Question

Gallium arsenide is one of the newer materials used to make semiconductor chips for use in supercomputers. Its composition is \(48.2 \%\) Ga and \(51.8 \%\) As. What is the empirical formula?

Short answer

The empirical formula is GaAs.

Step-by-step solution

Convert percentages to grams

Assume you have 100 grams of the compound. This means you have 48.2 grams of Ga and 51.8 grams of As.

Find the number of moles for Ga

To find the number of moles, use the formula \text{number of moles} = \frac{\text{mass}}{\text{molar mass}}. The molar mass of Ga (Gallium) is 69.72 g/mol. Thus, the moles of Ga are \[ \frac{48.2 \text{ grams}}{69.72 \text{ g/mol}} = 0.691 \text{ moles} \].

Find the number of moles for As

The molar mass of As (Arsenic) is 74.92 g/mol. Thus, the moles of As are \[ \frac{51.8 \text{ grams}}{74.92 \text{ g/mol}} = 0.691 \text{ moles} \].

Determine the mole ratio

To find the simplest whole number ratio, divide the number of moles of each element by the smallest number of moles found. Both Ga and As have 0.691 moles, so \[ \frac{0.691 \text{ moles of Ga}}{0.691} = 1 \] and \[ \frac{0.691 \text{ moles of As}}{0.691} = 1 \].

Write the empirical formula

The simplest whole number ratio is 1:1, so the empirical formula is GaAs.

Key concepts

Mole Calculation

Understanding mole calculations is key to mastering many topics in chemistry. A mole is a fundamental unit used to quantify the amount of substance. To convert from grams to moles, you need the compound's molar mass. The basic formula used is \(\text{number of moles} = \frac{\text{mass in grams}}{\text{molar mass}} \).

For instance, in the given problem, Gallium (Ga) has a molar mass of 69.72 g/mol, and Arsenic (As) has a molar mass of 74.92 g/mol. To find the number of moles of Gallium in 48.2 grams, you use the formula:
\[ \frac{48.2 \text{ grams}}{69.72 \text{ g/mol}} = 0.691 \text{ moles} \]For Arsenic, in 51.8 grams, it is calculated as:
\[ \frac{51.8 \text{ grams}}{74.92 \text{ g/mol}} = 0.691 \text{ moles} \]
These mole calculations are the stepping stones to understanding the rest of the problem.

Percent Composition

Percent composition indicates the percentage by mass of each element in a compound. It informs us how much of each element makes up a compound, by mass. When given percentages, always assume you have a 100g sample—making calculations easy.

For instance, Gallium arsenide (GaAs) has 48.2% Gallium (Ga) and 51.8% Arsenic (As). In a 100g sample, this means 48.2g of Ga and 51.8g of As. Recognizing percent composition transforms percentages into practical mass values—enabling further calculations like finding the moles. Knowing the percent composition is essential not only for mole calculation but also to determine the empirical formula.

Molecule Ratio

The molecule ratio represents the simplest whole-number ratio between the elements in a compound. To find this ratio, divide the moles of each element by the smallest number of moles among them.

In our given problem, both Gallium (Ga) and Arsenic (As) have 0.691 moles. This makes the ratio calculation straightforward:
\[ \frac{0.691 \text{ moles of Ga}}{0.691} = 1 \]\[ \frac{0.691 \text{ moles of As}}{0.691} = 1 \]
This ratio is simplified to 1:1.
Ratios bring clarity, translating molecular weight data into the simplest form—helping us determine the empirical formula. Here, with a 1:1 ratio, the empirical formula of the compound is GaAs. Understanding molecule ratios ensures accurate empirical formula determination, which represents the smallest whole-number ratio of atoms in the compound.