Question

A low-melting Sn-Pb-Cd alloy called eutectic alloy is analyzed. The mole ratio of tin to lead is 2.73: 1.00 and the mass ratio of lead to cadmium is 1.78: 1.00 . What is the mass percent composition of this alloy?

Short answer

The mass percent composition of the eutectic alloy is approximately: 50.04% Sn, 31.95% Pb, and 17.96% Cd.

Step-by-step solution

Convert the Mole Ratio to Mass Ratio

Firstly, it is necessary to convert the mole ratios into mass ratios. To accomplish this, the atomic masses of tin (Sn, 118.71 g/mol) and lead (Pb, 207.2 g/mol) are used. The mass of tin for the given mole ratio is \(2.73 \times 118.71 \, g/mol = 324.3773 \, g/mol\). For lead, the mass is \(1.00 \times 207.2 \, g/mol = 207.2 \, g/mol\). This gives a mass ratio of Sn to Pb of 324.3773 : 207.2.

Derive the Mass Ratio of Sn to Cd

The mass ratio of Lead to Cadmium is given as 1.78 : 1, which means for 1.78 g of Pb, there is 1 g of Cd. From step 1, we know that there are 207.2g of Pb, so the mass of Cd is calculated using cross multiplication as \( \frac{207.2 \, g}{1.78} = 116.4045 \, g\). Thus, the mass ratio of tin is to cadmium is 324.3773 : 116.4045.

Calculate the Mass Percent Composition

The last step is to calculate the percent composition by mass for each element. First add up all the masses to find the total mass of the alloy., which is \( 324.3773 \, g + 207.2 \, g + 116.4045 \, g = 647.9818 \, g\). Then the mass percent of each element is calculated as follows: for Sn \(\frac{324.3773}{647.9818} \times 100\%\), for Pb \(\frac{207.2}{647.9818} \times 100\%\), and for Cd \(\frac{116.4045}{647.9818} \times 100\%\).

Key concepts

Eutectic Alloy

A eutectic alloy is a special type of metal alloy with a unique composition. This specific composition allows it to melt or solidify at a single, sharp temperature. The term "eutectic" comes from the Greek word "eutektos", meaning "easily melted".
Eutectic alloys are fascinating because they exhibit a melting point that is lower than any of the individual components in their mixture.

• They have a precisely defined melting point, unlike other mixtures that melt over a range of temperatures.
• They are used extensively in industries for processes that require precise melting points, such as soldering.

In the context of our problem, the Sn-Pb-Cd alloy is a eutectic alloy, making it ideal for applications where low melting points are beneficial.

Mole Ratio

The mole ratio is a concept used to describe the relation between the amounts of substances in a chemical reaction or mixture. It defines the proportion of moles for each component, which is essential in understanding the composition of alloys, chemicals, and compounds.
For our eutectic alloy, the mole ratio of tin to lead is given as 2.73:1.00. This means for every 2.73 moles of tin, there is 1 mole of lead.

• Mole ratios help in converting the number of moles into the masses of each component when combined with atomic masses.
• This step is crucial when we transition from discussing moles (a measure of quantity) to mass, which is often more practical for chemical processes.

Understanding mole ratios is foundational in chemistry to determine relative quantities in a formula or reaction.

Mass Ratio

A mass ratio is used to infer the proportion of each constituent in terms of mass, instead of moles. It involves the comparison of the mass of one substance to the mass of another.
In our problem, we have two key mass ratios: the ratio of tin to lead based on mole ratios, and the direct mass ratio of lead to cadmium, 1.78:1.00. This gives:

• A mass ratio of tin to lead derived from the mole ratio.
• Using the mass ratio of lead to cadmium allows calculating cadmium's mass, as shown in the solution.

These ratios are essential as they help calculate the final mass percent composition of the alloy, which provides detailed insight into what the alloy is made of, in terms of weight.

Atomic Mass

Atomic mass refers to the mass of a single atom, commonly expressed in atomic mass units (amu or g/mol). It is average based on the isotopic composition of elements found naturally. Knowing the atomic mass is crucial in converting moles to grams, which is an everyday task in chemistry calculations.
For example, the atomic masses of tin (118.71 g/mol) and lead (207.2 g/mol) are used to derive the mass ratio from the mole ratio in our alloy problem.

• Atomic mass allows us to calculate how much of each element is present in an alloy when given mole ratios.
• It is the key to bridging the idea of number of atoms and their collective mass.

By utilizing atomic masses, one can transition from moles to a more tangible measure such as grams, which is often necessary in laboratory settings.