Problem 173 - General Chemistry

Question

A student gently drops an object weighing \(15.8 \mathrm{~g}\) into an open vessel that is full of ethanol, so that a volume of ethanol spills out equal to the volume of the object. The experimenter now finds that the vessel and its contents weigh \(10.5 \mathrm{~g}\) more than the vessel full of ethanol only. The density of ethanol is \(0.789 \mathrm{~g} / \mathrm{cm}^{3}\). What is the density of the object?

Step-by-Step Solution

Answer

The density of the object is approximately 1.19 g/cm³.

Step 1: Understand the Problem

We need to find the density of the object. The volume displaced by the object equals the volume of the ethanol spilled. Therefore, we need to calculate this displaced volume and use it along with the mass of the object to determine its density.

Step 2: Calculate the Volume Displaced

The additional weight observed (10.5 g more than the ethanol alone) is due to the object and not the ethanol itself, since the volume of ethanol displaced equals the object's volume. The weight of the ethanol displaced must also be equivalent to the additional weight observed. Therefore, the volume displaced which is also the volume of the object can be calculated using the density formula:\[ \text{Volume of the object} = \frac{\text{mass of ethanol displaced}}{\text{density of ethanol}} = \frac{10.5 \, \text{g}}{0.789 \, \text{g/cm}^3} \approx 13.31 \, \text{cm}^3 \]

Step 3: Calculate the Density of the Object

Density is defined as mass per unit volume. Using the mass of the object and the volume displaced, the density \( \rho \) of the object is:\[ \rho = \frac{\text{mass of object}}{\text{volume of object}} = \frac{15.8 \, \text{g}}{13.31 \, \text{cm}^3} \approx 1.19 \, \text{g/cm}^3 \]

Step 4: Conclusion

The density of the object is found by dividing its mass by the volume it displaced. This confirms the calculation is accurate and verifies through the relation between displacement and density.

Key Concepts

Density of Solids

Density is a fundamental concept in physics and chemistry, especially when it comes to understanding the properties of solids. It is defined as the amount of mass contained within a given volume. In simple terms, it tells us how tightly matter is packed together in a solid. To calculate the density (\( \rho \)) of a solid, use the formula: \( \rho = \frac{\text{mass}}{\text{volume}} \).
For example, consider a solid object with a mass of 15.8 grams and a volume of 13.31 cubic centimeters. By substituting these values into the density formula, we find its density to be approximately 1.19 grams per cubic centimeter.
Density is an intrinsic property, which means it does not depend on the quantity of material. This concept helps in identifying substances and understanding their physical characteristics.

Mass and Volume Relationship

The relationship between mass and volume is crucial in determining the density of substances. Mass refers to the quantity of matter in an object, while volume measures the space an object occupies. When these two are proportioned in a formula for density, we can determine whether an object is dense or less dense.
For solids and liquids, this relationship is direct and crucial for density calculation. Typically, increasing the mass while keeping the volume constant will result in a higher density, indicating a denser material. Conversely, if the volume increases while mass remains constant, the resulting lower density implies a lighter or less packed matter.
Mastering the interplay between mass and volume is essential for experiments in science and industry, especially when precise material properties are required.

Displacement Method

The displacement method is a reliable technique for measuring the volume of irregularly shaped objects. This method relies on the principle that an object submerged in a fluid will displace a volume of the fluid equal to its own volume.
To use this method, you simply submerge the object in a fluid, such as ethanol in this exercise, in a graduated vessel filled to the brim. Measure the fluid that spills over or the change in fluid level, and you'll have the volume of the object.

• It's easy and practical, especially without complex equipment.
• Perfect for solids that don't dissolve in the chosen fluid.

In our example, an object's volume was measured by noticing an additional 10.5 grams of ethanol overflow, which was calculated to correspond to a volume of 13.31 cubic centimeters. The displacement method is especially useful in laboratories and classrooms for students to visualize and calculate volume practically.

Density of Ethanol

Ethanol, a common liquid in chemistry and industry, has a specific density that is crucial for various applications. Its density is approximately 0.789 grams per cubic centimeter. Knowing this property helps when ethanol is used as a solvent or reagent.
The density of ethanol, as with other substances, is essential in processes like separation or purification because it affects how ethanol interacts with other materials. In the given exercise, ethanol's density helped determine the volume of a displaced liquid and, subsequently, the cut object's density calculation.
Being aware of the density of ethanol is particularly important in laboratory settings where precision and accuracy are vital for successful experiments. Hence, learning and applying these density values in real-world contexts is invaluable for students and professionals alike.