Problem 68 - Chemistry The Central Science

Question

One cup of fresh orange juice contains 124 \(\mathrm{mg}\) of ascorbic acid (vitamin \(\mathrm{C}, \mathrm{C}_{6} \mathrm{H}_{8} \mathrm{O}_{6} ) .\) Given that one cup \(=236.6 \mathrm{mL}\) calculate the molarity of vitamin \(\mathrm{C}\) in orange juice.

Step-by-Step Solution

Answer

The molarity of Vitamin C in orange juice can be calculated using the following steps: 1. Convert the given amount of Vitamin C (124 mg) to moles by dividing by the molar mass of Vitamin C, \(C_6H_8O_6\), which is approximately \(6 \times (12.01)\mathrm{g/mol} + 8 \times (1.01)\mathrm{g/mol} + 6 \times (16.00)\mathrm{g/mol} = 176.13\mathrm{g/mol}\). Therefore, the number of moles is \(\frac{124\mathrm{mg}}{176.13\mathrm{g/mol}} \times \frac{1\mathrm{g}}{1000\mathrm{mg}} \approx 0.000704\,\mathrm{mol}\). 2. Convert the volume of the solution (236.6 mL) to liters using the conversion factor \(1\,\mathrm{L} = 1000\,\mathrm{mL}\). Therefore, \(236.6\,\mathrm{mL} \times \frac{1\,\mathrm{L}}{1000\,\mathrm{mL}} = 0.2366\,\mathrm{L}\). 3. Finally, calculate the molarity using the formula \(M = \frac{\text{amount of solute in moles}}{\text{volume of solution in liters}}\). Thus, the molarity of Vitamin C in orange juice is \(\frac{0.000704\,\mathrm{mol}}{0.2366\,\mathrm{L}} \approx 0.00298\,\mathrm{M}\).

Step 1: Convert the given amount of Vitamin C in mg to moles

To convert the given amount of Vitamin C (124 mg) to moles, we need to find the molar mass of Vitamin C (C6H8O6) and then divide the given amount by the molar mass. Molar mass of Vitamin C: \( C_6H_8O_6 = 6 \times (12.01)\mathrm{g/mol} + 8 \times (1.01)\mathrm{g/mol} + 6 \times (16.00)\mathrm{g/mol} \) Calculate the molar mass and then divide the given amount (124 mg) by the molar mass.

Step 2: Convert the volume of the solution (orange juice) from mL to liters

We are given the volume of one cup of orange juice as 236.6 mL. To convert this volume to liters, we can use the following conversion factor: 1 L = 1000 mL Compute the volume in liters using this conversion factor.

Step 3: Calculate the molarity using the formula

Now that we have the amount of Vitamin C in moles and the volume of the solution in liters, we can use the formula to calculate molarity: M = (amount of solute in moles) / (volume of solution in liters) Substitute the values and compute the molarity of Vitamin C in orange juice.

Key Concepts

Vitamin C

Vitamin C, also known as ascorbic acid, is a vital nutrient for our bodies. It is renowned for its antioxidant properties and its role in supporting the immune system. Particularly prevalent in citrus fruits like oranges, Vitamin C helps repair tissues and enzymatic production of neurotransmitters. During this exercise, we focus on quantifying its concentration in a common source: orange juice.

To understand the amount of Vitamin C in a given solution, we must first convert it to the number of moles. Moles are a standard measuring unit in chemistry, representing a set number of molecules. This allows us to compare quantities of substances in reactions and solutions effectively. In the exercise, we're starting with a mass, which will be converted to moles using the molar mass of Vitamin C.

Molar Mass

Molar mass is a fundamental concept in chemistry, crucial for converting mass into moles. It is the mass of one mole of a substance—expressed in grams per mole (g/mol). Each element has a known atomic mass, and by adding these for each atom in a compound's formula, we find the compound's molar mass.

For Vitamin C (C₆H₈O₆), we calculate its molar mass by considering:

• The molar mass of carbon (C) is about 12.01 g/mol, and there are six carbon atoms.
• The molar mass of hydrogen (H) is about 1.01 g/mol, with eight hydrogen atoms.
• The molar mass of oxygen (O) is about 16.00 g/mol, and there are six oxygen atoms.

Gathering these together, the molar mass of Vitamin C equates to:
\( 6 \times 12.01 + 8 \times 1.01 + 6 \times 16.00 = 176.12 \ ext{g/mol} \)
This molar mass is significant, as it allows us to convert 124 mg of Vitamin C into moles using the relation:
\( \text{moles} = \frac{\text{mass in grams}}{\text{molar mass}} \)

Solution Volume

Solution volume is a key factor when calculating molarity, which describes the concentration of a solute in a solution. The volume must be expressed in liters when using the standard molarity formula.

In the problem, we begin with a volume given in milliliters (mL)—236.6 mL to be precise. Since 1 liter (L) equals 1000 milliliters, we convert the original volume to liters by dividing by 1000.
Hence, we have:
\( \text{Volume in Liters} = \frac{236.6 \text{ mL}}{1000} = 0.2366 \text{ L} \)

This conversion is essential as molarity requires an understanding of how much solute is present per unit of solution volume.

Unit Conversion

Unit conversion plays a crucial role in solving various chemistry problems, including this one. It's essential to seamlessly transform measurements into those required to use standard formulas accurately.

In this exercise, two primary conversions take place:

• Converting mass from milligrams to grams: Since the standard molar mass unit is g/mol, converting 124 mg of Vitamin C to grams is necessary. We accomplish this by dividing the mass in mg by 1000: \( \text{Mass in grams} = \frac{124 \text{ mg}}{1000} = 0.124 \text{ g} \)
• Converting volume from mL to Liters: Similarly, we needed the volume in liters to calculate molarity. The process involves dividing the milliliter value by 1000.

Understanding these unit conversions ensures precision and is vital in finding the correct molarity. It enables the application of the equation for molarity, which is the moles of solute divided by the liters of solution, offering insight into the concentration of Vitamin C in the orange juice.