Problem 49
Question
The standard entropies at \(298 \mathrm{~K}\) for certain group 14 elements are: \(\mathrm{C}(s,\) diamond \()=2.43 \mathrm{~J} / \mathrm{mol}-\mathrm{K}, \mathrm{Si}(s)=18.81 \mathrm{~J} /\) mol-K, Ge \((s)=31.09 \mathrm{~J} / \mathrm{mol}-\mathrm{K}, \quad\) a n d \(\quad \operatorname{Sn}(s)=51.818 \mathrm{~J} /\) mol-K. All but Sn have the same (diamond) structure. How do you account for the trend in the \(S^{\circ}\) values?
Step-by-Step Solution
Verified Answer
The trend in entropy increases down the group due to larger atomic size and Sn's unique structure.
1Step 1: Define Standard Entropy
Standard entropy, denoted as \( S^{\circ} \), is a measure of the disorder or randomness of a substance in the standard state (1 atm pressure and 298 K). It is usually given in units of J/mol-K.
2Step 2: Review Provided Data
The given standard entropies at 298 K are: \( \mathrm{C}(s, \text{diamond})=2.43 \mathrm{~J/mol-K} \), \( \mathrm{Si}(s)=18.81 \mathrm{~J/mol-K} \), \( \text{Ge}(s)=31.09 \mathrm{~J/mol-K} \), and \( \mathrm{Sn}(s)=51.818 \mathrm{~J/mol-K} \).
3Step 3: Consider Structural Differences
Identify that all elements except Sn have a diamond-like structure. Sn has a different, more metallic structure, which increases its entropy due to greater atomic vibrations and less ordered arrangement.
4Step 4: Analyze Trends in Entropy Values
Notice that the entropy increases as you move down the group from C to Sn. This is due to the increase in atomic size and corresponding increase in the number of accessible microstates, leading to higher entropy values.
5Step 5: Draw Conclusion about Entropy Trend
The trend is accounted for by the increase in atomic size from C to Sn and the difference in structural arrangement, with Sn having a higher entropy due to its unique structure compared to the other group 14 elements.
Key Concepts
Entropy TrendsGroup 14 ElementsAtomic Structure and Entropy
Entropy Trends
Entropy, symbolized as \( S^{\circ} \), reflects the amount of disorder or randomness in a substance. As elements move down the periodic table, there tends to be an increase in entropy. This is evident in Group 14 elements, from carbon to tin.
Whats drives this increase in entropy? Primarily, as the atomic size grows, so does the number of possible arrangements, or microstates, the atoms can have. This increase in arrangements results in higher entropy values. For instance, tin (Sn) has a higher standard entropy than carbon (C) because its atoms can occupy more positions due to a larger atomic size.
Whats drives this increase in entropy? Primarily, as the atomic size grows, so does the number of possible arrangements, or microstates, the atoms can have. This increase in arrangements results in higher entropy values. For instance, tin (Sn) has a higher standard entropy than carbon (C) because its atoms can occupy more positions due to a larger atomic size.
- Entropy is influenced by the structure of the substance.
- Structural differences result in variations in entropy values due to atomic arrangement possibilities.
Group 14 Elements
Group 14 elements include carbon (C), silicon (Si), germanium (Ge), and tin (Sn). These elements are known to form the backbone of many compounds and are fundamental in various materials. They display interesting trends in their standard entropies.
One key factor in these trends is their crystal structures. With the exception of tin, these elements share a diamond-like structure at room temperature, which is highly ordered. Tin, however, has a more metallic and less ordered structure, which allows for higher entropy. Here's a breakdown:
One key factor in these trends is their crystal structures. With the exception of tin, these elements share a diamond-like structure at room temperature, which is highly ordered. Tin, however, has a more metallic and less ordered structure, which allows for higher entropy. Here's a breakdown:
- Carbon (C), in diamond form, is very ordered, resulting in a much lower entropy value.
- Silicon (Si) and germanium (Ge), while having the same structure, have progressively higher entropies due to their increasing atomic sizes.
- Tin (Sn) diverges from the diamond structure, contributing to its higher entropy.
Atomic Structure and Entropy
The atomic structure is closely tied to a substance's entropy. In simple terms, the more ways atoms can be arranged in a material, the higher the entropy.
Atoms in elements like carbon, silicon, and germanium are bound in a tightly ordered diamond structure. This restricts their relative positions, resulting in lower entropy. However, as we progress from carbon to tin, we notice:
Atoms in elements like carbon, silicon, and germanium are bound in a tightly ordered diamond structure. This restricts their relative positions, resulting in lower entropy. However, as we progress from carbon to tin, we notice:
- Each element, with increasing atomic number, shows more available microstates, leading to higher entropy.
- Sn's unique structure breaks away from the diamond form, resulting in even more possible configurations.
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