(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.

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).

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.

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 \(\mathrm{Ar}(g)\).