Question

the four pieces of metal, and uses the water displacement method to measure the volume. data table. complete the table and graph the results of mass versus volume for the metal.
metal data table with rows for metal #, mass of metal, volume of water, volume of water + metal, volume of metal
graph grid
the density of the metal using the formula provided above. then, using the table below the identity of the metal.
metal density table: chromium (7.19 g/ml), copper (8.94 g/ml), gold (19.32 g/ml), platinum (21.40 g/ml), silver (10.49 g/ml), titanium (4.50 g/ml)

Answer:

To solve for the volume of the metal, we use the water displacement method, where the volume of the metal is the difference between the volume of water + metal and the volume of water. Then, we calculate the density for each piece of metal using the formula \(
ho = \frac{m}{V}\) (where \(
ho\) is density, \(m\) is mass, and \(V\) is volume), and then determine the average density to identify the metal.

Step 1: Calculate Volume of Each Metal Piece

The formula for the volume of the metal (\(V_{\text{metal}}\)) is:
\(V_{\text{metal}} = V_{\text{water + metal}} - V_{\text{water}}\)

• Metal 1:

\(V_{\text{metal}} = 41.9\ \text{mL} - 40.2\ \text{mL} = 1.7\ \text{mL}\)

• Metal 2:

\(V_{\text{metal}} = 42.2\ \text{mL} - 39.6\ \text{mL} = 2.6\ \text{mL}\)

• Metal 3:

\(V_{\text{metal}} = 45.6\ \text{mL} - 40.0\ \text{mL} = 5.6\ \text{mL}\)

• Metal 4:

\(V_{\text{metal}} = 51.8\ \text{mL} - 39.8\ \text{mL} = 12.0\ \text{mL}\)

Step 2: Calculate Density of Each Metal Piece

Using \(
ho = \frac{m}{V}\):

• Metal 1:

\(
ho_1 = \frac{16.49\ \text{g}}{1.7\ \text{mL}} \approx 9.70\ \text{g/mL}\) (Wait, let’s recalculate: \(16.49 \div 1.7 \approx 9.70\)? Wait, no—wait, maybe a miscalculation. Wait, \(16.49 \div 1.7 \approx 9.70\)? Wait, no, let’s check again. Wait, \(1.7 \times 9.7 = 16.49\)? Yes. But let’s do all four:

• Metal 2:

\(
ho_2 = \frac{28.61\ \text{g}}{2.6\ \text{mL}} \approx 11.00\ \text{g/mL}\) (Wait, \(28.61 \div 2.6 \approx 11.00\)? \(2.6 \times 11 = 28.6\), so yes, ~11.00)

• Metal 3:

\(
ho_3 = \frac{58.25\ \text{g}}{5.6\ \text{mL}} \approx 10.40\ \text{g/mL}\) (Wait, \(5.6 \times 10.4 = 58.24\), so ~10.40)

• Metal 4:

\(
ho_4 = \frac{126.00\ \text{g}}{12.0\ \text{mL}} = 10.50\ \text{g/mL}\)

Step 3: Calculate Average Density

Average density = \(\frac{
ho_1 +
ho_2 +
ho_3 +
ho_4}{4}\)

Wait, but let’s check the calculations again for accuracy:

• Metal 1: \(16.49 \div 1.7 = 9.7\) (exact: \(16.49 \div 1.7 = 9.7\))
• Metal 2: \(28.61 \div 2.6 = 11.0038 \approx 11.00\)
• Metal 3: \(58.25 \div 5.6 = 10.4018 \approx 10.40\)
• Metal 4: \(126 \div 12 = 10.5\)

Wait, maybe I made a mistake in Metal 1. Wait, the mass is 16.49 g, volume 1.7 mL. \(16.49 / 1.7 = 9.7\) g/mL. But Metal 2: 28.61 / 2.6 = 11.00 g/mL. Metal 3: 58.25 / 5.6 ≈ 10.40 g/mL. Metal 4: 126 / 12 = 10.5 g/mL.

Wait, maybe the intended approach is to use the total mass and total volume? Let’s try that.

Total mass = \(16.49 + 28.61 + 58.25 + 126.00 = 229.35\ \text{g}\)

Total volume = \(1.7 + 2.6 + 5.6 + 12.0 = 21.9\ \text{mL}\)

Average density = \(\frac{229.35\ \text{g}}{21.9\ \text{mL}} \approx 10.47\ \text{g/mL}\)

Step 4: Identify the Metal

Looking at the density table:

• Chromium: 7.19 g/mL
• Copper: 8.94 g/mL
• Gold: 19.32 g/mL
• Platinum: 21.40 g/mL
• Silver: 10.49 g/mL
• Titanium: 4.50 g/mL

The average density (\(\approx 10.47\ \text{g/mL}\)) is very close to Silver (10.49 g/mL).

Final Table (Completed Volume Column)

Metal #	Mass of Metal (g)	Volume of Water (mL)	Volume of Water + Metal (mL)	Volume of Metal (mL)
2	28.61	39.6	42.2	2.6
3	58.25	40.0	45.6	5.6
4	126.00	39.8	51.8	12.0

Graphing Mass vs. Volume

To graph mass (y-axis) vs. volume (x-axis), plot the points:

• (1.7, 16.49)
• (2.6, 28.61)
• (5.6, 58.25)
• (12.0, 126.00)

The graph should show a linear relationship (since density is constant), with the slope equal to the density (\(\approx 10.49\ \text{g/mL}\), matching Silver).

Final Answer (Metal Identity)

The metal is most likely Silver (density ~10.49 g/mL, matching the calculated average density).

For the table, the completed "Volume of Metal" column is:
1.7 mL, 2.6 mL, 5.6 mL, 12.0 mL.

(Note: If using individual densities, Metal 4 is exactly 10.5 g/mL, Metal 3 ~10.40, Metal 2 ~11.00, Metal 1 ~9.70—suggesting possible experimental error, but the total mass/volume gives ~10.47, closest to Silver.)