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 57

Chapter 20, Problem 57

FIGURE P20.57 shows the thermal energy of 0.14 mol of gas as a function of temperature. What is Cᵥ for this gas?

Verified step by step guidance

Step 1: Understand the problem. The graph shows the thermal energy (Eₜₕ) of 0.14 mol of gas as a function of temperature (T). The goal is to determine the molar heat capacity at constant volume (Cᵥ). Recall that Cᵥ is related to the change in thermal energy with respect to temperature.

Step 2: Use the formula for molar heat capacity at constant volume:

C v = \frac{\partial E_{th}}{\partial}

. This means you need to calculate the slope of the Eₜₕ vs. T graph.

Step 3: Identify two points on the graph to calculate the slope. For example, at T = 100°C, Eₜₕ = 1492 J, and at T = 200°C, Eₜₕ = 1892 J. The slope is given by

\frac{∆ E_{th}}{∆}

, where

∆ E_{th}

is the change in thermal energy and

∆ T

is the change in temperature.

Step 4: Calculate the slope using the values from the graph. Substitute

∆ E_{th}

= 1892 J - 1492 J and

∆ T

= 200°C - 100°C into the formula for the slope.

Step 5: Divide the slope by the number of moles (0.14 mol) to find the molar heat capacity Cᵥ. Use the relationship

C v = \frac{\partial E_{th}}{\partial T} \div n

, where n is the number of moles.

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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 the temperature of a gas increases, its thermal energy also increases. In the context of the provided graph, the thermal energy of the gas is plotted against temperature, illustrating how energy changes with temperature.

Heat Capacity (Cᵥ)

Heat capacity is a measure of the amount of heat energy required to change the temperature of a substance by a certain amount. For gases, the specific heat capacity at constant volume (Cᵥ) is particularly important, as it describes how much heat is needed to raise the temperature of one mole of gas while keeping its volume constant. It can be determined from the slope of the thermal energy versus temperature graph.

Moles of Gas

The concept of moles is fundamental in chemistry and physics, representing a specific quantity of particles, typically atoms or molecules. In this context, 0.14 mol of gas indicates the amount of gas being analyzed. The number of moles is crucial for calculating heat capacity, as it allows for the determination of how much energy is required to change the temperature of the gas based on its specific heat capacity.

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