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 \(\mathrm{Ar}(\mathrm{g})\) have nearly the same molar mass. At a given temperature, they will have the same number of microstates.

Short answer

Statement 1: True, because the Third Law of Thermodynamics allows us to know the absolute values of entropy (S). Statement 2: True, because heating a gas increases its kinetic energy, which in turn increases its translational, rotational, and vibrational motions. Statement 3: False, because the number of microstates depends not only on molar mass but also on factors such as molecular structure, internal complexity, and degrees of freedom.

Step-by-step solution

Statement 1: Entropy absolute values

The Third Law of Thermodynamics states that the entropy of a perfect crystal approaches zero as the temperature approaches absolute zero (0 K). Because there's a reference point (i.e., 0 K), we can indeed know the absolute values of entropy (S). Therefore, this statement is \( \boxed{\textbf{true}} \).

Statement 2: Heating a gas increases essential motions

When a gas is heated, it acquires more kinetic energy overall. This increase in kinetic energy will result in an increase in the translational, rotational, and vibrational motions of the gas particles. So, heating a gas like CO2 will increase its degrees of translational, rotational, and vibrational motions. This statement is \( \boxed{\textbf{true}} \).

Statement 3: Microstates of CO2(g) and Ar(g)

CO2(g) and Ar(g) have approximately the same molar mass. However, they have different molecular structures and complexities. CO2 is a linear molecule with 3 atoms, while Ar is an atomic gas (noble gas) with only one atom. The number of microstates is determined not only by molar mass but also by molecular structure, internal complexity, and degrees of freedom. Therefore, it is incorrect to say that they will have the same number of microstates at a given temperature, even if they have the same molar mass. This statement is \( \boxed{\textbf{false}} \).

Key concepts

Third Law of Thermodynamics

The Third Law of Thermodynamics stands as a foundational principle in physics, particularly with regard to temperature and disorder.

It stipulates that as a system approaches absolute zero temperature (\(0 \text{ K}\)), the entropy—or measure of disorder—of a perfect crystalline structure will also approach zero. This assertion hinges on the presumption that at absolute zero, a perfect crystal exists in its state of minimum possible motion, implying that its atoms are arranged in a perfectly ordered state.

At this ultimate lower bound temperature, the system possesses only one possible microstate, meaning there is only one way the particles can be arranged. Since entropy is a measure of disorder or randomness within the system, having only one microstate translates to minimal entropy. As temperature increases, the number of microstates increases due to additional possible arrangements of particles, thus increasing the entropy.

Importantly, the third law provides a reference point for the absolute measure of entropy. This is critical for many calculations in thermodynamics since we can define the absolute entropy at zero Kelvin and then measure changes from this reference point.

Kinetic Energy in Gases

Delving into the behavior of gases, kinetic energy plays an integral role. Kinetic energy is the energy associated with the motion of particles.

In a gas, particles are in continuous, random motion and the kinetic energy of these particles is directly related to their temperature. Understanding the molecular theory of gases lets us know that when the temperature of a gas rises, the average kinetic energy of the molecules also increases. This is simply because temperature is a measure of the average kinetic energy of the particles in a substance.

As a result of kinetic energy increase, the molecules move more rapidly and this movement translates into greater translational (straight-line movement), rotational (spinning), and vibrational (oscillation) motions. Each molecule's range of motion contributes to the overall kinetic energy and therefore, additional amounts of heat—energy transferred due to temperature difference—will increase these motions in gases like carbon dioxide (\( \text{CO}_2 \)).

Molecular Structure Impact on Microstates

Microstates are specific ways in which a system may be arranged, especially in terms of the positions and energies of its molecules. They have profound implications for entropy and are largely influenced by molecular structure.

The molecular structure dictates the number of microstates available due to varying degrees of freedom such as translational, rotational, and vibrational movements. For example, a more complex molecule like carbon dioxide (\( \text{CO}_2 \)), with three atoms arranged linearly, will have more microstates than a simpler, single-atom molecule like argon (\( \text{Ar} \)) because of the additional rotational and vibrational possibilities.CO2 can rotate and vibrate in more ways than monatomic argon can, which allows CO2 to have more potential arrangements of energy and therefore more microstates.

Even if two substances, such as \( \text{CO}_2 \) and \( \text{Ar} \) have similar molar masses, their molecular structure, and complexity lead to different quantities of microstates. Understanding the distinct difference between how molecular complexity relates to potential microstate variety is a fundamental aspect of thermodynamics that extends into understanding properties like entropy and the distribution of energy levels within a substance.