For \(5.7 \mathrm{~kg}\) of a magnesium-lead alloy of composition \(50 \mathrm{wt} \% \mathrm{~Pb}-50 \mathrm{wt} \% \mathrm{Mg}\), is it possible, at equilibrium, to have \(\alpha\) and \(\mathrm{Mg}_{2} \mathrm{~Pb}\) phases with respective masses of \(5.13\) and \(0.57 \mathrm{~kg} ?\) If so, what will be the approximate temperature of the alloy? If such an alloy is not possible, then explain why.

Understanding how to calculate the mass fraction is fundamental when dealing with different phases within an alloy. A mass fraction is essentially the ratio of the mass of a particular component to the total mass of all components in the mixture. In this context, our alloy is comprised of two phases, Mg (magnesium) and Pb (lead), both with distinct mass fractions corresponding to different phases present at equilibrium.

To calculate the mass fraction, you simply divide the mass of the specific phase by the total mass of the alloy. For example, the mass fraction of phase \( \alpha \) is calculated as \( mf_{\alpha} = \frac{5.13 \thinspace kg}{5.7 \thinspace kg} \) which equates to 0.90 or 90%. Similarly, the mass fraction for the \( Mg_2Pb \) phase is \( mf_{Mg2Pb} = \frac{0.57 \thinspace kg}{5.7 \thinspace kg} \) which is 0.10 or 10%.

Key points to remember about mass fraction calculation include:

• Mass fraction values are always between 0 and 1 (or 0% to 100% when expressed as a percentage).
• The sum of all mass fractions in a system must equal 1 (100%) as it represents the entire mass of the system.
• Mass fractions are used to check if the mass balance is correct as shown by the equality \( mf_{\alpha} + mf_{Mg2Pb} = 1.00 \) in our alloy example.

Through understanding mass fraction, students can ascertain the distribution of phases in an alloy and how this influences its properties and potential equilibrium states.

In the field of materials science, specifically when analyzing alloys, the \( Mg_2Pb \) phase refers to an intermetallic compound within a magnesium-lead alloy. This compound is significant due to its crystalline structure and stability at certain temperatures and compositions within the binary system of Mg and Pb.

The creation of the \( Mg_2Pb \) phase occurs when the two metals are combined in an alloy and reach equilibrium under the right thermal conditions. For our problem, we confirmed the possibility of this intermetallic phase coexisting with the \( \alpha \) phase by evaluating the mass fractions. The presence and quantity of the \( Mg_2Pb \) phase can alter properties like the alloy's electrical conductivity, melting point, and mechanical strength.

Considering the mass of \( Mg_2Pb \) at equilibrium relative to the mass of the alloy gives insight into the phase composition:

• The alloy studied is 10% \( Mg_2Pb \) by mass and 90% \( \alpha \) phase.
• These percentages indicate the presence of both phases and helps predict the material's overall behavior under various conditions.
• Finding the exact temperature for the formation of \( Mg_2Pb \) would typically require a binary phase diagram, which is not provided in the exercise.

Understanding the \( Mg_2Pb \) phase is crucial for engineers and materials scientists who design and work with Mg-Pb alloys for their unique attributes.

A binary phase diagram plots the phases present in a two-component system, typically with temperature and composition as the axes. It serves as a map for understanding how an alloy will behave under different conditions. The diagram displays lines or curves known as phase boundaries which indicate the temperature and composition at which the phases in the system are in equilibrium.

In practical applications, engineers and scientists use binary phase diagrams to determine mixtures of phases at various temperatures and compositions to manufacture materials with desired properties. For instance, if we had a binary phase diagram for a Mg-Pb alloy, we could easily identify at what temperature the alloy with a composition of 50 wt% Pb and 50 wt% Mg would consist of \( \alpha \) and \( Mg_2Pb \) phases as observed in our mass fraction calculations.

Here are some things to consider when using binary phase diagrams:

• Understanding the concept of tie-lines and the lever rule, which help quantify phase ratios in a two-phase region.
• Identifying eutectic, peritectic, and other special points that mark significant phase transformations.
• Using vertical lines at a constant composition to determine at which temperatures different phases will form.

While this exercise didn't include a phase diagram, in practice, these diagrams are indispensable tools for predicting material behavior, such as finding the temperature at which \( Mg_2Pb \) forms in equilibrium with other phases in a magnesium-lead alloy.