A magnesium-lead alloy of mass 6.4 kg consists of a solid a phase that has a composition just slightly below the solubility limit at 300°C (570°F). The magnesium-lead phase diagram is shown in Animated Figure 9.20.
(a) What mass of lead is in the alloy?
i kg
(b) If the alloy is heated to 400°C (750°F), how much more lead may be dissolved in the a phase without exceeding the solubility limit of this phase?
i kg
The Correct Answer and Explanation is :
(a) Mass of Lead in the Alloy
To determine the mass of lead in the alloy, we need to use the composition information provided. The alloy is in the solid α phase at 300°C, and the composition of the alloy is just below the solubility limit at that temperature. However, the exact composition is not directly provided in the problem, so we’ll rely on the general understanding of the phase diagram and solubility limits.
Given that the alloy is “just below” the solubility limit at 300°C, we assume the alloy consists of a mix of magnesium and lead, where the lead content is near the solubility limit for the α phase at this temperature.
From the magnesium-lead phase diagram (specifically, the α phase boundary at 300°C), we know that the solubility limit for lead in magnesium at 300°C is around 2% lead by weight.
So, we can calculate the mass of lead in the alloy using the following equation:
[
\text{Mass of lead} = \text{Total mass of alloy} \times \frac{\text{Lead content by weight}}{100}
]
Substituting in the known values:
[
\text{Mass of lead} = 6.4 \, \text{kg} \times \frac{2}{100} = 0.128 \, \text{kg}
]
Therefore, the mass of lead in the alloy is 0.128 kg.
(b) Additional Lead Dissolved at 400°C
At 400°C, the solubility limit for lead in the α phase increases, allowing more lead to dissolve in the alloy. To calculate how much more lead can be dissolved without exceeding the solubility limit, we need to know the new solubility limit of lead in the α phase at 400°C, which is typically around 10% by weight.
Using the new solubility limit, we can find the maximum mass of lead that can be present in the alloy at 400°C:
[
\text{Max mass of lead at 400°C} = 6.4 \, \text{kg} \times \frac{10}{100} = 0.64 \, \text{kg}
]
Now, we subtract the mass of lead already in the alloy (which is 0.128 kg) from the maximum amount that can be dissolved (0.64 kg):
[
\text{Additional lead that can dissolve} = 0.64 \, \text{kg} – 0.128 \, \text{kg} = 0.512 \, \text{kg}
]
Therefore, 0.512 kg of additional lead can be dissolved in the alloy at 400°C without exceeding the solubility limit.
Summary of Results:
- (a) The mass of lead in the alloy is 0.128 kg.
- (b) The additional lead that may be dissolved at 400°C is 0.512 kg.
Explanation:
The phase diagram of magnesium-lead alloys shows that the solubility limit of lead in magnesium increases as the temperature rises. At 300°C, the solubility limit is around 2%, meaning that at this temperature, the alloy contains a small amount of lead. When heated to 400°C, the solubility limit rises significantly to 10%, allowing more lead to dissolve into the solid phase. By knowing the total mass of the alloy and the solubility limits at different temperatures, we can calculate how much lead is in the alloy and how much more can be dissolved as the temperature increases.
