Question

One mole of element \(X\) has \(0.444\) times the mass of one mole of element \(Y .\) One atom of element \(X\) has \(2.96\) times the mass of one atom of \({ }^{12} \mathrm{C}\). What is the atomic weight of \(Y ?\) (a) 80 (b) \(15.77\) (c) \(46.67\) (d) \(40.0\)

Short answer

The atomic weight of element Y is approximately 80 amu.

Step-by-step solution

Calculate the Atomic Weight of Element X

Given that one mole of element X has a mass 0.444 times the mass of one mole of element Y, we can write this as a ratio: \( \frac{M_X}{M_Y} = 0.444 \). Also, one atom of element X has a mass 2.96 times the mass of one atom of Carbon-12. Since the atomic weight of Carbon-12 is defined to be exactly 12 atomic mass units (amu), the atomic weight of an atom of element X can be calculated by multiplying 2.96 with 12 amu, which gives us the atomic weight of X (\( A_X \)). \( A_X = 2.96 \times 12 \, \text{amu} \).

Calculate the Atomic Weight of Element Y

By solving the equation from Step 1 for \( A_X \), we get the atomic weight of element X. Then, using the calculated atomic weight of element X and the given ratio, we can find the atomic weight of element Y (\( A_Y \)). We can set up the equation \( A_Y = \frac{A_X}{0.444} \) to find the atomic weight of element Y.

Solve for the Atomic Weight of Element Y

Plug the value of \( A_X \), which was calculated in Step 1, into the equation from Step 2 to find the atomic weight of element Y, \( A_Y = \frac{A_X}{0.444} = \frac{2.96 \times 12}{0.444} \).

Key concepts

Understanding Atomic Weight Calculation

The atomic weight, also known as the atomic mass, is a fundamental concept in chemistry, reflecting the average mass of atoms of an element. It is measured in atomic mass units (amu), where one amu is defined as one-twelfth the mass of a single carbon-12 atom.

To calculate the atomic weight of an element, one can use the relative abundance of its isotopes and their respective masses. However, in the provided exercise, we use the comparison between the mass of one mole of an unknown element and the mass of one mole of carbon-12 to find the desired atomic weight.

Exercise Application

In the exercise, we're told that one atom of element X has 2.96 times the mass of carbon-12. Therefore, we can simply multiply the base atomic weight of carbon-12, which is precisely 12 amu, by 2.96 to find the atomic weight of element X. Calculations involving atomic weight are crucial in determining the proportions of elements in compounds and their reactions.

Molar Mass Comparison

Molar mass is a property that links the mass of a substance to its amount in moles. A mole is the SI unit of amount of substance, and one mole contains exactly 6.02214076×10²³ elementary entities (Avogadro's number). The molar mass tells us how many grams per mole a substance weighs and is numerically equal to the substance's atomic or molecular weight.

Different elements have different molar masses. The comparison of molar masses between elements allows us to understand the conversion of moles to grams for each element, which is a foundational step in stoichiometric calculations.

Comparing Molar Mass in the Exercise

When we say that one mole of element X has 0.444 times the mass of one mole of element Y, we are essentially comparing their molar masses directly. This allows us to set up a ratio and solve for the unknown molar mass. Molar mass comparisons are invaluable when predicting reaction yields, calculating empirical formulas, and balancing chemical equations.

Stoichiometry Explained

Stoichiometry is the quantitative relationship between reactants and products in a chemical reaction, grounded in the law of conservation of mass. It involves calculations that use balanced chemical equations to determine the amount of reactants required or products formed.

The stoichiometric coefficients indicate the number of moles of each substance involved. These coefficients enable conversions between moles of one substance to moles of another, utilizing the mole-to-mole ratios derived from the balanced equation.

Stoichiometry in Our Calculation

Our exercise doesn't involve a chemical reaction, but it exemplifies the concept of stoichiometric calculations. By establishing a relationship between the masses of two substances (in moles), we apply stoichiometric techniques to deduce the atomic weight of one element by referring to a known quantity of another element. This type of problem-solving is the essence of stoichiometry, where the mole concept serves as a bridge between the microscopic world of atoms and the macroscopic world of grams and kilograms.