The density of gold metal is 19.3 g/cm 3 , while that for silver is 10.5 g/cm 3 . Suppose you had the same size bar of each metal, with the dimensions 2.00 in x 10.0 cm x 760 mm. Volume of a cube is height x length x width.
Which bar would have the greatest mass?
Which bar contains the greater number of atoms?
If the two bars were melted and mixed together to form an alloy (or mixture) what would be the total mass?
The correct answer and explanation is :
To determine which metal bar—gold or silver—has the greatest mass, contains the greater number of atoms, and the total mass if both were melted together, we can follow these steps:
1. Calculate the Volume of Each Bar:
Given the dimensions:
- Length: 2.00 inches
- Width: 10.0 cm
- Height: 760 mm
First, convert all measurements to centimeters:
- Length: 2.00 inches × 2.54 cm/inch = 5.08 cm
- Width: 10.0 cm (already in cm)
- Height: 760 mm ÷ 10 = 76.0 cm
Volume = Length × Width × Height
= 5.08 cm × 10.0 cm × 76.0 cm
= 3860.8 cm³
2. Determine the Mass of Each Bar:
Using the densities provided:
- Density of gold (Au): 19.3 g/cm³
- Density of silver (Ag): 10.5 g/cm³
Mass = Density × Volume
For gold:
Mass_Au = 19.3 g/cm³ × 3860.8 cm³
= 74513.44 g
For silver:
Mass_Ag = 10.5 g/cm³ × 3860.8 cm³
= 40538.4 g
3. Identify Which Bar Has the Greatest Mass:
The gold bar has a mass of 74,513.44 grams, while the silver bar has a mass of 40,538.4 grams. Therefore, the gold bar has the greater mass.
4. Calculate the Number of Atoms in Each Bar:
To find the number of atoms, we use the formula:
Number of atoms = (Mass of the sample / Molar mass) × Avogadro’s number
- Molar mass of gold (Au): approximately 196.97 g/mol
- Molar mass of silver (Ag): approximately 107.87 g/mol
- Avogadro’s number: 6.022 × 10²³ atoms/mol
For gold:
Number of atoms_Au = (74513.44 g / 196.97 g/mol) × 6.022 × 10²³ atoms/mol
≈ 2.278 × 10²⁶ atoms
For silver:
Number of atoms_Ag = (40538.4 g / 107.87 g/mol) × 6.022 × 10²³ atoms/mol
≈ 2.263 × 10²⁶ atoms
5. Determine Which Bar Contains the Greater Number of Atoms:
The gold bar contains approximately 2.278 × 10²⁶ atoms, while the silver bar contains approximately 2.263 × 10²⁶ atoms. Thus, the gold bar contains a slightly greater number of atoms.
6. Calculate the Total Mass if Both Bars Were Melted Together:
Total mass = Mass_Au + Mass_Ag
= 74,513.44 g + 40,538.4 g
= 115,051.84 g
Explanation:
The mass of an object is determined by its volume and density. Given that both bars have the same volume, the metal with the higher density will have a greater mass. Gold’s density is 19.3 g/cm³, significantly higher than silver’s 10.5 g/cm³, resulting in the gold bar having a greater mass.
The number of atoms in a given mass depends on the molar mass of the element. Gold has a higher molar mass (196.97 g/mol) compared to silver (107.87 g/mol). However, due to the higher mass of the gold bar, the total number of atoms in the gold bar is slightly greater than in the silver bar.
When combining both bars into an alloy, the total mass is simply the sum of the individual masses, resulting in 115,051.84 grams.