(a) If you are told that the entropy of a certain system is...

(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.

Key Concepts

Comparison of Gases (Microstates vs. Molar Mass)

Even if two gases like CO2 and Ar have nearly the same molar mass, the number of microstates available to each at a given temperature can differ. This is because the internal structure and degrees of freedom of the molecules, such as the presence of rotational and vibrational modes in polyatomic CO2 versus the simpler monoatomic argon, result in different ways energy can be distributed among the particles' microstates, thereby affecting the entropy and overall statistical behavior of the gases.

Microstates and Energy Distribution

The concept of microstates refers to the different ways that a system's energy can be arranged among its particles and degrees of freedom. When energy is added to a gas, such as by heating, the additional energy increases the number of accessible microstates. The distribution of energy into translational, rotational, and vibrational modes directly affects the number of microstates available to a molecule, which is a key aspect of statistical mechanics.

Temperature

Temperature is a fundamental parameter in thermodynamics that is related to the average kinetic energy of the particles in a system. It determines how energy is distributed among the system's microstates. When entropy is zero, the system is in its absolute ground state, which implies it is at absolute zero temperature, because any thermal energy would allow access to additional microstates, thereby increasing entropy.

Entropy

Entropy is a measure of the number of microscopic configurations that a system can have, reflecting the degree of disorder or randomness. When a system has zero entropy, it means that there is a unique microstate with all particles in a well-defined, ordered state, typically corresponding to the ground state as described by the third law of thermodynamics. This condition, however, is idealized and practically attainable only at absolute zero temperature.

Molecular Degrees of Freedom

Molecular degrees of freedom are the independent modes by which a molecule can store energy, including translational, rotational, and vibrational motions. In polyatomic molecules like CO2, energy added by heating can be partitioned into these different degrees of freedom, leading to various ways of distributing the extra energy among the translational motion of the molecule as a whole, its rotations, and its internal vibrations.

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