Question

A $2.077-\mathrm{g}$ sample of an element, which has an atomic mass between 40 and $55,$ reacts with oxygen to form 3.708 g of an oxide. Determine the formula of the oxide (and identify the element).

Short answer

The unknown element is Tellurium (Te) with an approximate atomic mass of 52, and the formula of the oxide is Te2O3.

Step-by-step solution

Calculate the mass of oxygen

Since we know the mass of the element and the mass of the oxide, we can calculate the mass of oxygen that reacted with the element by subtracting the mass of the element from the mass of the oxide: Mass of oxygen = Mass of oxide - Mass of element = 3.708 g - 2.077 g = 1.631 g

Convert mass into moles

Let's assume that the atomic mass of the unknown element is "X". To find the moles of the element and oxygen, use the formula: Moles = (mass) / (atomic_mass) Moles of element = 2.077 g / X g/mol Moles of oxygen = 1.631 g / 16.00 g/mol

Determine the mole ratio of the element to oxygen

To find the mole ratio of the element to oxygen in the oxide, divide the moles of the element by the moles of oxygen: Mole Ratio = (Moles of element) / (Moles of oxygen) = (2.077/X) / (1.631/16.00) Simplify this equation: Mole Ratio = (2.077 * 16.00) / (1.631 * X) = 33.232 / X

Determine the formula of the oxide and identify the element

Given that the atomic mass of the unknown element is between 40 and 55, and we have the mole ratio as (33.232 / X), we can use this information to find the approximate atomic mass of the element and identify it. For a whole number ratio between the element and oxygen, X should be a factor of 33.232. We can try the atomic masses between 40 and 55 to check if any of them give a whole number ratio. For example, let's try X = 44: Mole Ratio = 33.232 / 44 = 0.755 This ratio doesn't give a whole number, and we can eliminate 44 as the atomic mass. Now let's try X = 52: Mole Ratio = 33.232 / 52 = 0.639 This ratio is very close to 2:3. So, the approximate atomic mass of the unknown element is 52, which belongs to Tellurium (Te). Therefore, the unknown element is Tellurium. Now that we know the element is Tellurium and the mole ratio is approximately 2:3, we can determine the formula of the oxide as Te2O3.

Key concepts

Atomic Mass Calculation

The atomic mass calculation is a fundamental concept in chemistry that involves determining the mass of a single atom of an element. It's usually expressed in atomic mass units (amu), where one amu is defined as one-twelfth the mass of a carbon-12 atom. In practical scenarios, such as the one in our exercise, we don't measure the mass of individual atoms. Instead, we look at a larger, measurable amount of an element and use the concept of moles.

In the given problem, the atomic mass of the element (which we will call 'X') is unknown, but we know it's between 40 and 55 amu. By utilizing the mass of an element sample and its oxide, we can predict the atomic mass through calculations involving moles and knowing from the periodic table that the atomic mass of oxygen is 16 amu. It's essential to understand that when calculating atomic masses, we're searching for the mass that fits with the empirical evidence, which in our case, is finding an atomic mass that gives a whole number mole ratio when reacting with oxygen.

Molar Mass Conversion

Molar mass is the weight of one mole of a substance and is typically measured in grams per mole (g/mol). Why is this useful? Because it connects the mass of a substance to its amount in moles, allowing for the conversion between the two. The molar mass of any element can be found on the periodic table as it's roughly equivalent to the element's atomic mass.

During our exercise, we convert the given mass of an element and oxygen into moles by dividing the mass by their respective molar masses. This conversion is crucial as it sets the stage for identifying the mole ratio between the element and oxygen. Remember, a critical step for students to apprehend is the correct interpretation of molar masses from the periodic table and the concept of converting mass to moles using the formula:
$$\text{Moles}=\frac{\text{mass}}{\text{molar mass}}$$

Mole Ratio

A mole ratio is an expression that provides the relative amounts of reactants and products in a chemical reaction. In the context of our exercise, the mole ratio allows us to express the proportion between the unknown element and oxygen in the oxide compound. Upon converting the masses of the element and oxygen to moles, we can set up a ratio comparing these amounts.

The formula to determine the mole ratio is essentially
$$\text{Mole Ratio}=\frac{\text{Moles of element}}{\text{Moles of oxygen}}$$
. Simplifying this expression gives us a number that represents how many moles of the element combine with one mole of oxygen. This ratio must make sense logically and result in whole numbers when dealing with empirical formulas. Because the formula must represent the smallest whole number ratio, understanding this concept is crucial for deciphering the actual formula of a compound.

Element Identification

The final step in many stoichiometric problems is often identifying the element or compound in question. To nail down the identity of an element, scientists use a combination of techniques, including calculations based on atomic mass, molar mass conversion, and mole ratios among other properties like spectroscopy and reactivity.

In our textbook exercise, once we compute the mole ratio and have an estimate for the atomic mass of the element, we match these values with known elements listed in the periodic table. Identifying the correct element involves making sure the estimated atomic mass fits within the given range and leads to an integer mole ratio with oxygen. The element Tellurium (Te), with an atomic mass of 52 amu, fits our requirement based on the calculated mole ratio close to 2:3. Thus, this method forms a bridge between raw observational data (like masses), mole concepts, and the vast database of element properties that is the periodic table, cementing the understanding of how we recognize chemical substances in a quantitative manner.