Question

Calculate Samples \(A, B,\) and \(C\) have masses of \(80 g, 12 g,\) and 33 g, and volumes of \(20 \mathrm{mL}, 4 \mathrm{cm}^{3},\) and 11 \(\mathrm{mL}\) , respectively. Which of the samples have the same density?

Short answer

The densities of samples A, B, and C are 4 g/mL, 3 g/mL, and 3 g/mL, respectively. Therefore, samples B and C have the same density of 3 g/mL.

Step-by-step solution

Calculate the density of sample A

Using the given mass of 80g and volume of 20mL for sample A, the density can be calculated as follows: Density of A = mass / volume = \( \frac{80g}{20mL} \) Density of A = 4 g/mL

Calculate the density of sample B

Using the given mass of 12g and volume of 4 cm³ for sample B, first, we need to convert the volume into mL, since 1 cm³ is equal to 1 mL, the conversion is straightforward: 4 cm³ = 4 mL Now we can calculate the density of sample B: Density of B = mass / volume = \( \frac{12g}{4mL} \) Density of B = 3 g/mL

Calculate the density of sample C

Using the given mass of 33g and volume of 11mL for sample C, the density can be calculated as follows: Density of C = mass / volume = \( \frac{33g}{11mL} \) Density of C = 3 g/mL

Compare densities to determine which samples have the same density

We have calculated the densities of all three samples: Sample A: 4 g/mL Sample B: 3 g/mL Sample C: 3 g/mL Samples B and C have the same density of 3 g/mL.

Key concepts

Mass-to-Volume Ratio

In the realm of physics and chemistry, the mass-to-volume ratio, commonly known as density, plays a pivotal role in identifying substances and understanding their physical properties. Density is measured as the amount of mass contained in a given volume. When we calculate density, we take the mass of an object or substance and divide it by its volume. This is a universal method for determining density and appears as a straightforward formula:
\texttt{Density (D) = \(\frac{Mass (m)}{Volume (V)}\)}

For example, in our exercise with samples A, B, and C, we determined each sample's density by using this formula. Nonetheless, to effectively employ this formula, it's essential to correctly understand the units used for both mass and volume. It's a common pitfall to mix units that aren't compatible, leading to inaccurate results. In this exercise, the mass was given in grams, while volume was provided in milliliters or cubic centimeters. It's crucial to remember that for fluids, 1 milliliter (1 mL) equals 1 cubic centimeter (1 cm³), which simplifies the conversion process for this exercise. The density, then, is expressed in grams per milliliter (g/mL). Understanding this relationship facilitates performing measures and comparisons accurately.

Units Conversion

Seamless conversion between different measurement units is essential for accuracy in scientific computations. In the exercise presented, understanding that 1 milliliter (1 mL) is equal to 1 cubic centimeter (1 cm³) permitted us to convert the volumes of the samples into a single unit, ensuring the densities could be compared fairly. This equality is specific to the milliliter and cubic centimeter due to the nature of water's density.When not dealing with mL and cm³, converting units usually requires multiplication or division by a conversion factor. Conversion factors are predetermined numerical values that translate a measurement in one unit to its equivalent in another unit. These factors are crucial when working with units of vastly different magnitudes, like converting kilometers to meters or pounds to kilograms. Keeping in mind the correct factors, and ensuring consistent units across substances you're comparing, is a must for precise density calculations.

Comparing Densities

The practice of comparing densities is a standard procedure to find out if substances are identical, how they will interact, or to determine their purity. For example, if two objects have different densities, one will float on the other if combined in a gravity-based system. This principle underlies many separation techniques in chemistry and is also a key factor in the design of objects that must float, such as boats or submarines.In the given exercise, after calculating the densities for samples A, B, and C, we compared them to identify which ones shared the same density. With the results being 4 g/mL for sample A and 3 g/mL for both samples B and C, we determined that samples B and C have matching densities. It's paramount to ensure all densities are calculated and expressed using the same units before comparing them, as inconsistencies here can lead to incorrect conclusions. Comparing densities is also an effective method to verify experimental results against known or theoretical values, offering a way to assess the precision and accuracy of lab work.