Question

What is the density of a metal sample if a \(37.51-\mathrm{g}\) sample placed into a graduated cylinder increased the liquid level from \(50.0 \mathrm{~mL}\) to \(57.5 \mathrm{~mL} ?\)

Short answer

The density of the metal is approximately 5.0013 g/mL.

Step-by-step solution

Understand the Problem

We need to find the density of the metal sample. We have the mass of the metal and the volume of the water it displaced.

Recall the Formula for Density

Density is calculated as mass divided by volume. The formula is: \[\text{Density} = \frac{\text{Mass}}{\text{Volume}}\] where mass is in grams and volume is in milliliters.

Calculate the Volume Displacement

The volume of the metal is the difference in the water level before and after placing the metal in the graduated cylinder. This is calculated as:\[\text{Volume of metal} = 57.5 \, \text{mL} - 50.0 \, \text{mL} = 7.5 \, \text{mL}\]

Calculate the Density

Using the formula for density we have:\[\text{Density} = \frac{37.51 \, \text{g}}{7.5 \, \text{mL}}\] Compute this to find the density.

Compute the Result

Perform the calculation:\[\text{Density} = \frac{37.51}{7.5} \approx 5.0013 \, \text{g/mL}\]

Key concepts

Volume Displacement

Volume displacement is a method used to determine the volume of an irregularly shaped object, like a metal sample. This is done by measuring how much liquid is displaced when the object is submerged. In our exercise, the metal was placed into a graduated cylinder containing liquid. The initial liquid level was at 50.0 mL. After placing the metal in the liquid, the level rose to 57.5 mL.

To find the volume of the metal, simply subtract the initial liquid level from the final liquid level. Here it is calculated as follows:

• Initial liquid level: 50.0 mL
• Final liquid level: 57.5 mL
• Volume of the object: 57.5 mL - 50.0 mL = 7.5 mL

This method is highly accurate and straightforward, making it especially useful in lab settings. Knowing the volume of the object is crucial for further calculations, like computing its density.

Mass Measurement

Determining the mass of an object is fundamental to finding its density. Mass is a measure of how much matter is in an object and is typically measured using a balance. It is expressed in units of grams for such exercises. In our specific example, the mass of the metal sample is given as 37.51 grams.

Understanding how mass is measured can be important when performing your experiments, as slight inaccuracies can affect your results. Thus, always ensure the balance is calibrated before measuring and avoid external influences like a draft or vibrations that might affect the accuracy.

Density Formula

The density formula is central to understanding how tightly matter is packed in a substance. Density is the ratio of mass to volume and helps identify materials based on how compact they are. It is given by:\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]This formula requires two inputs: the mass of the object and its volume, both of which should be in compatible units - grams and milliliters, respectively, in this context.

Applying this to our exercise, with a mass of 37.51 grams and a volume displacement of 7.5 milliliters, the density is calculated as:\[ \text{Density} = \frac{37.51\, \text{g}}{7.5\, \text{mL}} \approx 5.0013\, \text{g/mL} \]This density value tells us how much mass exists per unit volume of the metal. It is an intensive property, meaning it does not depend on the amount of substance or its size, making it ideal for material identification.