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Ch 18: Thermal Properties of Matter
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Non è quello che usi tu?Cambia libro di testo
Capitolo 18, Problema 27a

What is the total translational kinetic energy of the air in an empty room that has dimensions 8.008.00 m×12.00\(\times\)12.00 m×4.00\(\times\)4.00 m if the air is treated as an ideal gas at 1.001.00 atm?

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Calculate the volume of the room using the formula for the volume of a rectangular prism: \( V = \text{length} \times \text{width} \times \text{height} \). Substitute the given dimensions: \( V = 8.00 \text{ m} \times 12.00 \text{ m} \times 4.00 \text{ m} \).
Use the ideal gas law to find the number of moles of air in the room. The ideal gas law is \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature. Assume standard temperature (e.g., 298 K) and use \( R = 8.314 \text{ J/mol K} \). Rearrange to solve for \( n \): \( n = \frac{PV}{RT} \).
Determine the translational kinetic energy of the gas using the formula \( KE = \frac{3}{2} nRT \). This formula comes from the kinetic theory of gases, which states that the translational kinetic energy of an ideal gas is proportional to the number of moles and the temperature.
Substitute the values for \( n \), \( R \), and \( T \) into the kinetic energy formula to find the total translational kinetic energy of the air in the room.
Ensure all units are consistent, particularly checking that pressure is in pascals (1 atm = 101325 Pa) and volume is in cubic meters, to maintain consistency in the calculations.

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Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics that relates the pressure, volume, and temperature of an ideal gas. It is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the universal gas constant, and T is temperature. This law helps determine the amount of gas in a given space, crucial for calculating kinetic energy.
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Ideal Gases and the Ideal Gas Law

Translational Kinetic Energy

Translational kinetic energy refers to the energy possessed by an object due to its motion through space. For a gas, it is calculated using the formula KE = (3/2) nRT, where n is the number of moles and T is the temperature. This concept is essential for understanding how the motion of gas molecules contributes to the total energy in a system.
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Volume Calculation

Volume calculation is crucial for determining the amount of space occupied by a gas, which directly affects its pressure and kinetic energy. In this context, the volume of the room is calculated by multiplying its dimensions: 8.00 m * 12.00 m * 4.00 m, resulting in 384 cubic meters. This volume is used in conjunction with the Ideal Gas Law to find the number of moles of air in the room.
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