Question

Anthraquinone contains only carbon, hydrogen, and oxygen. When $4.80\mathrm{mg}$ anthraquinone is burned, $14.2\mathrm{mg}{\mathrm{CO}}_2$ and $1.65\mathrm{mg}{\mathrm{H}}_2\mathrm{O}$ are produced. The freezing point of camphor is lowered by ${22.3}^{\circ }\mathrm{C}$ when $1.32\mathrm{g}$ anthraquinone is dissolved in 11.4 g camphor. Determine the empirical and molecular formulas of anthraquinone.

Short answer

The empirical formula of anthraquinone is C_{14}H_{8}O_{2}, and its molecular formula is C_{28}H_{16}O_{4}.

Step-by-step solution

Calculate the masses of carbon and hydrogen in anthraquinone

From the given data, when 4.80 mg anthraquinone is burned, 14.2 mg CO2 and 1.65 mg H2O are produced. We can calculate the masses of carbon and hydrogen in anthraquinone as follows: Mass of carbon in CO2 = Mass of CO2 x (Mass of C in CO2 / Mass of CO2) Mass of hydrogen in H2O = Mass of H2O x (Mass of H in H2O / Mass of H2O) The atomic masses of C, H, and O are approximately 12, 1, and 16, respectively. The molecular mass of CO2 = 12 + 2(16) = 44 The molecular mass of H2O = 2(1) + 16 = 18

Calculate the mass of oxygen in anthraquinone

Now, we can determine the mass of oxygen in anthraquinone as follows: Mass of anthraquinone = Mass of carbon + Mass of hydrogen + Mass of oxygen => Mass of oxygen = Mass of anthraquinone - Mass of carbon - Mass of hydrogen

Calculate the moles of carbon, hydrogen, and oxygen

Now we will calculate the moles of each element: Moles of carbon = Mass of carbon / Atomic mass of carbon Moles of hydrogen = Mass of hydrogen / Atomic mass of hydrogen Moles of oxygen = Mass of oxygen / Atomic mass of oxygen

Determine the empirical formula of anthraquinone

We will now determine the empirical formula for anthraquinone by finding the lowest whole number ratio of moles of carbon, hydrogen, and oxygen. Divide each mole value by the smallest mole value obtained in step 3. If necessary, multiply the resulting ratios by the smallest whole number to obtain whole number ratios. The empirical formula is written using these whole number ratios as subscripts.

Determine the molecular formula of anthraquinone

To find the molecular formula of anthraquinone, we will first find the molecular weight using the freezing point depression data provided. We know that: Freezing point depression = Kf x molality x i Where Kf is the molal freezing point depression constant for camphor, molality is the molality of the solution and i is the van't Hoff factor (equal to 1 for molecular solutes). Given the freezing point depression of camphor is 22.3 °C when 1.32 g anthraquinone is dissolved in 11.4 g camphor, we can calculate the molality of the solution. Molality of solution = (Weight of anthraquinone in g) / (Weight of camphor in g) The molal freezing point depression constant for camphor (Kf) is 40.0 ºC/m. Now, we can find the molecular weight of anthraquinone. Molecular weight of anthraquinone = (Freezing point depression / (Kf x molality)) x 1000 Using the molecular weight of anthraquinone and the empirical formula, we can now determine the molecular formula by finding the whole number ratio of the empirical formula mass to the molecular weight. The molecular formula is found by multiplying the empirical formula by this whole number ratio.

Key concepts

Anthraquinone

Anthraquinone is an organic compound consisting of carbon, hydrogen, and oxygen atoms. It is an important chemical used in the synthesis of dyes, particularly in textiles and inks. The structure of anthraquinone makes it highly stable and non-polar, which is why it is insoluble in water but soluble in organic solvents. In chemical contexts, understanding anthraquinone involves determining its empirical and molecular formulas. These offer insight into its composition and structure, which are crucial for its application in various industrial processes. By thoroughly analyzing the combustion and freezing point depression data, we can deduce both the empirical and molecular formulas to comprehend better the compound's proportionate content and actual molar mass.

Combustion Analysis

Combustion analysis is a method used to determine the empirical formula of a compound containing carbon and hydrogen. The process involves burning the compound to convert it into carbon dioxide (CO₂) and water (H₂O). The amounts of CO₂ and H₂O produced can be measured to determine how much carbon and hydrogen were in the original compound.
Here's how it works:

• Measure the mass of the CO₂ produced, which is used to calculate the mass of carbon (C) it contained using the formula: $$\text{Mass of Carbon}=\frac{{\text{Mass of CO}}_2\times 12}{44}$$
• Similarly, measure the mass of the H₂O produced to calculate the mass of hydrogen (H) using the formula: $$\text{Mass of Hydrogen}=\frac{{\text{Mass of H}}_2\text{O}\times 2}{18}$$

By conducting the combustion analysis, we can identify the specific amounts of carbon and hydrogen, which when subtracted from the total mass of the compound, leaves the mass of oxygen (O) to complete the picture.

Freezing Point Depression

Freezing point depression is an important property in determining the molecular weight of a solute. It describes how the freezing point of a solvent decreases when a solute is dissolved in it. For anthraquinone, this property helps us find its molar mass using camphor as the solvent.
Here's a breakdown:

• The extent of freezing point depression is given by $$\mathrm{\Delta }{T}_f=K_f\cdot m\cdot i$$ Where:
  • $\mathrm{\Delta }{T}_f$ is the change in freezing point,
  • $K_f$ is the freezing point depression constant for the solvent,
  • $m$ is the molality of the solution, and
  • $i$ is the van 't Hoff factor. For molecular substances like anthraquinone, $i=1$.
• The molality $m$ is calculated as $$m=\frac{\text{mass of solute (g)}}{\text{mass of solvent (kg)}}$$

By rearranging the freezing point depression equation, we solve for the molar mass of the anthraquinone to determine the molecular formula.

Molecular Weight Determination

Molecular weight determination is crucial for identifying the molecular formula of a compound. By understanding a compound’s empirical formula, which tells us the simplest ratio of elements, we can calculate the molecular weight using experimental data, such as those from freezing point depression.
The process involves:

• Calculating the empirical formula mass, which is the sum of the atomic masses for all atoms in the empirical formula.
• Determining the experimental molar mass from methods like freezing point depression. From this, find the ratio of the experimental molar mass to the empirical formula mass:$$\text{Whole number ratio}=\frac{\text{Molar mass}}{\text{Empirical formula mass}}$$
• Using the whole number ratio to scale the empirical formula to find the molecular formula, which presents the actual number of each type of atom in a molecule of anthraquinone.

This process helps us accurately identify substances like anthraquinone in scientific study and industrial applications.