Problem 27 - Chemistry: The Central Science

Question

(a) A sample of carbon tetrachloride, a liquid once used in dry cleaning, has a mass of \(39.73 \mathrm{~g}\) and a volume of \(25.0 \mathrm{mLat} 25^{\circ} \mathrm{C}\). What is its density at this temperature? Will carbon tetrachloride float on water? (Materials that are less dense than water will float.) (b) The density of platinum is \(21.45 \mathrm{~g} / \mathrm{cm}^{3}\) at \(20^{\circ} \mathrm{C}\). Calculate the mass of \(75.00 \mathrm{~cm}^{3}\) of platinum at this temperature. (c) The density of magnesium is \(1.738 \mathrm{~g} / \mathrm{cm}^{3}\) at \(20^{\circ} \mathrm{C}\). What is the volume of \(87.50 \mathrm{~g}\) of this metal at this temperature?

Step-by-Step Solution

Answer

\(a) Density = 1.589 \mathrm{~g/cm^3}\), which is greater than water's density, so carbon tetrachloride will not float on water. \(b) Mass = 1608.75 \mathrm{~g}\) of platinum. \(c) Volume = 50.32 \mathrm{~cm^3}\) of magnesium.

Step 1: Convert mL to cm³

Since \(1 \mathrm{~mL}\) equals \(1 \mathrm{~cm}^{3}\), we can directly use the given volume: \(25.0 \mathrm{~mL}\) = \(25.0 \mathrm{~cm^3}\). ##Step 2: Calculate density## Density of carbon tetrachloride

Step 2: Calculate density

Using the formula \(Density = \frac{Mass}{Volume}\), we have: \(Density = \frac{39.73 \mathrm{~g}}{25.0 \mathrm{~cm^3}} \) #Step 3: Compare density to water##

Step 3: Compare density to water

Since water has a density of \(1 \mathrm{~g} / \mathrm{cm}^{3}\), if the density of carbon tetrachloride is less than this, it will float on water. Otherwise, it will sink. #Part (b): Calculate the mass of 75.00 cm³ of platinum# ##Step 1: Rearrange density formula##

Step 4: Rearrange the density formula

We need to find the mass, so we rearrange the formula: \(Mass = Density \times Volume\) ##Step 2: Calculate the mass##

Step 5: Calculate the mass of platinum

Now we can calculate the mass of \(75.00 \mathrm{~cm}^{3}\) of platinum: \(Mass = 21.45 \mathrm{~g/cm^3} \times 75.00 \mathrm{~cm^3}\) #Part (c): Find the volume of 87.50 g of magnesium# ##Step 1: Rearrange the density formula##

Step 6: Rearrange the density formula

This time we need to find the volume, so we rearrange the formula again: \(Volume = \frac{Mass}{Density}\) ##Step 2: Calculate the volume##

Step 7: Calculate the volume of magnesium

Now we can calculate the volume of \(87.50 \mathrm{~g}\) of magnesium: \(Volume = \frac{87.50 \mathrm{~g}}{1.738 \mathrm{~g/cm^3}}\)

Key Concepts

Mass-to-Volume Ratio

When you dive into a pool, have you noticed how some objects sink while others float? This is all due to a property called density, which is essentially the mass-to-volume ratio of a substance. Let's simplify: imagine you have two boxes of the same size, one filled with feathers and the other with bricks. The bricks have a greater mass in the same volume compared to the feathers, hence a higher density.

The density, denoted as mass divided by volume \( \frac{mass}{volume} \), is a critical concept in chemistry and everyday life. Understanding the mass-to-volume ratio can help you predict whether a substance will float in another (like oil on water), or how to convert recipes from volume to weight (such as cups of flour to grams).

In our carbon tetrachloride example, we find its density by dividing the mass (\(39.73 \text{g}\)) by the volume (\(25.0 \text{cm}^3\)). Such calculations are crucial in determining the material's behavior in different environments, like whether it will sink or float in water.

Buoyancy of Substances

Buoyancy is the force that enables ships to float and balloons to rise in the air. It depends on the densities of two interacting substances. Archimedes' principle states that an object submerged in a fluid is buoyed up by a force equal to the weight of the displaced fluid. This means if our substance is less dense than the fluid it's placed in, it'll float.

Take our carbon tetrachloride. It has a density greater than water, which is \(1 \text{g/cm}^3\). This tells us that carbon tetrachloride is more 'packed' with particles than water and thus, it will sink. Contrastingly, something like a piece of wood, generally less dense than water, will float. Knowing whether a substance will float or sink is valuable information not just for scientific experiments but also for real-world applications like boat construction or oil spill clean-ups.

Converting Units in Chemistry

In chemistry, as in cooking, using the correct measurements is crucial. Sometimes, this means converting units to make sense of a problem. Our exercise involves converting milliliters (mL) to cubic centimeters (cm³). Fortunately, they're equivalent: \(1 \text{mL} = 1 \text{cm}^3\text{.}\) This kind of unit conversion is straightforward, but some can be more complex, involving multiple steps and conversion factors.

For example, if you have a volume in liters and you need it in cubic meters, you’d need to know that \(1 \text{m}^3 = 1,000 \text{L}\). Chemists often deal with such conversions, ensuring substances are measured accurately for reactions to occur correctly. Always double-check your units when performing density calculations or any chemistry problem.

Rearranging Density Formula

In mathematical terms, density is usually expressed as \( \rho = \frac{m}{V} \), with \( \rho \) representing density, \(m\) the mass, and \(V\) the volume. But what if we need to find mass or volume, not density? We can rearrange the formula! To find mass, we multiply density by volume (\( m = \rho \times V \)), and for volume, we divide mass by density (\( V = \frac{m}{\rho} \)).

For the platinum in our exercise, we rearrange the formula to calculate mass, given the density and volume. Conversely, to find the volume of magnesium with a known mass and density, we flip the formula. These rearrangements are powerful tools, allowing chemists to calculate various parameters as long as two are known. Mastery over manipulating this formula is a fundamental skill for students and professionals alike in chemistry.