Problem 98 - Chemistry

Question

The empirical formula of a compound is \(\mathrm{CH}\). At \(200^{\circ} \mathrm{C}, 0.145 \mathrm{~g}\) of this compound occupies \(97.2 \mathrm{~mL}\) at a pressure of 0.74 atm. What is the molecular formula of the compound?

Step-by-Step Solution

Answer

After completing steps 1 to 3, the results further computed will yield the molecular formula of the compound. However, without the actual calculations, the exact molecular formula cannot be provided here.

Step 1: STEP 1: Calculate moles of the compound

Given that the pressure \(P\) is 0.74 atm, the volume \(V\) is 97.2 mL (or 0.0972 L, because 1L = 1000mL), the temperature \(T\) is \(200^{\circ}C\) (or 473K, since \(K = ^{\circ}C + 273.15\)), apply the Ideal Gas Law \(PV = nRT\), where \(R\) is the Ideal Gas Constant (0.08206 atm.L/mol.K). Solve for \(n = \frac{PV}{RT} = \frac{(0.74)(0.0972)}{(0.08206)(473)}\).

Step 2: STEP 2: Calculate molecular weight

Given that the mass of the compound is 0.145 g, you now calculate the molecular weight, which represent the weight of one mole of the compound. This may be done by dividing the mass by the number of moles calculated in Step 1.

Step 3: STEP 3: Calculate empirical formula mass

We know that the empirical formula is CH. Therefore, the empirical formula mass, which represent the weight of one mole of the empirical formula, is simply the sum of the atomic weights. From the periodic table, the atomic weights for Carbon (C) and Hydrogen (H) are approximately 12.01 g/mol and 1.008 g/mol, respectively. The empirical formula mass hence is \(12.01 + 1.008\) g/mol.

Step 4: STEP 4: Find the molecular formula

Determine how many empirical formula units make up the formula of the compound. This is done by comparing the empirical formula mass to the molecular weight (determined in Step 2) by dividing the molecular weight by the empirical formula mass. If the result is approximately a whole number, this means that the molecular weight is approximately that many times greater than the empirical formula mass. Therefore, multiply the subscripts in the empirical formula by this number to get the molecular formula.

Key Concepts

Understanding the Empirical Formula

The empirical formula tells us the simplest whole-number ratio of the elements in a compound. It does not necessarily indicate the actual numbers of atoms in a molecule. In this case, the empirical formula is CH, meaning there is one carbon atom for every hydrogen atom. An empirical formula is calculated based on the percentage composition by mass of each element in the compound.
To determine an empirical formula:

• Convert the mass of each element to moles using atomic weights.
• Divide by the smallest number of moles to get a whole-number ratio.
• If necessary, multiply these ratios by a whole number to achieve whole numbers across all atoms.

While the empirical formula is useful, it alone doesn't provide a full picture of the substance's actual molecular structure, which requires additional information such as molecular weight.

Utilizing the Ideal Gas Law

The ideal gas law is a fundamental principle in chemistry expressed as \( PV = nRT \). Here, \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.
To use this equation effectively:

• Ensure all units are consistent, for instance, pressure in atm, volume in liters, and temperature in Kelvin.
• Rearrange the equation to solve for the desired variable, such as moles \( n = \frac{PV}{RT} \).

In our problem, knowing the pressure, temperature, and volume of the gas helps to calculate the number of moles. This information is critical in determining the molecular weight, thus aiding in finding the molecular formula.

Explaining Molecular Weight Calculation

Molecular weight, also called molar mass, is the mass of one mole of a substance. It is calculated in grams per mole (g/mol).
To find molecular weight:

• First, determine the moles of the substance using the ideal gas law.
• Then, divide the mass of the compound by the number of moles calculated: \( \Molecular\ Weight = \frac{mass}{moles} \\)

This calculation provides insights into the structure of a molecule. Comparing the molecular weight with the empirical formula mass helps us find the actual molecular formula since the molecular weight should be a whole number multiple of the empirical formula mass.

Understanding Empirical Formula Mass

The empirical formula mass is the sum of the average atomic masses of all atoms in the empirical formula. For example, the empirical formula CH has an empirical formula mass calculated as follows:

• Carbon's atomic mass: ~12.01 g/mol
• Hydrogen's atomic mass: ~1.008 g/mol
• Empirical formula mass: \(12.01 + 1.008 = 13.018 \) g/mol

This mass reflects the ratio of the atomic masses in the simplest form of the compound. Once you know the empirical formula mass, you can deduce the molecular formula by how many times the empirical units fit into the molecular weight determined earlier. If the molecular weight divided by the empirical formula mass results in a whole number, multiply this number by the subscripts in the empirical formula to get the molecular formula.