Problem 29 - Chemistry An Atoms First Approach

Question

Calculate the molarity of each of these solutions. a. A 5.623 -g sample of NaHCO, is dissolved in enough water to make \(250.0 \mathrm{mL}\) of solution. b. A \(184.6-\mathrm{mg}\) sample of \(\mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7}\) is dissolved in enough water to make \(500.0 \mathrm{mL}\) of solution. c. A \(0.1025-\mathrm{g}\) sample of copper metal is dissolved in \(35 \mathrm{mL}\) of concentrated HNO_to form \(\mathrm{Cu}^{2+}\) ions and then water is added to make a total volume of \(200.0 \mathrm{mL}\). (Calculate the molarity of \(\mathrm{Cu}^{2+}\).)

Step-by-Step Solution

Answer

The molarities of the given solutions are as follows: a. The molarity of NaHCO3 solution is \(0.268 M\) b. The molarity of K2Cr2O7 solution is \(0.00126 M\) c. The molarity of Cu2+ ions in the solution is \(0.00808 M\)

Step 1: a. Calculate moles of NaHCO3

To convert the mass of NaHCO3 to moles, we first need to find the molar mass of NaHCO3. The molar mass of NaHCO3 is: 23 (Na) + 1 (H) + 12 (C) + 16x3 (O) = 84 g/mol Now we can convert 5.623 g of NaHCO3 to moles: 5.623 g / 84 g/mol = 0.067 moles of NaHCO3

Step 2: a. Convert volume to liters

The volume of the solution is given in milliliters (mL). We need to convert it to liters (L): 250.0 mL × (1 L / 1000 mL) = 0.250 L

Step 3: a. Calculate molarity

Now we can calculate the molarity using the formula M = moles of solute/liters of solution: M = 0.067 moles NaHCO3 / 0.250 L = 0.268 M

Step 4: b. Calculate moles of K2Cr2O7

To convert the mass of K2Cr2O7 to moles, we first need to find the molar mass of K2Cr2O7. The molar mass of K2Cr2O7 is: (2x39) (K) + (2x52) (Cr) + (7x16) (O) = 294 g/mol Now we can convert 184.6 mg of K2Cr2O7 to moles: 184.6 mg × (1 g / 1000 mg)= 0.1846 g 0.1846 g / 294 g/mol = 0.000628 moles of K2Cr2O7

Step 5: b. Convert volume to liters

The volume of the solution is given in milliliters (mL). We need to convert it to liters (L): 500.0 mL × (1 L / 1000 mL) = 0.500 L

Step 6: b. Calculate molarity

Now we can calculate the molarity using the formula M = moles of solute/liters of solution: M = 0.000628 moles K2Cr2O7 / 0.500 L = 0.00126 M

Step 7: c. Calculate moles of Cu

To convert the mass of Cu to moles, we first need to find the molar mass of Cu. The molar mass of Cu is: 63.5 g/mol Now we can convert 0.1025 g of Cu to moles: 0.1025 g / 63.5 g/mol = 0.001615 moles of Cu

Step 8: c. Convert volume to liters

The volume of the solution after adding water is given in milliliters (mL). We need to convert it to liters (L): 200.0 mL × (1 L / 1000 mL) = 0.200 L

Step 9: c. Calculate molarity

Now we can calculate the molarity of Cu2+ ions using the formula M = moles of solute/liters of solution: M = 0.001615 moles Cu / 0.200 L = 0.00808 M In summary: a. The molarity of NaHCO3 solution is 0.268 M b. The molarity of K2Cr2O7 solution is 0.00126 M c. The molarity of Cu2+ ions in the solution is 0.00808 M

Key Concepts

Moles of Solute

To find the molarity of a solution, you must first determine the number of moles of solute, which is the substance dissolved in the solvent.
This is done by using the formula: \[ \text{Moles of solute} = \frac{\text{Mass of solute (g)}}{\text{Molar mass of solute (g/mol)}} \]For instance, in example (a), we calculate the moles of NaHCO₃ by first finding its molar mass:

• Sodium (Na): 23 g/mol
• Hydrogen (H): 1 g/mol
• Carbon (C): 12 g/mol
• Oxygen (O): 16 g/mol × 3 = 48 g/mol

The total molar mass of NaHCO₃ is 84 g/mol.
So, the moles of NaHCO₃ are calculated as:\[\frac{5.623 \text{ g}}{84 \text{ g/mol}} = 0.067 \text{ moles} \]This process is essential for calculating molarity, as it connects mass with molecular quantities.

Volume Conversion to Liters

Before calculating molarity, it is crucial to convert the volume of the solution from milliliters (mL) to liters (L). This is because molarity is expressed in terms of moles per liter.
The conversion is straightforward: To convert from mL to L, divide by 1000:\[\text{Volume (L)} = \frac{\text{Volume (mL)}}{1000}\]For instance, if the volume is 250.0 mL (as seen in example (a)), you would convert it to liters as follows:\[\frac{250.0 \text{ mL}}{1000} = 0.250 \text{ L}\]Converting the volume ensures you use consistent units throughout the molarity calculation, which prevents errors in your final answer.

Molar Mass Calculation

Calculating the molar mass is an essential step in determining the moles of a substance.
Molar mass is calculated by adding the atomic masses of all the atoms in a molecule.
It allows us to translate the mass of a substance to moles, bridging the gap between the macroscopic and microscopic worlds.Consider example (b), where we calculate the molar mass of \(\text{K}_2\text{Cr}_2\text{O}_7\):

• Potassium (K): 39 g/mol × 2 = 78 g/mol
• Chromium (Cr): 52 g/mol × 2 = 104 g/mol
• Oxygen (O): 16 g/mol × 7 = 112 g/mol

The total molar mass is 294 g/mol.
With this value, you can easily convert any given mass of \(\text{K}_2\text{Cr}_2\text{O}_7\)to moles.

Cu2+ Ion Concentration

When calculating the concentration of ions, such as \(\text{Cu}^{2+}\) ions, it is important to recognize that the process includes the dissolution and conversion steps.In example (c), you first determine the moles of copper (Cu) using its molar mass of 63.5 g/mol:\[\frac{0.1025 \text{ g}}{63.5 \text{ g/mol}} = 0.001615 \text{ moles of Cu}\]Copper metal is then dissolved to form \(\text{Cu}^{2+}\) ions, and water is added to make the final solution.
The total volume of this solution is 200.0 mL, which you must convert to liters:\[\frac{200.0 \text{ mL}}{1000} = 0.200 \text{ L}\]The molarity, or concentration, of \(\text{Cu}^{2+}\) ions is then calculated:\[\frac{0.001615 \text{ moles}}{0.200 \text{ L}} = 0.00808 \text{ M}\]Understanding this conversion is key in determining the concentration of ions in a solution, reflecting the steps from metal dissolution to ion presence.