A magnesium–lead alloy of mass 7.5 kg consists of a solid α phase that has a composition just slightly below the solubility limit at 300?C (570?F).
(a) What mass of lead is in the alloy?
(b) If the alloy is heated to 400?C (750?F), how much more lead may be dissolved in the α phase without exceeding the solubility limit of this phase?
The Correct Answer and Explanation is :
To solve this problem, we need to break it down into two parts, as the question asks about the composition of a magnesium-lead alloy at different temperatures.
Part (a): Mass of Lead in the Alloy at 300°C
At 300°C (570°F), the alloy is a solid α phase that is just slightly below the solubility limit. The solubility limit of lead in the α phase of magnesium can be found in phase diagrams, which indicate the amount of lead (in percentage) that can dissolve in magnesium at a specific temperature.
From phase diagrams of magnesium-lead alloys, the solubility limit of lead in magnesium at 300°C is about 2.5% by weight. This means that at 300°C, the maximum percentage of lead that can be dissolved in the α phase is 2.5%.
Given that the total mass of the alloy is 7.5 kg, the mass of lead in the alloy can be calculated using the percentage composition:
[
\text{Mass of Lead} = \text{Total mass of alloy} \times \frac{\text{Lead percentage}}{100}
]
[
\text{Mass of Lead} = 7.5 \, \text{kg} \times \frac{2.5}{100} = 0.1875 \, \text{kg}
]
So, 0.1875 kg of lead is present in the alloy at 300°C.
Part (b): Additional Lead Dissolved at 400°C
At 400°C (750°F), the solubility limit of lead in magnesium increases. According to the phase diagram for the magnesium-lead system, the solubility limit of lead in magnesium at 400°C is approximately 5% by weight. This means that at 400°C, the α phase can dissolve up to 5% lead.
Now, we need to calculate how much additional lead can be dissolved in the α phase without exceeding the solubility limit:
[
\text{Maximum mass of lead at 400°C} = 7.5 \, \text{kg} \times \frac{5}{100} = 0.375 \, \text{kg}
]
The additional lead that can be dissolved is the difference between the maximum lead content at 400°C and the lead content at 300°C:
[
\text{Additional lead} = \text{Maximum mass of lead at 400°C} – \text{Mass of lead at 300°C}
]
[
\text{Additional lead} = 0.375 \, \text{kg} – 0.1875 \, \text{kg} = 0.1875 \, \text{kg}
]
Thus, 0.1875 kg of additional lead can be dissolved in the alloy when it is heated to 400°C without exceeding the solubility limit.
Summary:
- (a) The mass of lead in the alloy at 300°C is 0.1875 kg.
- (b) The additional mass of lead that can be dissolved in the α phase at 400°C is 0.1875 kg.