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Ch 18: A Macroscopic Description of Matter
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 18: A Macroscopic Description of MatterProblem 45
Chapter 18, Problem 45

A 6.0-cm-diameter, 10-cm-long cylinder contains 100 mg of oxygen (O₂) at a pressure less than 1 atm. The cap on one end of the cylinder is held in place only by the pressure of the air. One day when the atmospheric pressure is 100 kPa, it takes a 184 N force to pull the cap off. What is the temperature of the gas?

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Step 1: Begin by identifying the key variables in the problem. The diameter of the cylinder is 6.0 cm, so the radius is half of that, 3.0 cm or 0.03 m. The length of the cylinder is 10 cm or 0.1 m. The mass of oxygen is 100 mg or 0.1 g, which can be converted to kilograms as 0.0001 kg. The atmospheric pressure is given as 100 kPa, and the force required to pull the cap off is 184 N.
Step 2: Calculate the area of the circular cap using the formula for the area of a circle: \( A = \pi r^2 \). Substitute \( r = 0.03 \) m into the formula to find the area of the cap.
Step 3: Determine the pressure exerted by the gas inside the cylinder. The total force acting on the cap is the sum of the force due to the gas pressure and the atmospheric pressure. Use the relationship \( F = P \cdot A \), where \( F \) is the force, \( P \) is the pressure, and \( A \) is the area. Rearrange to solve for the pressure of the gas: \( P_{gas} = \frac{F}{A} - P_{atm} \). Substitute the values for \( F \), \( A \), and \( P_{atm} \) to find \( P_{gas} \).
Step 4: Use the ideal gas law \( PV = nRT \) to find the temperature of the gas. First, calculate the volume of the cylinder using \( V = A \cdot h \), where \( h \) is the length of the cylinder. Then, calculate the number of moles of oxygen \( n \) using \( n = \frac{m}{M} \), where \( m \) is the mass of the gas and \( M \) is the molar mass of oxygen (approximately 32 g/mol or 0.032 kg/mol). Substitute \( P_{gas} \), \( V \), \( n \), and the gas constant \( R = 8.314 \, \text{J/(mol·K)} \) into the ideal gas law and rearrange to solve for \( T \).
Step 5: Perform unit checks and ensure all values are consistent (e.g., pressure in pascals, volume in cubic meters, mass in kilograms, etc.). This ensures the calculated temperature is accurate and consistent with the SI unit system.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. Here, P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature. This law is fundamental for understanding the behavior of gases under various conditions and is essential for solving problems involving gas properties.
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Pressure and Force Relationship

Pressure is defined as force per unit area (P = F/A). In this scenario, the force required to remove the cap is related to the pressure exerted by the gas inside the cylinder and the atmospheric pressure outside. Understanding this relationship helps in calculating the effective pressure acting on the cap and is crucial for determining the gas's temperature using the Ideal Gas Law.
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Volume of a Cylinder

The volume of a cylinder can be calculated using the formula V = πr²h, where r is the radius and h is the height. In this problem, the cylinder's dimensions are given, and calculating its volume is necessary to apply the Ideal Gas Law effectively. Knowing the volume allows for the determination of the number of moles of gas present, which is essential for finding the temperature.
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Related Practice
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