Question

A cube of metal weighs \(1.45 \mathrm{kg}\) and displaces \(542 \mathrm{mL}\) of water when immersed. Calculate the density of the metal.

Short answer

The density of the metal is approximately \(2673.8 \mathrm{kg/m^3}\).

Step-by-step solution

Convert the given values to the appropriate units

To calculate the density, we need the mass in kilograms (kg) and the volume in cubic meters (m³). We already have the mass of the cube (1.45 kg), and we need to convert the volume of water displaced from milliliters (mL) to cubic meters (m³). To convert milliliters to cubic meters, we can use the following conversion formula: 1 milliliter = 1.0 x 10^(-6) cubic meters So, in this case: Volume (m³) = 542 mL * (1.0 x 10^(-6) m³/mL)

Calculate the Volume in cubic meters

Now, let's calculate the volume of the cube in cubic meters using the given volume conversion: Volume (m³) = 542 * (1.0 x 10^(-6)) Volume (m³) ≈ 0.000542 m³

Calculate the Density of the Metal

Now that we have the volume of the cube, we can use the density formula to find the density of the metal: Density = Mass / Volume Density = 1.45 kg / 0.000542 m³

Solve for Density

Let's solve for the density of the cube: Density ≈ 2673.8 kg/m³ The density of the metal is approximately 2673.8 kg/m³.

Key concepts

Mass and Volume Conversion

When dealing with problems involving density, converting units of mass and volume is crucial for obtaining accurate results. In our example, the mass of the metal cube is already given in kilograms, which is a standard unit for mass in scientific calculations. However, the volume needs conversion from milliliters (mL) to cubic meters (m³) because density is typically expressed in kilograms per cubic meter (kg/m³).

To convert milliliters to cubic meters, use the relationship that 1 mL equals \(1.0 \times 10^{-6}\) m³. This is because "milli" implies a thousandth, so a milliliter is a thousandth of a liter, or one-millionth of a cubic meter. In our specific case, 542 mL is converted to m³ by multiplying by \(1.0 \times 10^{-6}\):

• Volume in m³ = 542 mL \( \times 1.0 \times 10^{-6} \) m³/mL
• Resulting in 0.000542 m³ for the volume of the cube

Density Formula

Density is a critical physical property of matter, which helps identify substances and understand their behavior in different environments. The general formula for density is given by:

\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]

This formula shows the relationship between the mass of an object and the volume it occupies.

In this exercise, we have a metal cube with a mass of 1.45 kg and a converted volume of 0.000542 m³. To find the density of the metal, we simply divide its mass by its volume using the density formula above. This makes solving for density straightforward and efficient.

Using the given values, we perform the calculation:

• Density = \((1.45 \text{ kg}) / (0.000542 \text{ m³})\)
• Thus obtaining a density of approximately 2673.8 kg/m³

Unit Conversion

Unit conversion is an essential skill, particularly in science and engineering, where different measurement systems are often used.

In our density calculation task, we had to convert from milliliters to cubic meters to ensure compatibility with the density formula's standard form (kg/m³).

Typically you may encounter different units depending on the context:

• Milliliters (mL) to liters, where 1000 mL = 1 L
• Liters to cubic meters, where 1 L = 0.001 m³
• Grams to kilograms, where 1000 g = 1 kg

Each conversion involves multiplying by specific factors, which correspond to the relationship between units. Understanding these relationships is vital for precise scientific work.

Metal Density Calculation

Calculating the density of metals is often performed in various fields, including metallurgy, engineering, and physics. It helps in identifying an unknown metal or verifying its purity.

In this particular example, we calculated the density of a cube of metal using its mass and volume.

• First, ensure correct unit conversions to match the formula requirements.
• Use the density formula \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \) for calculations.
• Apply the corresponding values of mass and volume.

This systematic approach gave us the precise density measurement of approximately 2673.8 kg/m³.

By comparing this density with standard density values for known metals, one can identify the metal or check for consistency in alloy production.