Problem 36 - Chemistry and Chemical Reactivity

Question

You can identify a metal by carefully determining its density (d). An unknown piece of metal, with a mass of \(2.361 \mathrm{g},\) is \(2.35 \mathrm{cm}\) long, \(1.34 \mathrm{cm}\) wide, and \(1.05 \mathrm{mm}\) thick. Which of the following is this element? (a) Nickel, \(d=8.90 \mathrm{g} / \mathrm{cm}^{3}\) (b) Titanium, \(d=4.50 \mathrm{g} / \mathrm{cm}^{3}\) (c) Zinc, \(d=7.13 \mathrm{g} / \mathrm{cm}^{3}\) (d) \(\operatorname{tin}, d=7.23 \mathrm{g} / \mathrm{cm}^{3}\)

Step-by-Step Solution

Answer

The metal is Zinc.

Step 1: Understand the Formula for Density

Density is calculated using the formula \( d = \frac{m}{V} \), where \( d \) is the density, \( m \) is the mass, and \( V \) is the volume of the object.

Step 2: Convert Units

Ensure that all measurements are in centimeters. The thickness is given as \(1.05 \mathrm{mm}\), which should be converted to centimeters by dividing by 10, resulting in \(0.105 \mathrm{cm}\).

Step 3: Calculate the Volume

The volume \( V \) of the rectangular solid can be found by multiplying its dimensions: \( V = \text{length} \times \text{width} \times \text{thickness} = 2.35 \mathrm{cm} \times 1.34 \mathrm{cm} \times 0.105 \mathrm{cm} \). Calculating this gives \( V = 0.330045 \mathrm{cm}^3 \).

Step 4: Calculate the Density

Now calculate the density of the metal using the measurements provided. Substitute the known values into the density formula: \( d = \frac{2.361 \mathrm{g}}{0.330045 \mathrm{cm}^{3}} \). This results in a density of approximately \( 7.15 \mathrm{g} / \mathrm{cm}^{3} \).

Step 5: Determine the Metal

Compare the calculated density \( 7.15 \mathrm{g} / \mathrm{cm}^{3} \) to the given densities of metals. Zinc has a density of \( 7.13 \mathrm{g} / \mathrm{cm}^{3} \), which is closest to the calculated density.

Key Concepts

Volume Calculation

When determining the volume of an object, especially in the context of determining density, it's important to know the shape of the object.
A common formula used for volume calculation is based on the dimensions of the object. For rectangular shapes, the formula is\[ V = ext{length} \times ext{width} \times ext{thickness} \]
To calculate the volume of the given metal piece, input its dimensions: length = 2.35 cm, width = 1.34 cm, and thickness = 0.105 cm.

• First, ensure that all measurements are in consistent units. Here, all dimensions are converted to centimeters before calculation.
• Substitute the values into the volume formula:
  \[ V = 2.35 \times 1.34 \times 0.105 \]

After performing the multiplication, we get a volume approximately equal to 0.330045 cm³.

Mass Measurement

Mass measurement is a crucial step in density calculation. The mass of the object determines how much matter exists within it.
In this exercise, the mass of the unknown metal piece is given as 2.361 grams.

• When measuring mass, always ensure that the value is accurate and double-check with a calibrated scale.
• The given mass should be in grams to match the common unit of density, which is grams per cubic centimeter (g/cm³).

Having accurate mass measurements helps in calculating the object's density accurately.

Unit Conversion

Unit conversion is often necessary in physics and chemistry to ensure all values are compatible. In the exercise, the thickness was initially given in millimeters.
To convert this to centimeters, remember that 1 cm equals 10 mm. Therefore, if a measurement is given in millimeters, divide by 10 to convert to centimeters:

• Given thickness: 1.05 mm
• Converted thickness: 1.05 mm / 10 = 0.105 cm

This step is crucial because you want all dimensions in your volume calculation to be in the same unit—centimeters, in this case—to ensure precision in further calculations like determining density.

Identifying Metals

Identifying an unknown metal requires comparing its calculated density against the known densities of possible candidates.
In this case, after calculating the density, you should have a value that you can match to a list of known metal densities.

• Calculate density using the formula: \[ d = \frac{m}{V} \]
  where \( m = 2.361 \ g \) and \( V = 0.330045 \ cm^3 \).
• Perform the division to find the density:
  \[ d = \frac{2.361}{0.330045} \approx 7.15 \ g/cm^3 \]
• Compare this result to the provided densities:
  • Nickel: 8.90 g/cm³
  • Titanium: 4.50 g/cm³
  • Zinc: 7.13 g/cm³
  • Tin: 7.23 g/cm³
• The calculated density is closest to zinc, which has a density of approximately 7.13 g/cm³.

This systematic approach helps you correctly identify the metal based on its physical properties.