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 29

Chapter 20, Problem 29

The thermal energy of 1.0 mol of a substance is increased by 1.0 J. What is the temperature change if the system is (a) a monatomic gas, (b) a diatomic gas, and (c) a solid?

Verified step by step guidance

Step 1: Understand the relationship between thermal energy, temperature change, and heat capacity. The formula to use is ΔQ = n * C * ΔT, where ΔQ is the thermal energy added, n is the number of moles, C is the molar heat capacity, and ΔT is the temperature change.

Step 2: For a monatomic gas, the molar heat capacity at constant volume (C_v) is given by C_v = (3/2) * R, where R is the universal gas constant (R ≈ 8.314 J/(mol·K)). Substitute C_v into the formula ΔQ = n * C_v * ΔT and solve for ΔT.

Step 3: For a diatomic gas, the molar heat capacity at constant volume (C_v) is given by C_v = (5/2) * R. Substitute this value into the formula ΔQ = n * C_v * ΔT and solve for ΔT.

Step 4: For a solid, the molar heat capacity is typically approximated using the Dulong-Petit law, which states C ≈ 3 * R. Substitute this value into the formula ΔQ = n * C * ΔT and solve for ΔT.

Step 5: For each case (monatomic gas, diatomic gas, and solid), substitute the given values: ΔQ = 1.0 J and n = 1.0 mol into the respective formulas derived in steps 2, 3, and 4 to calculate the temperature change ΔT for each system.

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Key Concepts

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

Specific Heat Capacity

Specific heat capacity is the amount of heat required to change the temperature of a unit mass of a substance by one degree Celsius. It varies for different materials and phases (solid, liquid, gas) and is crucial for calculating temperature changes when thermal energy is added or removed.

Degrees of Freedom

Degrees of freedom refer to the number of independent ways in which a system can store energy. For gases, monatomic gases have three translational degrees of freedom, while diatomic gases have five (three translational and two rotational). This affects how they respond to added thermal energy.

Heat Transfer in Phases

Heat transfer in phases describes how different states of matter (solid, liquid, gas) absorb and transfer heat. Solids typically have lower heat capacities compared to gases, and the way energy is distributed among particles varies significantly, influencing the temperature change for a given amount of thermal energy.

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Related Practice

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

The vibrational modes of molecular nitrogen are 'frozen out' at room temperature but become active at temperatures above ≈1500 K. The temperature in the combustion chamber of a jet engine can reach 2000 K, so an engineering analysis of combustion requires knowing the thermal properties of materials at these temperatures. What is the expected specific heat ratio γ for nitrogen at 2000 K?

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Textbook Question

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A 6.0 m ✕ 8.0 m ✕ 3.0 m room contains air at 20℃. What is the room's thermal energy?

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