Question

In chemical research we often send newly synthesized compounds to commercial laboratories for analysis. These laboratories determine the weight percent of \(\mathrm{C}\) and \(\mathrm{H}\) by burning the compound and collecting the evolved \(\mathrm{CO}_{2}\) and \(\mathrm{H}_{2} \mathrm{O}\) They determine the molar mass by measuring the osmotic pressure of a solution of the compound. Calculate the empirical and molecular formulas of a compound, \(\mathrm{C}_{x} \mathrm{H}_{y} \mathrm{Cr},\) given the following information: (a) The compound contains \(73.94 \%\) C and \(8.27 \%\) \(\mathrm{H} ;\) the remainder is chromium. (b) At \(25^{\circ} \mathrm{C},\) the osmotic pressure of a solution containing \(5.00 \mathrm{mg}\) of the unknown dissolved in exactly \(100 \mathrm{mL}\) of chloroform solution is \(3.17 \mathrm{mm} \mathrm{Hg}\)

Short answer

Empirical formula is determined via mass ratios; molecular formula uses molar mass from osmotic pressure.

Step-by-step solution

Calculate the mass % of Chromium

Since the compound contains 73.94% C and 8.27% H, the remainder is Cr. Use the equation: \[\text{Mass \% of Cr} = 100 \% - 73.94 \% - 8.27 \%\]to find the mass percentage of Cr.

Determine moles of each element per 100 g

Assuming 100 g of the compound, convert the percentages into grams. Then, calculate moles of each element using their molar masses:- \( \text{Moles of C} = \frac{73.94 \text{ g}}{12.01 \text{ g/mol}} \)- \( \text{Moles of H} = \frac{8.27 \text{ g}}{1.008 \text{ g/mol}} \)- \( \text{Moles of Cr} = \frac{(100 - 73.94 - 8.27) \text{ g}}{51.9961 \text{ g/mol}} \)

Determine the Empirical Formula

Find the smallest number of moles among the elements calculated in Step 2, and divide all moles by this smallest amount to get their mole ratios. Use these ratios to establish the empirical formula.

Use the Osmotic Pressure to Find Molar Mass

Convert 5.00 mg to grams and use the osmotic pressure formula:\[\Pi = iMRT\]Where \( \Pi \) is the osmotic pressure, \( M \) is the molarity, \( R \) is the gas constant in appropriate units, and \( T \) is the temperature in Kelvin. Solve for \( M \) (molar mass).

Determine the Molecular Formula

Divide the molar mass obtained from Step 4 by the empirical formula mass (from Step 3) to find a whole number ratio. Multiply the subscripts in the empirical formula by this ratio to find the molecular formula.

Key concepts

Osmotic Pressure

Osmotic pressure is a fundamental concept in chemistry that helps in determining the molar mass of a substance. It refers to the pressure required to stop the flow of a solvent into a solution through a semi-permeable membrane. Understanding osmotic pressure is crucial because it relates to how substances behave in solutions. To calculate osmotic pressure, the formula \( \Pi = iMRT \) is commonly used, where:

• \( \Pi \) is the osmotic pressure.
• \( i \) is the Van't Hoff factor, which is typically 1 for non-ionizing solutes.
• \( M \) is molarity, representing the concentration of the solution.
• \( R \) is the gas constant, 0.0821 L·atm/mol·K.
• \( T \) is the temperature in Kelvin.

To find the molar mass from osmotic pressure, rearrange the equation to solve for molar mass, after first ensuring the temperature is in Kelvin and pressure is converted correctly. This allows the molarity and subsequently the molar mass of an unknown to be calculated, providing crucial information about the compound in question.

Elemental Analysis

Elemental analysis is the process of determining the elemental composition of a compound, specifically the weight percent of each element. Laboratories often perform elemental analysis during the investigation of new compounds. This involves burning the compound and measuring the resulting gases, such as CO\(_2\) and H\(_2\)O, to infer the composition.In our given problem, we calculate the mass percentage of elements like Carbon (C), Hydrogen (H), and Chromium (Cr). Knowing that the sum of percentages must equal 100%, we find the remainder of chromium after accounting for the other elements. Accurate elemental analysis provides the basis for calculating the empirical formula, as the moles of each element can be derived from their respective percentages by mass.

Mole Ratios

Mole ratios are derived from the quantities of moles of each element present in a compound, providing the simplest whole-number ratios between the elements. To calculate mole ratios, we assume a sample size—typically 100 grams—for ease, allowing the direct use of percentage compositions as grams. From this mass, we convert grams to moles by dividing by each element's respective molar mass. The smallest calculated mole value among the elements is then used to divide all the moles, which results in the mole ratios. These ratios are crucial in defining the empirical formula, thereby indicating the simplest possible formula that represents a compound's composition.

Molar Mass Calculation

The calculation of molar mass involves determining the mass of one mole of a compound, which is pivotal for converting between moles and grams in chemical equations. When dealing with osmotic pressure, the calculated molar mass allows for the determination of a compound's molecular formula. After finding the empirical formula and calculating its molar mass, the molar mass obtained from osmotic pressure measurements is used to find the molecular formula. The molecular formula is derived by dividing the experimental molar mass by the empirical formula's molar mass. If the result is a whole number, it indicates how many times the empirical formula must be multiplied to get the actual molecular formula. Thus, knowing molar mass is essential for ascertaining the correct molecular structure of a compound.