Problem 68 - Chemistry Matter and Change

Question

An object with a mass of 7.5 \(\mathrm{g}\) raises the level of water in a graduated cylinder from 25.1 \(\mathrm{mL}\) to 30.1 \(\mathrm{mL}\) . What is the density of the object?

Step-by-Step Solution

Answer

The density of the object is 1500 \( \mathrm{kg/m}^3 \).

Step 1: Convert Mass to Kilograms

We are given the mass of the object as \(7.5\ \mathrm{g}\). We need to convert this to kilograms for standard SI units. \(7.5\ \mathrm{g} = 0.0075\ \mathrm{kg}\).

Step 2: Calculate the Volume of the Object

The volume change in the graduated cylinder indicates the volume of the object. We find this by subtracting the initial water level from the final water level: \(30.1\ \mathrm{mL} - 25.1\ \mathrm{mL} = 5.0\ \mathrm{mL}\). We know that \(1\ \mathrm{mL} = 1\ \mathrm{cm}^3\), so the volume of the object is \(5.0\ \mathrm{cm}^3\) or \(5.0\times10^{-6}\ \mathrm{m}^3\) (since \(1\ \mathrm{cm}^3 = 1\times10^{-6}\ \mathrm{m}^3\)).

Step 3: Calculate the Density of the Object

Density is calculated using the formula, \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). Thus, the density of the object is \( \frac{0.0075\ \mathrm{kg}}{5.0\times10^{-6}\ \mathrm{m}^3} = 1500\ \mathrm{kg/m}^3 \).

Key Concepts

Mass Conversion

The process of mass conversion is essential when dealing with scientific measurements, especially when adhering to the International System of Units (SI units). Initially, the mass of the object in our exercise is given as 7.5 grams, but in scientific and engineering applications, it is often necessary to convert this to kilograms. The conversion is straightforward: knowing that 1 kilogram equals 1000 grams, we divide the mass in grams by 1000. Therefore, 7.5 grams becomes 0.0075 kilograms.

Why do we convert to kilograms? Kilograms are part of the SI units, which standardize measurements across various sciences, ensuring consistency when performing calculations or comparing results with other scientific data. Consistent use of SI units facilitates clear communication in the scientific community.

In summary, converting mass from grams to kilograms is a simple step of dividing by 1000, allowing us to work comfortably within the SI framework.

Volume Displacement

Volume displacement is a clever method for measuring the volume of an irregular object. In our exercise, we placed an object in a graduated cylinder filled with water. The water level rose, showing the volume of space the object occupies. This change in water level is the key to determining the volume. By subtracting the initial water level from the final level, we obtain the volume displaced by the object. Here, the calculation was: 30.1 mL - 25.1 mL = 5.0 mL.

What’s nice about this method is its simplicity and reliability. The principle is based on Archimedes' principle, which states that the volume of fluid displaced is equal to the volume of the object immersed.

• Initial water level: 25.1 mL
• Final water level: 30.1 mL
• Volume of object: 5.0 mL (or 5.0 cm³)

A good practice is to ensure measurements are precise, as small errors can impact results significantly.

SI Units

The International System of Units (SI) is a globally accepted set of units for scientific measurements, ensuring that anyone, anywhere, can understand and replicate scientific results. The core units include the meter for length, kilogram for mass, and second for time. In our example, we converted mass to kilograms and worked with volume in cubic meters, which are both SI units.

Standardizing units to the SI system is crucial because it eliminates ambiguity. Everyone utilizes the same baseline for calculations, thus preventing errors that can arise from unit conversion mistakes or misunderstandings. For instance, converting centimeters cubed (cm³) to meters cubed (m³) is key in our exercise: 1 cm³ equals 1 x 10⁻⁶ m³.

SI units make calculations like density concise and accurate. Density in this context is expressed in kg/m³, providing a standard understanding of how much mass of a substance is present in a given volume. Embracing SI units is not just about following convention but ensuring precision and clarity in scientific discourse.