Problem 36 - General Chemistry: Principles and Modern Applications

Question

How much (a) ethanol, \(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}(d=0.789 \mathrm{g} / \mathrm{mL}),\) in liters, must be dissolved in water to produce \(200.0 \mathrm{L}\) of 1.65 \(\mathrm{M} \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH} ?\) (b) concentrated hydrochloric acid solution \((36.0 \%\) HCl by mass; \(d=1.18 \mathrm{g} / \mathrm{mL}),\) in milliliters, is required to produce 12.0 L of 0.234 M HCl?

Step-by-Step Solution

Answer

a) 19.3 liters of Ethanol are required. b) 241.3 milliliters of HCl are required.

Step 1: Calculation for part a)

First, calculate the number of moles of ethanol in 200.0 L solution of 1.65 M ethanol using the formula: moles = Molarity * Volume(L). Substitute the given values and get \(moles = 1.65 mol/L * 200.0 L = 330.0 mol\) of ethanol. Then, convert this to grams using the molecular weight of \(C_2H_5OH = 46.07 g/mol\), by multiplying the number of moles with the molar mass, \(mass = 330.0 mol * 46.07 g/mol = 15202.1 g\). This is the mass of ethanol required.

Step 2: Convert the mass to volume for part a)

Then, convert the mass of ethanol to volume (in liters) using the density given, \(d = 0.789 g/mL\). So, \(volume = mass / density = 15202.1 g / 0.789 g/mL = 19274.9 mL = 19.3 L (to 2 decimal places)\). This is the volume of ethanol that must be dissolved in water.

Step 3: Calculation for part b)

First, calculate the number of moles in 12.0 L solution of 0.234 M HCl using the formula: moles = molarity * volume(L). Substitute these values, and get \(moles = 0.234 mol/L * 12.0 L = 2.808 mol\) of HCl. Since the concentrated HCl solution is 36% by mass, it means 100g of the solution contains 36g of HCl. Its molar mass is 36.5 g/mol, and therefore, the moles of HCl in 100g of solution is \(36.0 g / 36.5 g/mol = 0.986 mol\). Therefore, 1g of the solution contains \(0.986 mol /100 = 0.00986 mol\).

Step 4: Calculation of volume for part b)

To get the grams of the solution that will provide the required moles of HCl, divide the number of moles by moles per gram: \(grams = 2.808 mol / 0.00986 mol/g = 284.7 g\). Convert this mass to volume using the given density \(d = 1.18 g/mL\), so \(volume = mass/density = 284.7g/1.18 g/mL = 241.3 mL\). Hence, that's the volume required of the concentrated HCl solution.

Key Concepts

Molarity Calculations

Molarity, denoted as "M," is a way to express concentration in chemistry. It describes the number of moles of a solute per liter of solution. Calculating molarity is essential for solution preparation in various applications.
For example, in the problem: we needed to find out how many moles of ethanol were in a 200 L solution of 1.65 M. Here's the formula you would use:

• Moles of solute = Molarity × Volume (L)

Using this, we find that \(1.65 \, \text{mol/L} \times 200 \, \text{L} = 330 \, \text{mol}.\)
This molarity calculation helps determine how much of the substance you have in your solution.

Density and Volume Conversion

Density is a property that relates the mass of a substance to its volume. It's usually expressed in grams per milliliter (g/mL). Converting between mass and volume using density is crucial for many chemical calculations.
In the exercise, we calculated the volume of ethanol from its mass. First, we converted moles to mass using the molar mass, then used density for the conversion to volume:

• Mass = Moles × Molar Mass
• Volume = Mass / Density

For ethanol, given the density of 0.789 g/mL, this conversion helps you find that you need 19.3 liters of ethanol by dividing the mass by the density. Understanding this process is key to solving similar problems.

Moles and Molar Mass

Molar mass is the mass of one mole of a given substance, typically expressed in grams per mole (g/mol). It's a critical factor for converting between grams and moles in chemistry.

• Moles = Mass / Molar Mass

For ethanol, with a molar mass of 46.07 g/mol, the calculation helps you understand how many grams are needed for a specific number of moles. This connection between moles, molar mass, and grams is at the heart of many chemical calculations.

Concentration of Solutions

Concentration defines how much solute is present in a given volume of solution. It can be expressed in different ways, but molarity is one of the most common methods.

• Molarity (M) = Moles of Solute / Volume of Solution (L)

In part b of the exercise, you need to consider both the molarity and the percentage concentration by mass. For a solution that's 36% HCl by mass, understanding how to use this information alongside molarity helps determine the volume of solution required. This dual understanding of concentration is vital for creating solutions in the lab.