Problem 77 - Chemistry The Molecular Science

Question

You have a pure sample of apholate, \(\mathrm{C}_{12} \mathrm{H}_{24} \mathrm{~N}_{9} \mathrm{P}_{3},\) a highly effective commercial insecticide. (a) Calculate the molar mass of apholate. (b) Calculate the mass of \(\mathrm{N}\) in 100 . \(\mathrm{g}\) apholate. (c) A sample containing \(250.0 \mathrm{mg}\) apholate is sprayed on an agricultural field. Calculate the mass of phosphorus in this sample of apholate; express your result in grams. (d) Calculate the number of phosphorus atoms in this sample of apholate.

Step-by-Step Solution

Answer

(a) 387.36 g/mol; (b) 32.55 g; (c) 0.060 g; (d) 1.17 × 10¹² P atoms.

Step 1: Calculate Molar Mass

To find the molar mass of apholate, - Identify the atomic masses of each element (rounded to 2 decimal places): - Carbon (C): 12.01 g/mol - Hydrogen (H): 1.01 g/mol - Nitrogen (N): 14.01 g/mol - Phosphorus (P): 30.97 g/mol- Calculate the total mass contributions from each element in - C: 12.01 g/mol * 12 = 144.12 g/mol - H: 1.01 g/mol * 24 = 24.24 g/mol - N: 14.01 g/mol * 9 = 126.09 g/mol - P: 30.97 g/mol * 3 = 92.91 g/mol- Sum these values to get the total molar mass: \[ \text{Molar Mass of } \mathrm{C}_{12}\mathrm{H}_{24}\mathrm{N}_{9}\mathrm{P}_{3} = 144.12 + 24.24 + 126.09 + 92.91 = 387.36 \text{ g/mol} \]

Step 2: Mass of Nitrogen Calculation

Find the fraction of the mass of nitrogen in the compound and calculate its mass in 100 g of apholate.- Calculate the nitrogen's contribution to molar mass: fraction = \(\frac{\text{mass of N}}{\text{molar mass}}\) \[ \text{Mass fraction of N} = \frac{126.09}{387.36} \] - Multiply the fraction by the total mass: \[ \text{Mass of N in 100 g} = \frac{126.09}{387.36} \times 100 = 32.55 \text{ g} \]

Step 3: Mass of Phosphorus in 250 mg Apholate

Determine the mass of phosphorus in the sample of 250 mg of apholate.- Calculate the phosphorus contribution to molar mass: fraction = \(\frac{\text{mass of P}}{\text{molar mass}}\) \[ \text{Mass fraction of P} = \frac{92.91}{387.36} \] - Convert mg to g for the calculation: \[ 250 \text{ mg} = 0.250 \text{ g} \] - Calculate the mass of phosphorus in 0.250 g of apholate: \[ \text{Mass of P in 0.250 g} = \frac{92.91}{387.36} \times 0.250 = 0.060 \text{ g} \]

Step 4: Calculate Phosphorus Atoms Number

Calculate the number of phosphorus atoms in the 250 mg of apholate.- Find the moles of P in 0.250 g apholate using the molar mass: \[ \text{Mass of P in moles} = \frac{0.060 \text{ g}}{30.97 \text{ g/mol}} = 0.001937 \text{ mol P} \] - Use Avogadro's number to find the number of atoms: \[ 0.001937 \text{ mol} \times 6.022 \times 10^{23} \text{ atoms/mol} = 1.17 \times 10^{21} \text{ P atoms} \]

Key Concepts

Phosphorus Atoms Calculation

Calculating the number of phosphorus atoms within a chemical compound involves a few essential steps. First, you need to determine the number of moles of phosphorus in the sample. This is done by dividing the total mass of phosphorus in the apholate sample by the molar mass of phosphorus. Once you have the moles, you can find the number of atoms by multiplying the moles by Avogadro's number.

Here's a simple way to think about it:

• Find the mass of phosphorus in your sample, which you then convert into moles of phosphorus using the molar mass (30.97 g/mol for phosphorus).
• Using Avogadro's number, which is approximately \(6.022 \times 10^{23}\) atoms/mol, convert moles to atoms.

This provides the total count of phosphorus atoms in your sample.

Mass of Chemical Elements

When analyzing a compound like apholate, calculating the mass of each element is crucial for understanding its chemical composition. The mass of an element tells you how much of that element is present in a given amount of the compound.

This process generally follows these steps:

• First, calculate the mass contribution of each element based on its atomic mass and the number of atoms present in the chemical formula.
• For example, phosphorus (P) has an atomic mass of 30.97 g/mol, and apholate contains three phosphorus atoms.
• Multiply this atomic mass by the number of atoms to find the element's total mass in the compound.

Doing this for all elements in the formula ensures that you know precisely how much of each element contributes to the total mass of the apholate.

Avogadro's Number

Avogadro's number is a fundamental concept in chemistry that simplifies converting between moles and individual atoms or molecules. It states that one mole of any substance contains exactly \(6.022 \times 10^{23}\) atoms, molecules, or other particles.

This constant is used to make practical calculations in chemistry by:

• Allowing chemists to relate the macroscopic amounts of materials they work with to the microscopic number of atoms and molecules.
• Providing a bridge between a substance’s mass and its quantity in atomic or molecular terms.

Using Avogadro's number in calculations, you can easily determine the number of atoms present in a given sample by multiplying the moles by \(6.022 \times 10^{23}\). This is particularly helpful when dealing with elements like phosphorus in apholate, where you're interested in the number of atoms, not just their collective mass.

Chemical Composition Analysis

Chemical composition analysis is integral in determining what elements are present in a compound and their respective proportions. In the case of apholate, this involves analyzing its molecular formula, \(\mathrm{C}_{12}\mathrm{H}_{24}\mathrm{N}_{9}\mathrm{P}_{3}\), to ascertain the mass percent of each element.

To perform this analysis:

• Start by calculating the molar mass of the entire compound, as you've done previously for apholate.
• Next, find the mass contributions of each element based on their proportions in the molecular formula.
• Finally, express each element's mass as a percentage of the total molar mass.

This information allows you to understand how much of each element is present in apholate, which can be essential for various technological or agricultural applications, such as determining effective doses of the pesticide.