Problem 15 - Introductory Chemistry Atoms First

Question

How many milliliters of a 2.55 M solution of glucose, \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\), molar mass \(=180.155 \mathrm{~g} / \mathrm{mol}\), do you need to obtain \(25.0 \mathrm{~g}\) of glucose?

Step-by-Step Solution

Answer

To obtain $25.0 \text{ g}$ of glucose from a $2.55 \text{ M}$ solution, you will need a volume of \( \frac{\frac{25.0 \text{ g}}{180.155 \text{ g/mol}}}{2.55 \text{ M}} \) × 1000 mL.

Step 1: Calculate moles of glucose

To find the moles of glucose, we will use the formula: moles = mass / molar mass Given mass of glucose = 25.0 g Molar mass of glucose = 180.155 g/mol So, moles of glucose = \( \frac{25.0 \text{ g}}{180.155 \text{ g/mol}} \)

Step 2: Calculate the volume of the solution

Now, we have the moles of glucose and the molarity of the solution. We will use the formula of molarity to find the volume of the solution in liters: Molarity (M) = moles of solute / volume of solution (in liters) Given molarity of the solution = 2.55 M Substituting the values in the equation: 2.55 M = \( \frac{moles \, of \, glucose}{volume \, of \, solution} \) Solving for the volume of the solution, we get: volume of solution = \( \frac{moles \, of \, glucose}{2.55 \text{ M}} \)

Step 3: Convert volume to milliliters

Since 1 liter = 1000 milliliters (mL), we will convert the volume of solution (in liters) that we found in step 2 to milliliters. volume of solution (in milliliters) = volume of solution (in liters) × 1000 Now, we have all the necessary information to find the required volume of the glucose solution: volume of solution (in milliliters) = \( \frac{moles \, of \, glucose}{2.55 \text{ M}} \) × 1000 Putting the values from step 1, we get: volume of solution (in milliliters) = \( \frac{\frac{25.0 \text{ g}}{180.155 \text{ g/mol}}}{2.55 \text{ M}} \) × 1000 Calculate the above expression to get the required volume of the solution.

Key Concepts

Moles of Solute

To understand how to determine the amount needed from a solution, it's essential to learn about the concept of moles of solute. A solute is a substance that is dissolved in a solution. The number of moles of a solute expresses the amount of that substance, and it's calculated using the formula:

• moles = mass / molar mass.

For our exercise, the mass of glucose is given as 25.0 grams. With a specified molar mass of 180.155 g/mol for glucose, the calculation becomes:\[moles = \frac{25.0 \text{ g}}{180.155 \text{ g/mol}}.\]This calculation helps us understand how much of the glucose is actually present in terms of moles. Moles provide a way to determine how particles at the molecular level are involved in chemical reactions or solutions. When you know how many moles you have, it is easier to relate it to molarity and volume.

Molar Mass Calculation

Molar mass is a crucial concept when dealing with chemical substances. It is the mass of one mole of a given substance, typically expressed in g/mol. For glucose, \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\), it is important to find the molar mass to determine the moles from a known mass.Calculating the molar mass involves adding up the atomic masses of all the atoms present in the chemical formula. Glucose's formula tells us there are:

• 6 Carbon (C) atoms
• 12 Hydrogen (H) atoms
• 6 Oxygen (O) atoms

These are combined as:

• 6 \( \times 12.01 \text{ g/mol} \) for Carbon,
• 12 \( \times 1.008 \text{ g/mol} \) for Hydrogen,
• 6 \( \times 16.00 \text{ g/mol} \) for Oxygen.

Summing these gives a total molar mass, which in case of glucose resembles approximately 180.155 g/mol. This figure helps in calculating the moles from a known mass or vice versa, making it pivotal for working with chemical equations and solutions.

Volume Conversion

Chemistry often requires us to convert volumes, especially when dealing with solutions. In our calculations, we determine the volume of a solution from moles and molarity, initially in liters. However, practical lab scenarios or experiments often need volumes in milliliters (mL).To convert liters to milliliters, it is helpful to remember that 1 liter is equivalent to 1000 mL. This conversion can be represented by the equation:\[\text{volume in milliliters} = \text{volume in liters} \times 1000.\]In the step-by-step solution, once we calculate the volume of the glucose solution in liters using its moles and molarity, we use this conversion to translate that volume into milliliters, which can then easily be measured in a lab.Understanding volume conversion is essential in ensuring that the correct amount of solution is used, preventing errors in experimental work. It bridges the gap between theoretical calculations and physical measurement, making sure scientific investigations are precise and accurate.