Question

The following phosphorus sulfides are known: \(\mathrm{P}_{4} \mathrm{~S}_{3}\), \(\mathrm{P}_{4} \mathrm{~S}_{7},\) and \(\mathrm{P}_{4} \mathrm{~S}_{10} .\) Do these compounds obey the law of multiple proportions?

Short answer

No, these compounds don't obey the law of multiple proportions.

Step-by-step solution

Understanding the Law of Multiple Proportions

The law of multiple proportions states that when two elements form more than one compound with each other, the masses of one element that combine with a fixed mass of the other are in ratios of small whole numbers.

Finding the Mass Ratio of Sulfur for Each Compound

Let's consider phosphorus as the fixed element. Each compound has four phosphorus atoms. We'll compare the sulfur content of these compounds. We'll use atomic masses: P = 31 amu, S = 32 amu.

Calculating the Mass of Sulfur in Each Compound

For each compound, calculate the total mass of sulfur:1. \(\text{For } \mathrm{P}_4 \mathrm{S}_3, \text{ mass of S } = 3 \times 32 = 96 \text{ amu}\)2. \(\text{For } \mathrm{P}_4 \mathrm{S}_7, \text{ mass of S } = 7 \times 32 = 224 \text{ amu}\)3. \(\text{For } \mathrm{P}_4 \mathrm{S}_{10}, \text{ mass of S } = 10 \times 32 = 320 \text{ amu}\)

Comparing Ratios of Sulfur Masses

We will now compare the masses of sulfur for the three compounds, always considering the fixed mass of phosphorus (124 amu corresponding to four phosphorus atoms):1. \(\text{Ratio for } \mathrm{P}_4 \mathrm{S}_3 \text{ is } 96:124\)2. \(\text{Ratio for } \mathrm{P}_4 \mathrm{S}_7 \text{ is } 224:124\)3. \(\text{Ratio for } \mathrm{P}_4 \mathrm{S}_{10} \text{ is } 320:124\)

Simplifying Ratios

Simplify these ratios to check for whole number relation:1. For \(\mathrm{P}_4 \mathrm{S}_3\), the simplified ratio is \( \frac{96}{124} ≈ 0.774\)2. For \(\mathrm{P}_4 \mathrm{S}_7\), the simplified ratio is \( \frac{224}{124} ≈ 1.806\)3. For \(\mathrm{P}_4 \mathrm{S}_{10}\), the simplified ratio is \( \frac{320}{124} ≈ 2.581\)

Conclusion on the Law of Multiple Proportions

These ratios aren't exact simple whole numbers, so these compounds don't strictly obey the law of multiple proportions.

Key concepts

Phosphorus Sulfides

Phosphorus sulfides are compounds composed of phosphorus (P) and sulfur (S). These compounds are known to form in different combinations, which include

• diphosphorus trisulfide (\( \mathrm{P}_{4} \mathrm{S}_{3} \)),
• rapidly followed by diphosphorus heptasulfide (\( \mathrm{P}_{4} \mathrm{S}_{7} \)),
• and diphosphorus decasulfide (\( \mathrm{P}_{4} \mathrm{S}_{10} \)).

These substances are examples of how elements like phosphorus and sulfur can combine in diverse ways, resulting in different phosphorus sulfides. Each combination indicates a different ratio of phosphorus to sulfur atoms.
Phosphorus sulfides are particularly interesting because their structure and formation demonstrate how elements interact with each other at the atomic level.In chemistry, studying the variations in phosphorus sulfides helps scientists understand more about

• the bonding properties of phosphorus,
• as well as how it behaves in different chemical environments.

These variations also illustrate the chemical bonding that governs these interactions, often influenced by factors such as temperature and pressure.

Atomic Mass

Atomic mass is an essential concept in chemistry, giving us the weight of an atom relative to other atoms. It is measured in atomic mass units (amu). For instance, phosphorus has an atomic mass of 31 amu, while sulfur has an atomic mass of 32 amu.
These values help chemists calculate the total mass of elements in a compound. By doing this, they can better understand the proportions and relationships within chemical compounds. In the context of the phosphorus sulfides

• (\( \mathrm{P}_{4} \mathrm{S}_{3} \), \( \mathrm{P}_{4} \mathrm{S}_{7} \), and \( \mathrm{P}_{4} \mathrm{S}_{10} \)),

the atomic mass values enable the precise computation of sulfur content in each:

• In \( \mathrm{P}_{4} \mathrm{S}_{3} \), three sulfur atoms contribute to the mass.
• This continues with seven in \( \mathrm{P}_{4} \mathrm{S}_{7} \),
• and ten in \( \mathrm{P}_{4} \mathrm{S}_{10} \).

Understanding atomic mass is pivotal for any stoichiometric calculations needed in chemistry, making it a cornerstone of quantitative chemical analysis.

Mass Ratio

The mass ratio concept is central to understanding how elements combine to form compounds. When looking at the phosphorus sulfides, the mass of sulfur in each compound changes based on its interaction with phosphorus. To find these ratios, we fix the mass of phosphorus and only vary sulfur's mass per compound.
This is computed using atomic mass, summing up the contributions of each sulfur atom.In

• \( \mathrm{P}_{4} \mathrm{S}_{3} \), the sulfur to phosphorus mass ratio is 96 amu to 124 amu,
• in \( \mathrm{P}_{4} \mathrm{S}_{7} \), it's 224 amu to 124 amu,
• and for \( \mathrm{P}_{4} \mathrm{S}_{10} \), the ratio is 320 amu to 124 amu.

The law of multiple proportions states that the mass ratios of sulfur (in these compounds) should simplify to small whole numbers compared to phosphorus. Despite simplifying the ratios, the phosphorus sulfides here do not result in exact small whole numbers. This indicates that these particular phosphorus sulfides don't strictly follow the law of multiple proportions. This knowledge deepens the understanding of chemical bonding and composition.

Chemical Compounds

Chemical compounds are substances formed from two or more elements bonded together in specific ratios. These compounds form through chemical reactions, where atoms from different elements share or exchange electrons to achieve stability.
The phosphorus sulfides (\( \mathrm{P}_{4} \mathrm{S}_{3} \), \( \mathrm{P}_{4} \mathrm{S}_{7} \), and \( \mathrm{P}_{4} \mathrm{S}_{10} \)) are a great example of how varying ratios of phosphorus and sulfur can create distinct chemical compounds.Each compound has unique properties depending on how the elements are combined and their respective ratios. Understanding these compounds involves

• knowing the atomic structure,
• the mass of each participating element,
• and how these masses relate to each other.

Through examining different phosphorus sulfides, we see how complex and varied chemical compounds can be. Each compound offers insights into the principles governing chemical reactions and the resultant products, enhancing our comprehension of material science and chemistry as a whole.