Question

Wood's metal contains \(50.0 \%\) bismuth, \(25.0 \%\) lead, \(12.5 \%\) tin and \(12.5 \%\) cadmium by weight. What is the mole fraction of tin ? (Atomic weights : \(\mathrm{Bi}=209, \mathrm{~Pb}=207, \mathrm{Sn}=119, \mathrm{Cd}=112\) ) (a) \(0.202\) (b) \(0.158\) (c) \(0.176\) (d) \(0.221\)

Short answer

The mole fraction of tin (Sn) in Wood's metal is 0.158.

Step-by-step solution

- Calculate the mass of each metal in a sample

Assume you have 100 g of Wood's metal. Calculate the mass of each component based on the given percentages. Bismuth (Bi) = 50.0%, Lead (Pb) = 25.0%, Tin (Sn) = 12.5%, and Cadmium (Cd) = 12.5%. The mass of each component would be: Bi = 50.0 g, Pb = 25.0 g, Sn = 12.5 g, Cd = 12.5 g.

- Calculate the moles of each metal

Using the atomic weights, Bi = 209, Pb = 207, Sn = 119, Cd = 112, calculate the moles of each component by dividing the mass by the respective atomic weight. Moles of Bi = 50.0 g / 209, Moles of Pb = 25.0 g / 207, Moles of Sn = 12.5 g / 119, Moles of Cd = 12.5 g / 112.

- Calculate the total moles in the sample

Sum the moles of all components to find the total moles in the sample. Total moles = Moles of Bi + Moles of Pb + Moles of Sn + Moles of Cd.

- Calculate the mole fraction of tin

The mole fraction of tin (Sn) is the number of moles of tin divided by the total number of moles in the sample. Mole fraction of Sn = Moles of Sn / Total moles.

Key concepts

Wood's Metal Composition

Wood's metal is a unique alloy known for its low melting point, making it useful for various applications, such as in fusible plugs and for making molds and casts in dentistry. Its composition is integral to its distinct physical properties. By definition, Wood's metal contains four primary metals: bismuth, lead, tin, and cadmium. The specific percentages of each metal are critical, as they influence the alloy's overall behavior and melting point.

The exercise at hand uses these specific mass percentages to determine the mole fraction of tin in Wood's metal. Understanding the composition is the first step towards interpreting the mole fraction, an expression of the concentration of a particular component in the alloy, thereby providing insight on the relative amount of that metal in a given sample.

Converting Mass Percent to Moles

The process of converting mass percent to moles is fundamental in chemistry for comprehending the composition of mixtures and compounds. In this exercise, the mass percent of each component in Wood's metal is provided and must be converted into moles to progress towards the solution. The concept of moles links the macroscopic world to the atomic scale and allows chemists to count atoms and molecules practically.

In practical terms, to convert mass percent to moles, one typically assumes a convenient mass for the whole sample, like 100 grams, which aligns the mass percent directly with the mass in grams. From there, each component's mass, derived from its mass percent, is divided by its atomic weight to obtain the number of moles. This standardized approach, applied to each metal in Wood's metal in our example, leads us to the next step: understanding atomic weight.

Atomic Weight

Atomic weight, often referred to as relative atomic mass, is a pivotal concept in chemistry, representing the average mass of atoms of an element, measured in atomic mass units (amu). It takes into account the different isotopes of an element and their relative abundance. The atomic weight is vital for converting grams to moles, as it provides a bridge between a known mass of an element and the number of atoms or molecules present.

In the context of the exercise, the atomic weights for bismuth (Bi), lead (Pb), tin (Sn), and cadmium (Cd) are given. These weights are used to convert the mass of each metal component in Wood's metal into moles. It is the ratio of the mass of the metal to its respective atomic weight that yields the number of moles - a step that is essential for calculating mole fractions.

Molar Mass

Molar mass, closely related to atomic weight, is the mass of one mole of a substance (be it an element or a compound) and is expressed in grams per mole (g/mol). It is essential to understand that while atomic weight is a dimensionless quantity due to being an average based on a relative scale, molar mass carries units of mass per amount of substance (grams per mole).

For individual elements, the molar mass is numerically equivalent to the atomic weight but scaled to grams per mole. For compounds, the molar mass is the sum of the atomic weights of all atoms present in a molecule, also scaled to grams per mole. This concept is fundamental when performing the mole calculations in our exercise; the molar mass is effectively used to translate the given mass percentages into moles to calculate the mole fractions of each component in Wood's metal.