Ch 20: The Micro/Macro Connection

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 20: The Micro/Macro Connection

Problem 34a

Chapter 20, Problem 34a

2.0 mol of monatomic gas A initially has 5000 J of thermal energy. It interacts with 3.0 mol of monatomic gas B, which initially has 8000 J of thermal energy. Which gas has the higher initial temperature?

Verified step by step guidance

Step 1: Recall the formula for the thermal energy of a monatomic ideal gas: \( U = \frac{3}{2} nRT \), where \( U \) is the thermal energy, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature in kelvins.

Step 2: Rearrange the formula to solve for temperature \( T \): \( T = \frac{2U}{3nR} \). This equation will allow us to calculate the temperature of each gas based on its thermal energy and number of moles.

Step 3: For gas A, substitute \( U = 5000 \ \text{J} \), \( n = 2.0 \ \text{mol} \), and \( R = 8.314 \ \text{J/(mol·K)} \) into the formula \( T = \frac{2U}{3nR} \). Simplify the expression to find the temperature of gas A.

Step 4: For gas B, substitute \( U = 8000 \ \text{J} \), \( n = 3.0 \ \text{mol} \), and \( R = 8.314 \ \text{J/(mol·K)} \) into the same formula \( T = \frac{2U}{3nR} \). Simplify the expression to find the temperature of gas B.

Step 5: Compare the calculated temperatures of gas A and gas B. The gas with the higher temperature has the higher initial temperature.

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Key Concepts

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

Thermal Energy

Thermal energy refers to the total kinetic energy of the particles in a substance due to their motion. It is directly related to temperature, as higher thermal energy typically indicates higher temperatures. In this context, the thermal energy of each gas is crucial for determining their respective temperatures.

Ideal Gas Law

The Ideal Gas Law, expressed as PV = nRT, relates the pressure (P), volume (V), number of moles (n), gas constant (R), and temperature (T) of an ideal gas. For monatomic gases, this law helps in understanding how the number of moles and thermal energy can influence temperature, allowing us to compare the two gases in the question.

Temperature and Molar Heat Capacity

Temperature is a measure of the average kinetic energy of the particles in a substance. For monatomic gases, the molar heat capacity at constant volume (Cv) is a key factor, as it relates the change in thermal energy to the change in temperature. This relationship allows us to calculate and compare the initial temperatures of gases A and B based on their thermal energies and the number of moles.

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Textbook Question

Your calculator can't handle enormous exponents, but we can make sense of large powers of e by converting them to large powers of 10. If we write e = 10^α, then e^β = (10^α)^β = 10^αβ. What is the multiplicity of a macrostate with entropy S = 1.0 J/K? Give your answer as a power of 10.

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Textbook Question

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Textbook Question

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Textbook Question

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