Problem 1 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

\(0.5850 \mathrm{g}\) of NaCl were dissolved in \(100.0 \mathrm{cm}^{3}\) of water. \(10.0 \mathrm{cm}^{3}\) of this solution were made up with water to \(250.0 \mathrm{cm}^{3}\). Calculate the concentration of the resulting solution in: (a) moldm \(^{-3}\) (b) \(\left.\mathrm{molm}^{-3} ;(\mathrm{c}) \mathrm{mg} \mathrm{dm}^{-3} ;(\mathrm{d}) \text { ppm (by mass). (Section } 11.1\right)\)

Step-by-Step Solution

Answer

(a) 0.0040008 mol/dm³; (b) 4.0008 mol/m³; (c) 233.748 mg/dm³; (d) 233.748 ppm.

Step 1: Calculate Moles of NaCl

First, calculate the number of moles of NaCl dissolved. The molar mass of NaCl is approximately 58.44 g/mol. Use the formula: \( \text{moles} = \frac{\text{mass}}{\text{molar mass}} \). \[\text{moles of NaCl} = \frac{0.5850 \text{ g}}{58.44 \text{ g/mol}} \approx 0.010002 \text{ mol}\]

Step 2: Calculate Initial Concentration

Calculate the concentration of the initial solution (in mol/dm³) using the volume of the solution, which is 100 cm³ or 0.100 dm³.\[\text{concentration} = \frac{\text{moles}}{\text{volume (dm}^3\text{)}} = \frac{0.010002}{0.100} = 0.10002 \text{ mol/dm}^3\]

Step 3: Dilute the Solution

The initial solution is further diluted. 10.0 cm³ is taken and made up to 250.0 cm³. Calculate the new concentration using \( \text{C}_1\text{V}_1 = \text{C}_2\text{V}_2 \) with the initial volume \( V_1 = 0.010 \text{ dm}^3 \) and final volume \( V_2 = 0.250 \text{ dm}^3 \).\[\text{C}_2 = \frac{\text{C}_1 \times V_1}{V_2} = \frac{0.10002 \times 0.010}{0.250} = 0.0040008 \text{ mol/dm}^3\]

Step 4: Convert Concentration to mol/m³

Convert the concentration from mol/dm³ to mol/m³. We know that 1 dm³ = 0.001 m³.\[0.0040008 \text{ mol/dm}^3 = 0.0040008 \times 1000 \text{ mol/m}^3 = 4.0008 \text{ mol/m}^3\]

Step 5: Convert Concentration to mg/dm³

Convert the molar concentration to mg/dm³ using the molar mass of NaCl (58.44 g/mol). First convert moles to grams, then to mg.\[4.0008 \times 58.44 \text{ g/m}^3 = 233.748 \text{ g/m}^3 = 233748 \text{ mg/m}^3\]And since 1 m³ = 1000 dm³:\[233.748 \text{ mg/dm}^3\]

Step 6: Calculate Concentration in ppm

The concentration in mg/dm³ is equivalent to ppm when the solvent is water, as density is approximately 1 g/cm³. Therefore, concentration in ppm is:\[\text{ppm} = 233.748\]

Key Concepts

Molarity Calculation

Molarity is a way of expressing the concentration of a solute in a solution. It is defined as the number of moles of solute per liter (dm³) of solution. To calculate molarity, you need two key values: the moles of solute and the volume of the solution in liters.

• **Moles of Solute**: Use the formula \( \text{moles} = \frac{\text{mass}}{\text{molar mass}} \). For NaCl, with a molar mass of 58.44 g/mol, this helps us determine the moles of the salt dissolved. For instance, if 0.5850 g of NaCl dissolve, then the moles of NaCl amount to approximately 0.010002.
• **Volume of Solution**: Ensure the solution's volume is in liters (dm³), as molarity is expressed in mol/dm³. In the original problem, 100 cm³ equals 0.100 dm³.

Calculating molarity involves dividing the moles of solute by the volume in liters. Thus, \( \text{molarity} = \frac{0.010002}{0.100} = 0.10002 \text{ mol/dm}^3 \). This illustrates the concentration before any dilution.

Dilution Process

Dilution is the process of reducing the concentration of a solute in a solution, usually by adding more solvent. The relationship between the concentrations and volumes of a solution before and after dilution is represented by the formula: \( C_1 V_1 = C_2 V_2 \).
In this equation, \( C_1 \) and \( V_1 \) are the initial concentration and volume of the solution, while \( C_2 \) and \( V_2 \) are the concentration and volume after dilution.
**Steps in Dilution**:

• Start with the initial concentration and volume. For example from the given problem, if 10.0 cm³ (0.010 dm³) of a 0.10002 mol/dm³ solution is diluted to 250.0 cm³ (0.250 dm³), you need to find the new concentration.
• Apply the formula: \( C_2 = \frac{C_1 \times V_1}{V_2} \).
• Substitute the values: \( C_2 = \frac{0.10002 \times 0.010}{0.250} = 0.0040008 \text{ mol/dm}^3 \). This is the concentration after dilution.

This demonstrates how the concentration decreases proportionally to the increase in volume.

Unit Conversion

Unit conversion is crucial when working with different measurements of concentration. Often, you'll need to switch between units such as mol/dm³ and mol/m³, or between mass and volume units.
**Converting mol/dm³ to mol/m³**:

• Since 1 dm³ equals 0.001 m³, you can convert molarity to mol/m³ by multiplying by 1000, because there are 1000 dm³ in a m³.
• For example, 0.0040008 mol/dm³ is equivalent to 4.0008 mol/m³.

**Converting to mg/dm³ and ppm**:

• To convert moles to mass (specifically to mg), multiply by the molar mass, then adjust for the desired unit. As seen, multiplying by the molar mass of NaCl (58.44 g/mol) and adjusting for m³ leads to 233748 mg/m³.
• Choosing mg/dm³, 233.748 mg/dm³ directly translates to 233.748 ppm, because the volume of water is the same and its density is approximately 1 g/cm³.

Understanding these conversions ensures that you accurately switch between various units of concentration.