Alloy Density via Rule of Mixtures
Calculates the theoretical specific gravity / density of a multi-metal alloy based on the mass fractions and elemental densities of each constituent component.
\[ \frac{1}{\rho_{alloy}} = \sum_{i=1}^{n} \frac{w_i}{\rho_i} = \frac{w_{Au}}{\rho_{Au}} + \frac{w_{Ag}}{\rho_{Ag}} + \frac{w_{Cu}}{\rho_{Cu}} \]
Plain Text Formula:
1 / rho_alloy = (w_Au / rho_Au) + (w_Ag / rho_Ag) + (w_Cu / rho_Cu)
📐 Variable Definitions & Parameters
| Symbol | Variable Name | Standard Unit | Physical Definition |
|---|---|---|---|
| \[ \rho_{alloy} \] | Theoretical Alloy Density | g/cm³ | Composite density of the mixture |
| \[ w_i \] | Mass Fraction of Component i | decimal | Weight percentage divided by 100 (sum equals 1.0) |
| \[ \rho_i \] | Density of Component i | g/cm³ | Elemental density: Au=19.32, Ag=10.49, Cu=8.96 g/cm³ |
💡 Worked Engineering Example
Problem Statement:
Find the density of standard 14K yellow gold consisting of 58.3% Au, 20.8% Ag, and 20.9% Cu.
Step-by-Step Solution:
- Volume Au = 0.583 / 19.32 = 0.03017 cm³/g
- Volume Ag = 0.208 / 10.49 = 0.01983 cm³/g
- Volume Cu = 0.209 / 8.96 = 0.02333 cm³/g
- Total specific volume = 0.07333 cm³/g
- Density = 1 / 0.07333 = 13.64 g/cm³
🎯 Result: Theoretical density of 14K yellow gold is 13.64 g/cm³.