Question

The standard entropies at \(298 \mathrm{~K}\) for certain of the group \(4 \mathrm{~A}\) elements are as follows: \(\mathrm{C}(s,\) diamond \()=2.43 \mathrm{~J} / \mathrm{mol}-\mathrm{K},\) \(\mathrm{Si}(s)=18.81 \mathrm{~J} / \mathrm{mol}-\mathrm{K}, \mathrm{Ge}(s)=31.09 \mathrm{~J} / \mathrm{mol}-\mathrm{K},\) and \(\operatorname{Sn}(s)=\) \(51.818 \mathrm{~J} / \mathrm{mol}-\mathrm{K} .\) All but \(\mathrm{Sn}\) have the diamond structure. How do you account for the trend in the \(S^{\circ}\) values?

Short answer

The trend in the standard entropy values of group 4A elements can be accounted for by considering the increasing atomic mass and structural differences between the elements. The diamond structure of carbon (C), silicon (Si), and germanium (Ge) gives them lower entropy values due to its stability and rigidity, while the non-diamond structure of tin (Sn) allows for more configurational freedom, resulting in a higher entropy. Additionally, the increase in atomic mass leads to more atoms and electrons in a sample, contributing to the rise in entropy down the group.

Step-by-step solution

Identify Trends and Properties of Elements in Group 4A

Group 4A elements include carbon (C), silicon (Si), germanium (Ge), tin (Sn), and lead (Pb). The given values of standard entropies at 298 K indicate that the entropy values increase down the group from carbon to tin. Let's organize this information in a table. | Element | Standard Entropy (J/mol·K) | Structure | |---------|---------------------------|-----------| | C | 2.43 | Diamond | | Si | 18.81 | Diamond | | Ge | 31.09 | Diamond | | Sn | 51.818 | Non-Diamond | From the table, note that all elements except tin (Sn) have the diamond structure.

Explain The Effect of Atomic Mass

As we go down the group 4A, the atomic mass of the elements increases. This increase in atomic mass means more atoms in a given sample, which subsequently results in a higher number of microstates and greater entropy. Additionally, elements with larger atomic masses also have more electrons, and these electrons are distributed over a larger volume. This can lead to a higher number of possible electronic configurations, further contributing to the increase in entropy as we go down the group from carbon (C) to tin (Sn).

Explain The Effect of Crystal Structure

The diamond structure has a very stable and rigid lattice that is held together by strong covalent bonds. This leads to a more ordered structure with limited vibrational freedom, which typically results in lower entropy values. On the other hand, tin (Sn) does not have the diamond structure, meaning that it might have a less ordered and more flexible lattice, leading to higher entropy.

Relate Atomic Mass and Crystal Structure to the Entropy Trend

The combination of increasing atomic mass and the structural differences between the elements in group 4A results in the observed trend of increasing entropy from carbon (C) to tin (Sn). The diamond structure of carbon (C), silicon (Si), and germanium (Ge) gives them lower entropy values, whereas the non-diamond structure of tin (Sn) allows for more configurational freedom, resulting in a higher entropy value. In conclusion, the trends in the standard entropy values of group 4A elements can be accounted for by considering the increasing atomic mass and structural differences between the elements, mainly the presence of the diamond structure in all elements except tin (Sn).

Key concepts

Group 4A Elements

Group 4A elements are an interesting batch on the periodic table. They include carbon (C), silicon (Si), germanium (Ge), tin (Sn), and lead (Pb). These elements are known for the diversity in their structural forms and uses. As we move down the group, each element showcases different characteristics due to changes in their atomic structures and masses.

• Carbon: Known for its diamond structure, carbon can also exist as graphite, which has drastically different properties.
• Silicon: Widely used in electronics, it forms the backbone of modern computing technologies.
• Germanium: Another semiconductor, crucial in the fabrication of different kinds of electronic devices.
• Tin: Known for its corrosion resistance, often used in alloys like bronze.
• Lead: Heavier and much softer, used in batteries and radiation shielding.

As each element progresses, their applications and physical behaviours evolve due to differences in how their atoms are bound and interact with each other.

Atomic Mass

Atomic mass plays a crucial role in understanding the properties of elements. It is the mass of an atom measured in unified atomic mass units (u). For the group 4A elements, atomic mass increases as you move down the group from carbon to lead.
This increase in atomic mass corresponds to an increase in the number of protons and neutrons in the atoms' nuclei, leading to heavier atoms. Larger atoms can introduce more vibrational and rotational microstates, which contribute to thermal entropy.
In chemistry, a higher atomic mass implies a larger number of potential atomic arrangements or "microstates". These microstates can be thought of as different ways the atoms in a sample can be arranged, influencing how disordered or "entropic" the material is. The possibility of more electron configurations in heavier elements like tin compared to lighter ones like carbon adds complexity and hence increases the standard entropy as you move down the group.

Crystal Structure

One of the defining features of materials is their crystal structure, which greatly affects their physical properties. The group 4A elements predominantly exhibit the diamond structure, a testament to strong covalent bonds forming a robust and stable lattice.
In the diamond structure:

• Each atom is bonded tetrahedrally to four other atoms.
• This leads to a highly ordered and compact structure.
• The rigidity restricts atomic movement, resulting in low entropy.

Tin (Sn), however, does not share this diamond lattice structure, which results in a looser atomic arrangement. This less ordered structure results in increased atomic freedom and higher entropy. Tin's rearrangement is more flexible, allowing for an increase in vibrational states, contributing to its higher standard entropy compared to other group 4A elements.

Entropy Trend

Entropy is a measure of disorder and randomness in a system, often linked to the number of microstates available to a system. For group 4A elements, as you move from carbon to tin, there is an observable increase in standard entropy.
This trend is primarily driven by two factors:

• The increase in atomic mass, meaning more particles present in a given volume, thus increasing possible microstates.
• The variation in crystal structure, particularly the absence of the rigid diamond structure in tin, which allows more freedom of movement.

The entropy differences thus accurately reflect both the atomic mass and differing crystal structures. In essence, as elements become heavier and their structures become less rigid, the entropy naturally increases. Understanding these trends helps in predicting the behaviors of these materials in various practical applications like electronics and metallurgy.