Ch.19 - Chemical Thermodynamics

Problem 90f

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Chapter 19, Problem 90f

Consider a system that consists of two standard playing dice, with the state of the system defined by the sum of the values shown on the top faces. (f) Calculate the absolute entropy of the two-dice system.

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Identify the possible outcomes for the sum of the top faces when two dice are rolled. The sums can range from 2 (1+1) to 12 (6+6).

Calculate the number of microstates (Ω) for each macrostate (sum). For example, the sum of 2 can only occur in one way (1+1), while the sum of 7 can occur in six ways (1+6, 2+5, 3+4, 4+3, 5+2, 6+1).

Sum up all the microstates for each possible sum to find the total number of microstates for the system.

Use the formula for absolute entropy, S = k * ln(Ω), where k is the Boltzmann constant and Ω is the total number of microstates.

Substitute the total number of microstates into the entropy formula to calculate the absolute entropy of the system.

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Key Concepts

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

Entropy

Entropy is a measure of the disorder or randomness in a system. In thermodynamics, it quantifies the number of microscopic configurations that correspond to a thermodynamic system's macroscopic state. Higher entropy indicates a greater number of possible arrangements, reflecting a more disordered state. In the context of dice, the entropy can be calculated based on the number of possible outcomes for the sums of the dice.

Statistical Mechanics

Statistical mechanics is a branch of physics that connects the microscopic properties of individual atoms and molecules to the macroscopic properties of materials. It provides the framework for calculating thermodynamic quantities, such as entropy, by considering the probabilities of different microstates. For the two-dice system, statistical mechanics helps determine the likelihood of each possible sum and thus the overall entropy of the system.

Microstates and Macrostates

In statistical mechanics, a microstate refers to a specific detailed configuration of a system, while a macrostate is defined by macroscopic properties like temperature and pressure. The relationship between microstates and macrostates is crucial for calculating entropy, as the entropy of a macrostate is related to the number of microstates that correspond to it. For the two-dice system, each unique sum represents a macrostate, with multiple microstates contributing to that sum.

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The Boltzmann Equation

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Related Practice

Open Question

Indicate whether each of the following statements is true or false. If it is false, correct it. (a) The feasibility of manufacturing NH3 from N2 and H2 depends entirely on the value of ΔH for the process N2(g) + 3 H2(g) → 2 NH3(g). (e) Spontaneous processes are those that are exothermic and that lead to a higher degree of order in the system.

Textbook Question

For each of the following processes, indicate whether the signs of ΔS and ΔH are expected to be positive, negative, or about zero. (e) A piece of charcoal is combusted to form CO₂(g) and H₂O(g).

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Textbook Question

The reaction 2 Mg(s) + O₂(g) ⟶ 2 MgO(s) is highly spontaneous. A classmate calculates the entropy change for this reaction and obtains a large negative value for ΔS°. Did your classmate make a mistake in the calculation? Explain.

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Open Question

Ammonium nitrate dissolves spontaneously and endothermally in water at room temperature. What can you deduce about the sign of ΔS for this dissolution process?

Textbook Question

A standard air conditioner involves a refrigerant that is typically now a fluorinated hydrocarbon, such as CH₂F₂. An air-conditioner refrigerant has the property that it readily vaporizes at atmospheric pressure and is easily compressed to its liquid phase under increased pressure. The operation of an air conditioner can be thought of as a closed system made up of the refrigerant going through the two stages shown here (the air circulation is not shown in this diagram).

During expansion, the liquid refrigerant is released into an expansion chamber at low pressure, where it vaporizes. The vapor then undergoes compression at high pressure back to its liquid phase in a compression chamber. (c) In a central air-conditioning system, one chamber is inside the home and the other is outside. Which chamber is where, and why?

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Textbook Question

During expansion, the liquid refrigerant is released into an expansion chamber at low pressure, where it vaporizes. The vapor then undergoes compression at high pressure back to its liquid phase in a compression chamber. (e) Suppose that a house and its exterior are both initially at 31 °C. Some time after the air conditioner is turned on, the house is cooled to 24 °C. Is this process spontaneous or nonspontaneous?

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