Question

Indicate whether each statement is true or false. (a) Unlike enthalpy, where we can only ever know changes in \(H,\) we can know absolute values of \(S\) . (b) If you heat a gas such as \(\mathrm{CO}_{2},\) you will increase its degrees of translational, rotational and vibrational motions. (c) \(\mathrm{CO}_{2}(g)\) and \(\mathrm{Ar}(g)\) have nearly the same molar mass. At a given temperature, they will have the same number of microstates.

Short answer

(a) True: Absolute values of entropy S can be determined, unlike enthalpy H, where only changes can be known. (b) True: Heating a gas like CO₂ increases its translational, rotational, and vibrational motions. (c) False: CO₂ and Ar have different numbers of microstates at a given temperature, even though they have nearly the same molar mass.

Step-by-step solution

Statement (a)

This statement says that unlike enthalpy, where we can only ever know changes in H, we can know absolute values of S. Enthalpy, H, is a state function which means that only its change (∆H) in any process can be measured. However, for entropy, S, it is possible to determine an absolute value for a substance as it's based on the thermodynamic probabilities. In particular, we can find the zero point of entropy, i.e., the entropy of a perfect crystals at 0 K according to the Third Law of Thermodynamics. So, statement (a) is True.

Statement (b)

This statement says that if you heat a gas such as CO₂, you will increase its degrees of translational, rotational and vibrational motions. When you heat any gas, its molecules gain energy. This increase in energy results in an increase in the motion of all its molecules. Molecules in gases have three types of motion: translational (straight-line motion), rotational (spinning motion around an axis), and vibrational (back-and-forth motion). So when we heat a gas like CO₂, its molecules gain energy, increasing their translational, rotational, and vibrational motions. Therefore, statement (b) is True.

Statement (c)

This statement says that CO₂(g) and Ar(g) have nearly the same molar mass. At a given temperature, they will have the same number of microstates. While it's true that CO₂ and Ar have nearly the same molar mass, this fact does not lead to them having the same number of microstates at a given temperature. Microstates represent all the possible ways in which energy can be distributed among the molecules in a system. The number of microstates depends on both the type and complexity of the molecule. CO₂ is a more complex molecule than Ar, as it exhibits three types of molecular motion (translational, rotational, and vibrational) while Ar, being a monatomic gas, only exhibits translational motion. Consequently, CO₂ will generally have more microstates than Ar at any given temperature. Thus, statement (c) is False.

Key concepts

Entropy (S)

Entropy, commonly denoted as S, is a fundamental concept in thermodynamics that quantifies the disorder or randomness in a system. It's often associated with the number of ways a system can be arranged, which we call microstates. Unlike other thermodynamic properties, the Third Law of Thermodynamics allows scientists to determine the absolute entropy of a perfect crystal at absolute zero (0 K), making it unique.

As a system gains energy, for instance when heated, its entropy typically increases because the molecules have more energy to adopt a greater number of configurations or microstates. Understanding entropy is pivotal as it helps predict whether processes will occur spontaneously based on an increase in the overall entropy of the universe.

Enthalpy (H)

Enthalpy, represented by H, is another crucial thermodynamics term. It refers to the total heat content of a system at constant pressure. Unlike entropy, we can’t measure the absolute enthalpy of a system; we can only measure changes in enthalpy (H).

These changes, H, are indicative of the energy absorbed or released during a reaction or process, which is why enthalpy changes become a crucial component in evaluating reaction energetics. Endothermic reactions absorb heat (H > 0), whereas exothermic reactions release heat (H < 0), and changes in enthalpy are directly related to the energy of molecular motions such as translational, rotational, and vibrational movements.

Microstates

Microstates are essentially the different ways in which a system can arrange its components while still maintaining its macroscopic properties. These properties include temperature, pressure, and volume. In the context of entropy, the greater the number of microstates, the higher the entropy.

In gases, for example, microstates relate to the positions and velocities of all the molecules. Complex molecules with multiple atoms, like carbon dioxide (CO_2), inherently have more potential microstates than simpler, monatomic gases like argon (Ar) due to additional modes of movement such as vibrational and rotational motions alongside translational.

Translational Motion

Translational motion refers to the straightforward, linear movement of particles within a substance from one location to another. Imagine gas particles zipping around in a container - that's translational motion in action. This type of motion contributes to the pressure exerted by the gas.

When a gas is heated, particles move faster as they gain kinetic energy, increasing the degree of translational motion and hence the entropy of the system. The kinetic theory of gases mathematically links translational motion to temperature, with higher temperatures leading to more vigorous translational activity.

Rotational Motion

Rotational motion in molecules is akin to the spinning of a top. Molecules, particularly those with more than one atom, can rotate around various axes. This rotation isn't just random—it’s quantized and has discrete energy levels that contribute to the molecule’s overall energy.

With a rise in temperature, molecules have more energy to access higher rotational energy levels, creating more potential microstates. Consequently, this increase in accessible rotational states leads to an increase in entropy. In the case of linear molecules like CO_2, they have two rotational axes, thus expanding the ways the energy can be distributed.

Vibrational Motion

In vibrational motion, atoms in a molecule oscillate about their average positions. They can stretch and compress like springs, which is another mode wherein energy is stored and released. Just as with rotational motion, vibrational motion is also quantized, possessing discrete energy levels.

Vibrational motion comes into play in more complex molecules and is highly sensitive to temperature changes. When we heat a substance, we're essentially pumping more energy into its vibrational modes, which increases the number of ways that the atoms can move and the number of potential microstates. This increase in complexity and energy distribution adds to the entropy of the system.