Question

Calculate the volume in milliliters for each of the following solids. (a) $1.00\mathrm{kg}$ of silicon $(d=2.33\mathrm{g}/{\mathrm{cm}}^3)$ (b) $1.00\mathrm{kg}$ of titanium $(d=4.51\mathrm{g}/{\mathrm{cm}}^3)$

Short answer

(a) 429 mL of silicon, (b) 222 mL of titanium.

Step-by-step solution

Calculate Mass in Grams

Convert the given mass from kilograms to grams. Since 1 kg = 1000 g, multiply the mass in kg by 1000 to get grams.

Apply Volume Formula

Use the formula for volume based on density: $V=\frac{m}{d}$, where $V$ is volume, $m$ is mass, and $d$ is density. Substitute the values for mass (in grams) and density (in $\mathrm{g}/{\mathrm{cm}}^3$).

Convert to Milliliters

Remember that $1{\mathrm{cm}}^3=1\mathrm{mL}$. Thus, the calculated volume in cubic centimeters is equal to the volume in milliliters.

Key concepts

Volume Formula

When tackling problems involving the calculation of volume for solids, understanding the formula is fundamental. The volume formula you primarily use in these scenarios is derived from the relationship between mass and density. This formula is expressed as:$$V=\frac{m}{d}$$In this equation, $V$ represents the volume, $m$ indicates the mass of the solid, and $d$ stands for its density. Knowing this formula allows you to calculate the space that a specific amount of a solid material occupies.
To find the volume, you divide the mass of the material by its density. This formula is straightforward yet incredibly powerful when dealing with various solid materials and their properties.

Unit Conversion

In calculations, it's often necessary to convert units to maintain consistency and accuracy. In the provided exercise, the mass is given in kilograms, but density is usually expressed with grams and centimeters. This conflict necessitates a unit conversion.
You'll start by converting kilograms to grams. This conversion is simple: multiply the mass in kilograms by 1000, as 1 kilogram equals 1000 grams. After this step, you have a suitable unit of grams to use with density values given in $\mathrm{g}/{\mathrm{cm}}^3$. Once the mass is in grams, calculations become smoother and the subsequent steps more straightforward.

Solid Materials

Solid materials, due to their fixed shapes and volumes, behave uniquely in density calculations. Each solid material has a distinct density, which plays a crucial role in determining its volume based on mass. Solid materials demonstrate consistent physical states, unlike gases or liquids, making them easier to measure for exercises like the one described.

• Silicon: Often used in technology and manufacturing, has a density of $2.33\mathrm{g}/{\mathrm{cm}}^3$. This value is important for understanding how much space a given mass of silicon will occupy.
• Titanium: Known for its strength and light weight, it has a density of $4.51\mathrm{g}/{\mathrm{cm}}^3$.

Understanding these density values aids in precisely calculating how much volume each material will occupy.

Density and Mass Relationship

The relationship between density and mass is central to understanding how to calculate volume. Density is defined as the mass per unit volume of a substance, captured by the equation $d=\frac{m}{V}$. Rearranging this formula to solve for volume gives you $V=\frac{m}{d}$, which is the key formula used in volume calculations.
Density tells you how tightly matter is packed within a substance. A higher density means mass is concentrated in a smaller volume, while a lower density indicates that the same mass is spread over a larger space.
Understanding these concepts ensures you can accurately assess how volume, mass, and density interact across different solid materials.