Question

Phosphorus trichloride is important in the manufacture of insecticides, fuel additives, and flame retardants. Phosphorus has only one naturally occurring isotope, \({ }^{31} \mathrm{P},\) whereas chlorine has two, \({ }^{35} \mathrm{Cl}(75 \%)\) and \({ }^{37} \mathrm{Cl}(25 \%) .\) (a) What different molecular masses (in amu) can be found for \(\mathrm{PCl}_{3} ?\) (b) Which is the most abundant? (c) What is the ratio of the effusion rates of the heaviest and the lightest \(\mathrm{PCl}_{3}\) molecules?

Short answer

The molecular masses are 136, 138, 140, and 142 amu. The most abundant is 136 amu. The ratio of effusion rates is approximately 1.02.

Step-by-step solution

- Identify Atomic Masses

Phosphorus (P) has only one isotope, \(^{31}P\), with an atomic mass of 31 amu. Chlorine (Cl) has two isotopes: \(^{35}Cl (75\%)\) with an atomic mass of 35 amu and \(^{37}Cl (25\%)\) with an atomic mass of 37 amu.

- Calculate Molecular Masses of Different Combinations

The molecular masses of \PCl_3 \ can be calculated by combining P with the different isotope combinations of Cl: \[ 31 + 3(35) = 136 \text{ amu} \] \[ 31 + 2(35) + 37 = 138 \text{ amu} \] \[ 31 + 35 + 2(37) = 140 \text{ amu} \] \[ 31 + 3(37) = 142 \text{ amu} \]

- Determine Most Abundant Molecular Mass

To find the most abundant molecular mass, consider the abundance of Cl isotopes: \(35Cl 75\%\) and \(37Cl 25\%\). The most probable combination will be the one with the isotopes appearing most often. The molecular mass \ 136 \text{ amu} \ ( three \(35Cl\) atoms) is most probable.

- Calculate Effusion Rates

Use Graham's law of effusion: \ \frac{Rate_1}{Rate_2} = \frac{\text{Square root of M_2}}{\text{Square root of M_1}} \ to find the ratio of effusion rates for the heaviest and lightest \(PCl_3\) molecules. \ \frac{Effusion Rate_{136}}{Effusion Rate_{142}} = \frac{\text{Square root of 142}}{\text{Square root of 136}} = \frac{\text{11.91}}{\text{11.662}} \approx 1.02 \.

Key concepts

isotope abundance

Isotopes are atoms of the same element that have different numbers of neutrons. An element can have multiple isotopes with varying abundance in nature. For phosphorus trichloride (PCl3), we focus on chlorine which has two stable isotopes:

• ³⁵Cl with a 75% abundance
• ³⁷Cl with a 25% abundance

Phosphorus itself has only one stable isotope, ³¹P. The abundance of these isotopes affects the molecular mass calculations for compounds. The higher the natural abundance of an isotope, the more it will contribute to the average molecular mass of a compound.

molecular mass calculation

Calculating the molecular mass of a compound involves summing the average atomic masses of the atoms in the molecule. For phosphorus trichloride (PCl₃), we consider different combinations of isotopes:

• ³¹P + 3(³⁵Cl): 31 + 3(35) = 136 amu
• ³¹P + 2(³⁵Cl) + ³⁷Cl: 31 + 2(35) + 37 = 138 amu
• ³¹P + ³⁵Cl + 2(³⁷Cl): 31 + 35 + 2(37) = 140 amu
• ³¹P + 3(³⁷Cl): 31 + 3(37) = 142 amu

These computed values represent the possible molecular masses of PCl₃ based on different isotope combinations. Understanding these molecular mass combinations is crucial for determining the properties and behavior of the compound.

Graham's law of effusion

Graham's law of effusion states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass. This can be expressed as: \[\frac{Rate_1}{Rate_2} = \frac{\sqrt{M_2}}{\sqrt{M_1}}\]In the context of PCl₃, we want to compare the effusion rates of the heaviest and lightest molecules. Substituting the molecular masses: \[\frac{Effusion Rate_{136}}{Effusion Rate_{142}} = \frac{\sqrt{142}}{\sqrt{136}} = \frac{11.91}{11.662} \approx 1.02\]This shows that the lightest PCl₃ molecules effuse slightly faster than the heaviest ones due to their lower mass. This concept is essential in understanding molecular motion and behavior under different conditions.

phosphorus trichloride

Phosphorus trichloride (PCl₃) is a chemical compound used in various industrial applications like manufacturing insecticides, fuel additives, and flame retardants. Its chemical properties and behavior are influenced by the isotopic composition of its constituent atoms. Understanding these isotopic influences helps in tailoring its usage in specific chemical processes. For example:

• Knowing which isotopic combination is most prevalent (i.e., ³⁵Cl-based) helps predict its average molecular mass and reactivity.
• Using Graham's law, we can understand how the molecule will behave under different physical conditions, such as in effusion or diffusion processes.

Overall, a deep understanding of PCl₃ starts with grasping its molecular mass, isotopic composition, and the laws governing its physical behavior.