Question

(a) 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. (b) You are told that the number of microstates for a system increases. What does this tell you about the entropy of the system?

Short answer

(a) When the energy of a CO2 gas is increased by heating it, the additional energy can be distributed among the molecules in the following ways: 1. Translational Motion, 2. Rotational Motion, and 3. Vibrational Motion. The energy distribution is not uniform, and molecules can be in various energy states simultaneously. (b) If the number of microstates for a system increases, it indicates an increase in its entropy, leading to a higher degree of disorder or randomness within the system.

Step-by-step solution

(a) Distribution of Energy

When the energy of a gas (CO2 in this case) is increased by heating it, the additional energy can be distributed among the molecules in several ways: 1. Translational Motion: The energy is divided among the molecules in the form of kinetic energy in their motion through space. Each molecule moves in a straight line until it collides with another molecule or the walls of a container, changing its direction and/or speed. 2. Rotational Motion: The energy is used to increase the rotational energy of the molecules. For CO2 molecules, they can rotate around two perpendicular axes since it is a linear molecule. Each rotation will increase their rotational kinetic energy. 3. Vibrational Motion: The energy can also be distributed among the vibrational modes of the molecules. Each vibration mode corresponds to a specific way the atoms within a molecule move relative to each other. CO2 has four vibrational modes: two asymmetric stretching modes, one symmetric stretching mode, and one bending mode. When additional energy is added to the gas, it can be distributed among these different modes of energy transfer. It is important to note that the energy distribution is not uniform, and molecules in a gas sample can be in various energy states simultaneously.

(b) Microstates and Entropy

The number of microstates refers to the number of different ways the energy of a system can be arranged among its particles. If the number of microstates for a system increases, that means there are more possible configurations or arrangements for the energy within the system. Entropy is a measure of disorder or randomness in a system. The higher the number of microstates, the more disordered or random the system is. Consequently, if the number of microstates for a system increases, the entropy of the system also increases. In summary, an increase in the number of microstates in a system indicates an increase in its entropy, which leads to a higher degree of disorder or randomness within the system.