Problem 40 - Chemistry The Central Science

Question

(a) If you are told that the entropy of a certain system is zero, what do you know about the system and the temperature? (b) The energy of a gas is increased by heating it. Using \(\mathrm{CO}_{2}\) as an example, illustrate the different ways in which additional energy can be distributed among the molecules of the gas. (c) \(\mathrm{CO}_{2}(g)\) and \(\mathrm{Ar}(g)\) have nearly the same molar mass. At a given temperature, will they have the same number of microstates? Explain.

Step-by-Step Solution

Answer

(a) A system with zero entropy is a perfect crystal at absolute zero (0 K) temperature, indicating perfect order and no uncertainty in energy distribution. (b) In a \(\mathrm{CO}_2\) gas, additional energy can be distributed as translational (linear motion), rotational (rotation around axes), and vibrational (molecular vibrations) energy. (c) \(\mathrm{CO}_2(g)\) and \(\mathrm{Ar}(g)\) won't have the same number of microstates at a given temperature as \(\mathrm{CO}_2(g)\) has additional rotational and vibrational energy levels, providing more ways to distribute energy among the molecules.

Step 1: (a) Interpreting the Entropy of a System

In this case, the entropy of the system is zero. Entropy is a measure of the disorder or randomness of a system. When a system has zero entropy, it means that the system is perfectly ordered, and there is no uncertainty about its internal energy distribution. If we consider the Third Law of Thermodynamics, it states that the entropy of a perfect crystal at absolute zero (0 kelvin) is zero. Therefore, we can conclude that the system is a perfect crystal and the temperature of the system is absolute zero (0 K).

Step 2: (b) Energy Distribution in CO2 Molecules

When the energy of a \(\mathrm{CO}_2\) gas is increased by heating it, the additional energy can be distributed among the molecules of the gas in several ways: 1. Translational energy - As the gas is heated, the molecules will move faster in their linear paths. This increase in translational energy will result from an increase in their kinetic energy. 2. Rotational energy - The molecules of \(\mathrm{CO}_2\) can also rotate around their axes. By heating the gas, some of the energy will be used to increase the rate of rotation of the molecules. 3. Vibrational energy - The \(\mathrm{CO}_2\) molecules can vibrate in different modes, such as asymmetric stretch, symmetric stretch, and bending vibrations. With the increase in energy, the amplitude and frequency of these vibrations will also increase.

Step 3: (c) Microstates of CO2(g) and Ar(g)

At a given temperature, \(\mathrm{CO}_2(g)\) and \(\mathrm{Ar}(g)\) will not have the same number of microstates. Here's why: Microstates are the different possible arrangements of energy among the molecules in a system. While both \(\mathrm{CO}_2(g)\) and \(\mathrm{Ar}(g)\) have nearly the same molar mass, their structures are different and have different degrees of freedom for energy distribution. \(\mathrm{Ar}(g)\) is a monatomic gas and can only distribute energy in translational and electronic energy levels. On the other hand, \(\mathrm{CO}_2(g)\) is a linear polyatomic molecule and has more degrees of freedom due to additional rotational and vibrational energy levels. As a result, at a given temperature, \(\mathrm{CO}_2(g)\) will have more microstates (ways to distribute energy among the molecules) than \(\mathrm{Ar}(g)\).

Key Concepts

Third Law of Thermodynamics

The Third Law of Thermodynamics is a fundamental principle in physics. It states that the entropy of a perfect crystal approaches zero as the temperature approaches absolute zero (0 Kelvin). Entropy measures the disorder in a system, and a perfectly ordered crystal at absolute zero has no disorder. This is because the molecules are in their most organized state.
Understanding this concept helps explain why systems can't reach absolute zero. As temperature decreases, less thermal energy is available, and molecular motion reduces. However, reaching absolute zero is impossible because it would require removing all entropy, a theoretical challenge.

• Perfect crystals naturally have zero entropy at 0 Kelvin due to maximum order.
• Real systems can't achieve absolute zero, as removing all disorder is practically impossible.
• This principle explains why cooling becomes challenging as temperatures approach absolute zero.

Energy Distribution

When you heat a gas like CO_2, energy is distributed among molecules in different ways. This increase in energy leads to various changes within the system, enhancing both molecular motion and activity.

• Translational Energy: Molecules move faster along a path, increasing kinetic energy. This movement is typically straight-line motion.
• Rotational Energy: CO_2 molecules spin around their axes. Heating increases rotation rates, distributing additional energy.
• Vibrational Energy: Molecules vibrate in different modes, like stretching and bending. Extra energy increases these vibration rates and amplitudes.

Each way of energy distribution enhances the activity level of molecules differently. It's fascinating how heating changes molecular dynamics in gases!

Microstates

Microstates are the different ways in which energy can be arranged in a system at a given temperature. While it might sound complex, it's a core concept in understanding molecular behavior.
Imagine comparing CO_2(g) and Ar(g). Although they have similar molar masses, they differ in structure. Ar is a monatomic gas, allowing energy distribution only through translation. However, CO_2 is polyatomic, with additional vibrational and rotational modes.

• Ar(g): Simplistic energy distribution through translational moves only.
• CO_2(g): Rich energy distribution possibilities through extra rotations and vibrations.
• More complex molecules like CO_2 have far more microstates than simpler gases like Ar.

This variance in microstates explains why different gases have unique energy behaviors even under similar conditions. Unique structures lead to different energy arrangements and levels of molecular activity.