Question

Two students measure the density of gold. One works with a \(100-g\) bar of pure gold. The other works with a \(200-g\) bar of pure gold. Which student measures the larger density? Explain your answer.

Short answer

Both students measure the same density of 19.32 g/cm\(^3\) for their respective gold bars. Density is a property that remains constant for a pure substance regardless of mass, and since both bars are made of pure gold, their densities are equal.

Step-by-step solution

Recap the formula for density

Density is a measure of mass per unit volume of a substance. The formula for density is given by: \(Density = \frac{Mass}{Volume}\)

Given information

We are given the following information about the two students: - Student 1: working with a 100-g bar of pure gold - Student 2: working with a 200-g bar of pure gold

Calculate density for Student 1

Let's calculate the density for the 100-g gold bar. We need to know the volume of the gold bar, and to find this, we will use the known density of pure gold, which is approximately 19.32 g/cm\(^3\). We can rearrange the density formula to solve for volume: \(Volume = \frac{Mass}{Density}\) Volume of 100-g gold bar = \(\frac{100}{19.32}\) = 5.17 cm\(^3\) (approximately) Now, we can calculate the density of the 100-g gold bar: Density of 100-g gold bar = \(\frac{100}{5.17}\) = 19.32 g/cm\(^3\)

Calculate density for Student 2

Now, let's calculate the density for the 200-g gold bar. Volume of 200-g gold bar = \(\frac{200}{19.32}\) = 10.34 cm\(^3\) (approximately) Now, we can calculate the density of the 200-g gold bar: Density of 200-g gold bar = \(\frac{200}{10.34}\) = 19.32 g/cm\(^3\)

Compare the densities

Now that we have calculated the densities for both gold bars, we can compare them: Density of 100-g gold bar = 19.32 g/cm\(^3\) Density of 200-g gold bar = 19.32 g/cm\(^3\) As we can see, the density of both gold bars is the same, regardless of their mass. This means that neither student measures a larger density than the other, as they are both measuring the density of pure gold, which remains constant at 19.32 g/cm\(^3\).

Key concepts

Mass-Volume Relationship

When discussing the physical properties of a material, the mass-volume relationship is essential. It refers to the correlation between how much matter is present in an object (mass) and the amount of space that object occupies (volume). Understanding this relationship is fundamental, especially when delving into concepts like density.

This relationship is easily seen in everyday life. For instance, consider a sponge and a stone of the same size. Despite their similar volume, their mass is markedly different due to the sponge's porous nature. This comparison is a practical demonstration of how different materials have different mass to volume ratios.

When measuring these properties, it's important to use consistent units, such as grams for mass and cubic centimeters for volume, to derive meaningful results. In the context of our gold density exercise, we see that the mass of gold changes (100g versus 200g), but since gold is a material with a consistent mass-to-volume ratio, the volume changes proportionally, keeping the density constant.

Density Formula

The density formula is a concise way to describe how closely packed the particles of a substance are. It is denoted mathematically as:
\[Density = \frac{Mass}{Volume}\].

This equation is intuitive: density increases as the mass of the material increases for a given volume, or as the volume decreases for a given mass. The units of density are typically expressed as mass per unit volume, like g/cm³ or kg/m³.

In our exercise, we're examining gold, a material known for its high density, which indicates that it has a lot of mass packed into a small volume. The formula is useful both as a way to measure density directly and as a means of finding mass or volume when the other variables are known, as we did with volume in the solution steps.

Density Calculation

Calculating the density of a substance often requires taking a measured mass and volume and plugging these values into the density formula. The key to accurate density calculation is precise measurement and consistent units.

Let's take the exercise provided, where the known density of gold allows us to find the volume of each gold bar by rearranging the density formula to \(Volume = \frac{Mass}{Density}\). Once we have the volume, we can use the original formula to check our work, ensuring that the calculated density matches the known density of gold.

This reassurance step, where after finding the volume we return to calculate the density and verify the constant value, is vital. It serves as a check for calculation errors and confirms our understanding that density is an intrinsic property of a substance, unchanged by the quantity of material present.