Problem 77 - Chemistry: The Central Science in SI Units, Global Edition

Question

What is the osmotic pressure formed by dissolving \(50.0 \mathrm{mg}\) of acetylsalicylic acid \(\left(\mathrm{C}_{9} \mathrm{H}_{8} \mathrm{O}_{4}\right)\) in \(0.100 \mathrm{~L}\) of water at \(37^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

The osmotic pressure formed by dissolving 50.0 mg of acetylsalicylic acid in 0.100 L of water at 37°C is approximately \(7.08 × 10⁻²\) atm.

Step 1: Calculate the molar mass of acetylsalicylic acid

To calculate the molar mass, we sum up the molar masses of all the atoms in acetylsalicylic acid (\(\mathrm{C}_{9} \mathrm{H}_{8} \mathrm{O}_{4}\)). Molar mass of C: 12.01 g/mol Molar mass of H: 1.01 g/mol Molar mass of O: 16.00 g/mol Molar mass of acetylsalicylic acid = 9(12.01) + 8(1.01) + 4(16.00) = 180.16 g/mol

Step 2: Determine the number of moles in the 50.0 mg sample

First, convert 50.0 mg to grams: 50.0 mg × 1 g/1000 mg = 0.0500 g Now, use the molar mass to find the number of moles: Number of moles = mass / molar mass = 0.0500 g / 180.16 g/mol = 2.77 × 10⁻⁴ mol

Step 3: Calculate the molarity of the solution

Molarity (M) = number of moles / volume of solution in liters M = 2.77 × 10⁻⁴ mol / 0.100 L = 2.77 × 10⁻³ mol/L

Step 4: Convert temperature from Celsius to Kelvin

To convert temperature from Celsius to Kelvin, we add 273.15 to the Celsius temperature: T(K) = 37°C + 273.15 = 310.15 K

Step 5: Calculate the osmotic pressure

Use the osmotic pressure formula, remembering that van't Hoff factor (i) is 1 for non-electrolyte substances like acetylsalicylic acid: \( \Pi = iMRT \) Plug in the values: (M = 2.77 × 10⁻³ mol/L, R = 0.0821 L atm/mol K, and T = 310.15 K) \( \Pi = 1 × (2.77 × 10⁻³ \: \text{mol/L}) × (0.0821 \: \text{L atm/mol K}) × (310.15 \: \text{K}) \) \( \Pi = 7.08 × 10⁻² \: \text{atm} \) The osmotic pressure formed by dissolving 50.0 mg of acetylsalicylic acid in 0.100 L of water at 37°C is approximately 7.08 × 10⁻² atm.

Key Concepts

Molar Mass Calculation

When calculating the molar mass of a compound such as acetylsalicylic acid, it is crucial to understand the atomic composition of the substance. Each element in the molecule contributes to the total molar mass. For acetylsalicylic acid ( C₉H₈O₄), we calculate it as follows:

• Carbon (C): 9 atoms × 12.01 g/mol = 108.09 g/mol
• Hydrogen (H): 8 atoms × 1.01 g/mol = 8.08 g/mol
• Oxygen (O): 4 atoms × 16.00 g/mol = 64.00 g/mol

Adding these values together gives a total molar mass of 180.16 g/mol. This basic calculation is essential for converting between mass and moles, which is a common need in chemistry.
By breaking down the molecular formula into individual atoms and using their molar masses, you can calculate the molar mass of any substance.

Molarity Determination

Molarity, an important concept in chemistry, refers to the concentration of a solute in a solution. It is expressed as the number of moles of solute per liter of solution (mol/L). Calculating molarity involves knowing both the amount of solute and the total volume of the solution.
In our example:

• Number of moles of acetylsalicylic acid: After converting the mass from mg to g, the number of moles is calculated using the formula: \[ \text{Number of moles} = \frac{\text{mass in grams}}{\text{molar mass}} \]
• Volume of solution: Given in liters, in this case, it is 0.100 L.
  
• Molarity (M): Calculated using: \[ M = \frac{\text{number of moles}}{\text{volume in liters}} \]

This gives us a molarity of 2.77 × 10⁻³ mol/L. Accurately determining molarity is essential for predicting the behavior of substances in solutions.

Temperature Conversion

Temperature conversion is often necessary when dealing with scientific formulas that require absolute temperature values, usually in Kelvin. Converting Celsius to Kelvin is straightforward. The Celsius scale needs to be adjusted by adding 273.15 to align with the absolute scale of Kelvin, which begins at absolute zero.
In this example, we convert 37°C to Kelvin as follows:

• Formula: \[ T(\text{K}) = T(\text{°C}) + 273.15 \]

Applying this conversion: 37°C + 273.15 = 310.15 K
Using the Kelvin temperature is crucial when applying it to calculations, such as those involving the gas constant in physical chemistry.

Osmotic Pressure Formula

Osmotic pressure is the pressure required to prevent the flow of solvent into a solution through a semipermeable membrane. It is a fundamental concept in solutions, particularly biology and chemistry. The osmotic pressure ( \( \Pi \)) can be calculated using van't Hoff's formula:

• Formula: \[ \Pi = iMRT \]
• Van't Hoff factor (i): Represents the number of particles the solute forms in solution. For acetylsalicylic acid, it is 1 because it doesn't dissociate.
• Molarity (M): As calculated, 2.77 × 10⁻³ mol/L.
• Gas Constant (R): 0.0821 L atm/mol K, a universal constant used for ideal gas calculations.
• Temperature (T): The absolute temperature in Kelvin (310.15 K).

Substituting the values into the formula, we find:\[ \Pi = 1 \times (2.77 \times 10^{-3} \text{ mol/L}) \times (0.0821 \text{ L atm/mol K}) \times (310.15 \text{ K}) \]
This results in an osmotic pressure of approximately 7.08 × 10⁻² atm, demonstrating how solute concentration and temperature affect it.