Problem 105 - Chemistry

Question

Vinaigrette salad dressing consists mainly of oil and vinegar. The density of olive oil is \(0.918 \mathrm{~g} / \mathrm{cm}^{3}\), the density of vinegar is \(1.006 \mathrm{~g} / \mathrm{cm}^{3}\), and the two do not mix. If a certain mixture of olive oil and vinegar has a total mass of \(397.8 \mathrm{~g}\) and a total volume of \(422.8 \mathrm{~cm}^{3}\), what is the volume of oil and what is the volume of vinegar in the mixture?

Step-by-Step Solution

Answer

The volume of olive oil is approximately 314 cm³, and the volume of vinegar is approximately 108.8 cm³.

Step 1: Define Variables

Let the volume of olive oil be denoted as \( V_o \) and the volume of vinegar be \( V_v \). We have two primary equations in this problem: the sum of the volumes and the sum of the masses of these substances.

Step 2: Set Up Volume Equation

Given that the total volume of the mixture is the sum of the individual volumes, we write: \[ V_o + V_v = 422.8 \text{ cm}^3 \]

Step 3: Use Densities to Set Up Mass Equation

Using the known densities, we calculate the mass of each component:- The mass of olive oil: \( m_o = 0.918 \times V_o \)- The mass of vinegar: \( m_v = 1.006 \times V_v \)The total mass equation is: \[ 0.918V_o + 1.006V_v = 397.8 \text{ g} \]

Step 4: Solve the System of Equations

We have the system of equations:\[ V_o + V_v = 422.8 \]\[ 0.918V_o + 1.006V_v = 397.8 \]First, solve the first equation for \( V_v \):\[ V_v = 422.8 - V_o \]Substitute \( V_v \) in the second equation:\[ 0.918V_o + 1.006(422.8 - V_o) = 397.8 \]This simplifies to:\[ 0.918V_o + 425.4568 - 1.006V_o = 397.8 \]Combine like terms:\[ -0.088V_o = -27.6568 \]Solve for \( V_o \):\[ V_o \approx 314 \text{ cm}^3 \]

Step 5: Calculate Vinegar Volume

Substitute \( V_o \) back into the equation for \( V_v \):\[ V_v = 422.8 - 314 \]\[ V_v \approx 108.8 \text{ cm}^3 \]

Key Concepts

volume of oil

The volume of oil in a mixture can be determined through the relationship between its mass and density. In this problem, olive oil has a density of \(0.918 \, \text{g/cm}^3\). Using density as the bridge, you can calculate the mass of the oil if the volume is known, and vice versa.
Density is a fundamental concept that relates how much mass is contained in a given volume. When dealing with fluids like oil, it is often assumed to spread evenly, allowing us to compute its volume by understanding its density and total mass.
To find the volume of oil, use the formula:

• **Mass**: How heavy the oil is (in grams).
• **Density**: Given as \(0.918 \, \text{g/cm}^3\).
• **Volume Calculation**: Look to solve for the volume \(V_o\) using equations derived from these parameters.

From the system of equations given, the calculation for the volume of oil \(V_o\) comes to approximately \(314 \, \text{cm}^3\). This is determined by using the sequential approach where these equations are solved stepwise.

volume of vinegar

The volume of vinegar in a mixture can also be calculated by using its density and understanding the total mass constraints. Here, vinegar has a density of \(1.006 \, \text{g/cm}^3\).
When computing the volume of vinegar, the density plays a key role much like how it did with the oil. It defines how concentrated vinegar is in terms of mass per unit volume. Since vinegar is heavier per cubic centimeter compared to oil, given its higher density, this aspect is important for such calculations.
To deduce the vinegar's volume:

• **Total Volume Constraint**: The sum of oil and vinegar volumes is \(422.8 \, \text{cm}^3\).
• **Density Factor**: Using \(1.006 \, \text{g/cm}^3\) to find the mass of vinegar if needed.
• **Equation Utilization**: Plug the solved volume of oil into the total volume equation to determine vinegar's volume \(V_v\).

After finding the value of \(V_o\), substituting it back gives \(V_v\) approximately as \(108.8 \, \text{cm}^3\).

system of equations

Systems of equations are a powerful tool easy enough to handle multiple unknowns at once, typically in problems like this where two conditions must be simultaneously satisfied. Here, the goal was to resolve the volumes of oil and vinegar with known mass totals.
In this scenario, each equation carried unique information:

• **Volume Equation**: This simply knows that the combined volume \(V_o + V_v = 422.8\.\text{cm}^3\).
• **Mass Equation**: Incorporates the densities and states \(0.918 V_o + 1.006 V_v = 397.8 \, \text{g}\).

Solving these involves substitution or elimination methods. Begin solving by simplifying one equation, substituting guessed or calculated variables, and logically proceeding from known to unknowns.
The crux is to manipulate one equation to isolate one variable, substitute back into another equation, thereby unraveling each quantity step-by-step. This structured approach allows solving for both volumes effectively.