Question

The density of osmium (the densest metal) is $22.57\mathrm{g}/{\mathrm{cm}}^3.$ If a $1.00-\mathrm{kg}$ rectangular block of osmium has two dimensions of $4.00\mathrm{cm}\times 4.00\mathrm{cm}$, calculate the third dimension of the block.

Short answer

The third dimension (height) of the 1.00-kg rectangular block of osmium with dimensions of $4.00\mathrm{cm}\times 4.00\mathrm{cm}$ is 2.79 cm.

Step-by-step solution

Write down the given information.

We are given the following information: Density of osmium, $\rho =22.57$ g/cm³. Mass of osmium block, m = 1.00 kg (We need to convert this to grams). Dimensions of osmium block (Length × Width × Height): $4.00\mathrm{cm}\times 4.00\mathrm{cm}\times h$. (Note: The third dimension (height) is what we need to calculate.)

Convert the mass into grams.

Given that 1 kg = 1000 g, we can convert the mass of the osmium block into grams: Mass, m = 1.00 kg × (1000 g/kg) = 1000 g

Write down the formula relating mass, density, and volume.

The formula we will use is: mass = density × volume. In this case, mass (m) = 1000 g, density (ρ) = 22.57 g/cm³, and volume (V) = Length × Width × Height.

Calculate the volume of the osmium block.

Using the mass and density values, we can calculate the volume of the osmium block. Rearrange the density formula to solve for volume: Volume, V = mass / density = m / ρ Now, plug in the values for mass and density: V = 1000 g / 22.57 g/cm³ = 44.31 cm³

Calculate the third dimension (height).

Now, we know the volume (44.31 cm³) and two dimensions (4.00 cm and 4.00 cm) of the block. We can find the third dimension (height) by dividing the volume by the product of the known dimensions: Height, h = V / (Length × Width) Plugging in the values, we get: h = 44.31 cm³ / (4.00 cm × 4.00 cm) = 2.79 cm

State the final answer.

The third dimension (height) of the 1.00-kg rectangular block of osmium with dimensions of $4.00\mathrm{cm}\times 4.00\mathrm{cm}$ is 2.79 cm.

Key concepts

Mass to Volume Ratio

Understanding the 'mass to volume ratio' is crucial in executing various scientific calculations, especially when dealing with substances’ physical properties. This ratio, most commonly referred to as density, is a measure of how much mass is contained within a unit volume of a material. It can be intuitively understood as how tightly packed the matter within a substance is.

For example, in the given exercise, we deal with osmium, which is known as the densest metal. Its mass to volume ratio is given as 22.57 g/cm³. This number tells us that for every cubic centimeter of osmium, there is 22.57 grams of mass. Knowing this ratio enables us to determine one of the most enigmatic properties: the actual volume that a certain mass of a substance will occupy, which is foundational in solving problems related to material properties.

Let's put this into practice. With the osmium example, if we have a 1.00 kg block, we can find out how much space it occupies through the density value. Using the formula for volume (V = m/ρ), where m is mass and ρ (rho) is density, we can calculate the volume in cubic centimeters for the given mass. This part of the calculation is central since the mass to volume ratio is directly linked to the concept of density.

Converting Mass Units

In many scientific problems, 'converting mass units' from one system to another is an essential skill, as it allows for the consistent use of formulas that require specific units. Conversions are necessary because measurements can be recorded in different units, and using them incorrectly could lead to errors in calculations and results.

For example, in the exercise, the mass of the osmium block is provided in kilograms (kg), but the density is given in terms of grams per cubic centimeter (g/cm³). It’s important to convert the mass into grams to use the density formula effectively because the units must match for proper computation. This translates to multiplying the mass by 1000 because one kilogram equals 1000 grams.

In this context, after converting 1.00 kg into 1000 g, we align the units with those of density, thus enabling us to apply the formula for density without hitches. Mastery of unit conversion is a fundamental skill in not just chemistry or physics, but across all sciences.

Density Formula Application

Applying the 'density formula' is at the core of many problems in physics and chemistry. The formula is simply density (ρ) = mass (m) / volume (V). It is used to find any of the three variables when the other two are known. Applying this formula correctly is vital for both theoretical understanding and practical experiments.

In our problem, we use the formula to find the unknown dimension of a block of osmium. With the mass and density known, we rearrange the formula to find the volume (V = m/ρ). After determining the volume, if the material is of a regular shape, like the rectangular block in our problem, we can then calculate any missing dimension by manipulating the formula further. We divide the volume by the product of the known dimensions to tease out the unknown third dimension.

The iterative application of the density formula is a powerful tool, allowing students to navigate complex problems by breaking them down into more manageable steps. This particular application brings in a sense of accomplishment as students use it not only to comprehend theoretical concepts but also to tackle everyday practical questions.