The physical world is composed of matter, and to understand it, scientists use fundamental properties like mass, volume, and density. Mass is a measurement of the total amount of matter contained within an object, while volume quantifies the three-dimensional space that object occupies. Density serves as the intrinsic property linking these two measurements, describing how tightly the matter is packed. Since these three properties are related, knowing any two allows for the calculation of the third.
Understanding Density and Mass
The mathematical relationship begins with the definition of density as the ratio of an object’s mass to its volume. This relationship is expressed by the formula $D = M/V$. Mass ($M$) is typically measured in units like grams (g) or kilograms (kg). Volume ($V$) is measured in cubic units, such as cubic centimeters ($\text{cm}^3$) for solids or milliliters (mL) for liquids.
Density ($D$) is categorized as a derived unit because its measurement is formed by combining the units of mass and volume. For instance, if mass is measured in grams and volume in cubic centimeters, the resulting density unit is grams per cubic centimeter ($\text{g/cm}^3$). This base equation, $D = M/V$, establishes the starting point for any calculation involving these three properties.
Rearranging the Formula to Find Volume
Calculating volume when mass and density are known requires algebraic manipulation of the base density equation. The goal is to isolate the volume variable ($V$). Starting with $D = M/V$, the first step is to multiply both sides by $V$ to move it out of the denominator, resulting in the expression $D \times V = M$.
The final step in isolating volume is to divide both sides of the modified equation by density ($D$). This rearrangement yields the final formula: $V = M/D$. To perform this calculation, one must have the object’s mass and its known density. Density values for pure substances are often found in scientific reference tables. By substituting the known values for mass and density into the formula, the unknown volume can be determined.
Working Through a Calculation Example
Applying the derived formula, $V = M/D$, can be demonstrated using a sample of pure iron. Assume the iron sample has a mass of 156 grams ($M = 156 \text{ g}$). Iron’s density is a known property, approximately $7.8 \text{ g/cm}^3$ ($D = 7.8 \text{ g/cm}^3$). The calculation involves dividing the mass by the density, ensuring that the units are consistent across both measurements.
Substituting the values into the volume formula gives $V = 156 \text{ g} / 7.8 \text{ g/cm}^3$. The numerical calculation is $156$ divided by $7.8$, which equals $20$. The units must also be processed, which involves dividing grams (g) by grams per cubic centimeter ($\text{g/cm}^3$). The gram units cancel out, leaving only the cubic centimeter unit ($\text{cm}^3$). Therefore, the volume of the iron sample is $20 \text{ cm}^3$.