Chapter 14: Problem 81
How many grams of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) must be dissolved into 155 \(\mathrm{g}\) of water to create a solution with a molality of 8.20 \(\mathrm{mol} / \mathrm{kg}\) ?
Short Answer
Expert verified
To create a solution with a molality of 8.20 \(\mathrm{mol}/\mathrm{kg}\), 134.76 grams of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) must be dissolved into 155 g of water.
Step by step solution
01
Understand the concept of molality
Molality (\(m\)) is a measure of the concentration of a solute (in this case, \(\mathrm{Na}_{2} \mathrm{CO}_{3}\)) in a solution, defined as the number of moles of solute per kilogram of solvent (in this case, water). The formula for molality is:
\(m = \frac{\text{moles of solute}}{\text{mass of solvent (kg)}}\)
02
Determine the molar mass of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\)
To find the molar mass of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\), we need to add up the molar masses of all the elements in the compound:
Molar mass of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\):
= (2 × molar mass of Na) + (1 × molar mass of C) + (3 × molar mass of O)
= (2 × 22.99 g/mol) + (1 × 12.01 g/mol) + (3 × 16.00 g/mol)
= 105.98 g/mol
03
Calculate the mass of solvent in kilograms
In order to use the molality formula, we need the mass of the solvent (water) in kilograms. We are given that the mass of water is 155 g, so we just need to convert this to kg:
mass of water (kg) = 155 g × (1 kg / 1000 g) = 0.155 kg
04
Calculate the moles of solute needed for the desired molality
We are given the desired molality as 8.20 \(\mathrm{mol} / \mathrm{kg}\). Using the molality formula, we can find the required moles of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\):
moles of solute = molality × mass of solvent (kg)
= 8.20 mol/kg × 0.155 kg
= 1.2710 mol
05
Calculate the mass of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) required
Now that we have the moles of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) needed, we can multiply it by its molar mass to find the mass of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) required:
mass of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) = moles of solute × molar mass of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\)
= 1.2710 mol × 105.98 g/mol
= 134.76 g
So, 134.76 grams of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) must be dissolved into 155 g of water to create a solution with a molality of 8.20 \(\mathrm{mol} / \mathrm{kg}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Molar Mass
Molar mass is key to understanding how much of a substance you need to use in a chemical reaction or solution. It represents the mass of one mole of a substance and is usually expressed in grams per mole (g/mol). For any compound, it is calculated by adding together the atomic masses of all constituent elements. This information is often found on the periodic table.
For example, to find the molar mass of sodium carbonate (\(Na_{2}CO_{3}\)), follow these steps:
For example, to find the molar mass of sodium carbonate (\(Na_{2}CO_{3}\)), follow these steps:
- Look at the periodic table for the atomic mass of sodium (Na), carbon (C), and oxygen (O).
- The molar mass of Na is approximately 22.99 g/mol.
- Add these up: \( (2 \times 22.99\ g/mol) + (12.01\ g/mol) + (3 \times 16.00\ g/mol) = 105.98\ g/mol \).
Solute
In chemistry, a solute is a substance that is dissolved in a solvent to create a solution. The solute can be in a solid, liquid, or gas state, but it is the component in a lesser amount compared to the solvent. It is key in determining the concentration of a solution, which is often expressed in terms of molality, molarity, or percentage.
For the given problem, the solute is sodium carbonate (\(Na_{2}CO_{3}\)), which is dissolved in water to create a specific molality. Remember:
For the given problem, the solute is sodium carbonate (\(Na_{2}CO_{3}\)), which is dissolved in water to create a specific molality. Remember:
- The solute is the component that determines the solution's behavior and properties.
- To calculate how much solute you need, you use its molar mass and desired molality.
Solvent
The solvent is the medium in which the solute is dissolved to form a solution. In most cases, the solvent is present in a greater quantity than the solute. It plays a crucial role by allowing the solute to disperse and interact at a molecular level. Water is often termed the "universal solvent" because it dissolves many substances.
For this exercise, water is the solvent.
For this exercise, water is the solvent.
- The mass of the solvent affects the concentration of the solution.
- In this problem, the mass of water is given as 155 g, or 0.155 kg when converted for calculation purposes.