Calculate the theoretical density of Iron and Aluminum using the structure, edge length, and atomic weight. 1b) Compare the theoretical density of each metal with the experimental density that you calculated from the mass and volume (Iron: 7.9 g/cm³ and Aluminum: 2.7 g/cm³). Remember that when comparing numbers, it is best to use percent difference. Provide a sentence to explain why these densities differ.
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The Correct Answer and Explanation is:
The Correct Answer and Explanation is:
) Theoretical Density Calculation
The theoretical density (ρ\rhoρ) of a metal is calculated using the formula:ρ=n⋅AVc⋅NA\rho = \frac{n \cdot A}{V_c \cdot N_A}ρ=Vc⋅NAn⋅A
Where:
- nnn = number of atoms per unit cell
- AAA = atomic weight (g/mol)
- VcV_cVc = volume of the unit cell (cm³)
- NAN_ANA = Avogadro’s number (6.022×1023 atoms/mol)(6.022 \times 10^{23} \, \text{atoms/mol})(6.022×1023atoms/mol)
For Iron (Fe)
- Structure: Body-Centered Cubic (BCC)
- Edge length (a): 2.866 Å = 2.866×10−82.866 \times 10^{-8}2.866×10−8 cm
- nnn = 2 atoms per unit cell
- Atomic weight = 55.85 g/mol
Vc=a3=(2.866×10−8)3=2.35×10−23 cm3V_c = a^3 = (2.866 \times 10^{-8})^3 = 2.35 \times 10^{-23} \, \text{cm}^3Vc=a3=(2.866×10−8)3=2.35×10−23cm3ρ=2⋅55.852.35×10−23⋅6.022×1023=111.71.415×101≈7.89 g/cm3\rho = \frac{2 \cdot 55.85}{2.35 \times 10^{-23} \cdot 6.022 \times 10^{23}} = \frac{111.7}{1.415 \times 10^1} \approx 7.89 \, \text{g/cm}^3ρ=2.35×10−23⋅6.022×10232⋅55.85=1.415×101111.7≈7.89g/cm3
For Aluminum (Al)
- Structure: Face-Centered Cubic (FCC)
- Edge length (a): 4.049 Å = 4.049×10−84.049 \times 10^{-8}4.049×10−8 cm
- nnn = 4 atoms per unit cell
- Atomic weight = 26.98 g/mol
Vc=a3=(4.049×10−8)3=6.64×10−23 cm3V_c = a^3 = (4.049 \times 10^{-8})^3 = 6.64 \times 10^{-23} \, \text{cm}^3Vc=a3=(4.049×10−8)3=6.64×10−23cm3ρ=4⋅26.986.64×10−23⋅6.022×1023=107.923.996≈2.70 g/cm3\rho = \frac{4 \cdot 26.98}{6.64 \times 10^{-23} \cdot 6.022 \times 10^{23}} = \frac{107.92}{3.996} \approx 2.70 \, \text{g/cm}^3ρ=6.64×10−23⋅6.022×10234⋅26.98=3.996107.92≈2.70g/cm3
1b) Comparison with Experimental Densities
- Iron:
Theoretical = 7.89 g/cm³
Experimental = 7.9 g/cm³
Percent difference=∣7.89−7.9∣7.9×100%≈0.13%\text{Percent difference} = \frac{|7.89 – 7.9|}{7.9} \times 100\% \approx 0.13\%Percent difference=7.9∣7.89−7.9∣×100%≈0.13%
- Aluminum:
Theoretical = 2.70 g/cm³
Experimental = 2.7 g/cm³
Percent difference=∣2.70−2.7∣2.7×100%=0%\text{Percent difference} = \frac{|2.70 – 2.7|}{2.7} \times 100\% = 0\%Percent difference=2.7∣2.70−2.7∣×100%=0%
Explanation
The theoretical and experimental densities are nearly identical. Slight differences may result from defects in the metal crystal, impurities, or rounding in measured values. Real materials may contain vacancies or dislocations that lower the actual density compared to the ideal crystal used in theoretical calculations.
