematical relationships and physical behavior of gases.

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Most popular questions from this chapter

You and your team have been given the task of preparing a gas piston device that is to be deployed to the bottom of the Southern Sea, beneath the Brunt Ice Shelf off the coast of Antarctica. The inner volume of the cylinder has a diameter of \(9.00 \mathrm{cm}\) and a length of \(25.0 \mathrm{cm},\) and is fitted with a movable piston that compresses the argon (Ar) gas that it contains. Before deployment, the piston is located and held at the very top of the cylinder, providing the maximum volume for the gas. When the device is submerged, however, the piston will move inward due to the water pressure created by the depth, and the change in temperature of the gas. The device is to be taken to a depth of \(350 \mathrm{m}\) in salt water (density \(\rho=1.03\) \(\left.\mathrm{g} / \mathrm{cm}^{3}\right)\) at a temperature of \(33.0^{\circ} \mathrm{F} .\) If the maximum allowable compression distance of the piston is \(13.0 \mathrm{cm},\) what is the minimum gas pressure that must be attained when the device is filled at the surface (at atmospheric pressure and \(T=59.0^{\circ} \mathrm{F}\) ) to prevent the maximum displacement of the piston from exceeding its maximum ( \(13.0 \mathrm{cm}\) ) when submerged to a depth of \(350 \mathrm{m} ?\)

A cubical box with each side of length \(0.300 \mathrm{m}\) contains 1.000 moles of neon gas at room temperature \((293 \mathrm{K}) .\) What is the average rate (in atoms/s) at which neon atoms collide with one side of the container? The mass of a single neon atom is \(3.35 \times 10^{-26} \mathrm{kg}\).

In a diesel engine, the piston compresses air at \(305 \mathrm{K}\) to a volume that is one-sixteenth of the original volume and a pressure that is 48.5 times the original pressure. What is the temperature of the air after the compression?

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A tank contains \(11.0 \mathrm{g}\) of chlorine gas \(\left(\mathrm{Cl}_{2}\right)\) at a temperature of \(82^{\circ} \mathrm{C}\) and an absolute pressure of \(5.60 \times 10^{5} \mathrm{Pa} .\) The mass per mole of \(\mathrm{Cl}_{2}\) is \(70.9 \mathrm{g} / \mathrm{mol}\) (a) Determine the volume of the tank. (b) Later, the temperature of the tank has dropped to \(31^{\circ} \mathrm{C}\) and, due to a leak, the pressure has dropped to \(3.80 \times 10^{5}\) Pa. How many grams of chlorine gas have leaked out of the tank?
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