Problem 102 - Chemistry: An Atoms-Focused Approach

Question

Gasoline and water are immiscible. Regular-grade (87 octane) gasoline has a lower density \((0.73 \mathrm{g} / \mathrm{mL})\) than water \((1.00 \mathrm{g} / \mathrm{mL}) .\) A \(100-\mathrm{mL}\) graduated cylinder with an inside diameter of \(3.2 \mathrm{cm}\) contains \(34.0 \mathrm{g}\) of gasoline and \(34.0 \mathrm{g}\) of water. What is the combined height of the two liquid layers in the cylinder? The volume of a cylinder is \(\pi r^{2} b,\) where \(r\) is the radius and \(b\) is the height.

Step-by-Step Solution

Answer

Answer: The combined height of the two liquid layers in the graduated cylinder is 10.03 cm.

Step 1: Calculate the volume of gasoline and water

First, we need to calculate the volume of both gasoline and water using their densities and masses. We can use the formula: Volume = mass/density For gasoline: Volume\(_{gasoline}\) = \(\frac{34.0\mathrm{g}}{0.73\mathrm{g/mL}}\) = \(46.58\,\mathrm{mL}\) For water: Volume\(_{water}\) = \(\frac{34.0\mathrm{g}}{1.00\mathrm{g/mL}}\) = \(34.0\,\mathrm{mL}\)

Step 2: Calculate the individual heights of gasoline and water layers

Next, we will use the cylinder volume formula and the given diameter to calculate the heights of the gasoline and water layers. The formula for the volume of a cylinder is: Volume = \(\pi r^2 b\) The radius of the cylinder can be found by dividing the given diameter (3.2 cm) by 2: \(r = \frac{3.2\,\mathrm{cm}}{2}\) = \(1.6\,\mathrm{cm}\) Now, we can solve for the height of the gasoline and water layers: For gasoline: \(b_{gasoline}\) = \(\frac{Volume_{gasoline}}{\pi r^2}\) = \(\frac{46.58\,\mathrm{mL}}{\pi (1.6\,\mathrm{cm})^2}\) = \(5.80\,\mathrm{cm}\) For water: \(b_{water}\) = \(\frac{Volume_{water}}{\pi r^2}\) = \(\frac{34.0\,\mathrm{mL}}{\pi (1.6\,\mathrm{cm})^2}\) = \(4.23\,\mathrm{cm}\)

Step 3: Calculate the combined height of both layers

Finally, to find the combined height of both the gasoline and water layers, we simply add the two individual heights: Combined height = \(b_{gasoline}\) + \(b_{water}\) = \(5.80\,\mathrm{cm} + 4.23\,\mathrm{cm} = 10.03\,\mathrm{cm}\) Thus, the combined height of the two liquid layers in the graduated cylinder is \(10.03\,\mathrm{cm}\).

Key Concepts

Volume calculation

When dealing with liquids, understanding how to calculate their volume is crucial, especially if you need to determine how much space they occupy in a container. Calculating the volume of a liquid given its mass and density is straightforward and relies on the formula: \[\text{Volume} = \frac{\text{mass}}{\text{density}}\]To find the volume, you simply divide the mass of the liquid by its density. For example, in our exercise, we calculated the volume of gasoline and water. The density of gasoline is lower than that of water, which directly affects the volume calculation. Since both liquids were given a mass of 34.0 g, using their respective densities allowed us to find their individual volumes:

• Gasoline: \(\text{Volume}_{\text{gasoline}} = \frac{34.0\,\text{g}}{0.73\,\text{g/mL}} = 46.58\,\text{mL}\)
• Water: \(\text{Volume}_{\text{water}} = \frac{34.0\,\text{g}}{1.00\,\text{g/mL}} = 34.0\,\text{mL}\)

This method is very useful when you don't have direct access to the measuring tools but only need to know the substance’s mass and density.

Immiscible liquids

Immiscible liquids are those that do not mix with each other upon combining, staying as separate layers due to differences in density. A classic example would be oil and water, or in this case, gasoline and water. These substances do not form a homogeneous mixture. Instead, each liquid maintains its separate identity. In any container where immiscible liquids are combined, the denser liquid will sink to the bottom. In our given exercise, the gasoline and water in the cylinder stay as two distinct layers. The water, being denser, settles beneath the gasoline layer. When calculating the total height of the liquid column, these inherent properties of immiscible liquids ensure that each liquid's height is considered independently before summing them for a total height. This property of liquid layers helps in understanding and handling mixtures in both chemical applications and everyday life.

Cylinder volume formula

The mathematical formula for the volume of a cylinder is essential when dealing with problems involving cylindrical containers. The formula is expressed as:\[\text{Volume} = \pi r^2 b\]where \(r\) is the radius of the base of the cylinder and \(b\) is the height of the cylinder.In practice, this formula helps determine how much a cylinder can hold. By rearranging the formula, you can find unknown parameters such as height when the volume and radius are known. In the exercise, we calculated the height of each liquid in the cylinder by rearranging the formula to solve for \(b\):\[b = \frac{\text{Volume}}{\pi r^2}\]Given the cylinder's diameter, the radius was calculated by dividing the diameter by two, leading to a radius of 1.6 cm. We used this to find the heights of both gasoline and water separately:

• Gasoline: \(b_{\text{gasoline}} = \frac{46.58\,\text{mL}}{\pi (1.6\,\text{cm})^2} = 5.80\,\text{cm}\)
• Water: \(b_{\text{water}} = \frac{34.0\,\text{mL}}{\pi (1.6\,\text{cm})^2} = 4.23\,\text{cm}\)

Adding these heights provided the combined height of the liquid layers, demonstrating the practical application of the cylinder volume formula.