Problem 58 - General Chemistry: Principles and Modern Applications

Question

Determine (a) the number of \(\mathrm{Kr}\) atoms in a 5.25 -mg sample of krypton (b) the molar mass, \(M,\) and identity of an element if the mass of a \(2.80 \times 10^{22}\) -atom sample of the element is \(2.09 \mathrm{g}\) (c) the mass of a sample of phosphorus that contains the same number of atoms as \(44.75 \mathrm{g}\) of magnesium

Step-by-Step Solution

Answer

a) There are approximately \(3.78 \times 10^{20}\) atoms of Kr in the 5.25-mg sample. b) The molar mass of the element is 45 g/mol, and thus the element is Scandium (Sc). c) The mass of a phosphorus sample that contains the same number of atoms as a 44.75-g sample of magnesium is approximately 57 g.

Step 1: a) Determine the number of Kr atoms

First, convert the mass of krypton from milligrams to grams: \[5.25 \, \text{mg} = 5.25 \times 10^{-3} \, \text{g}\]. Since the molar mass of \(\mathrm{Kr}\) is 83.8 g/mol, we can calculate the number of moles by dividing the mass by its molar mass: \[ n = \frac{Mass}{Molar \, mass} = \frac{5.25 \times 10^{-3}\,g}{83.8 g/mol} \text{ mol} \]. To find the number of atoms, multiply the number of moles by Avogadro’s number: \[ \text{Number of atoms} = n \times (\text{Avogadro’s number}) = \left( \frac{5.25 \times 10^{-3}\,g}{83.8 g/mol} \right) \times 6.022 \times 10^{23} atoms/mol\]

Step 2: b) Find the molar mass and identity of an element

We have the number of atoms and the mass of a sample. Let’s find the amount in moles using Avogadro’s number: \[ n = \frac{\text{Number of atoms}}{\text{Avogadro’s number}} = \frac{2.80 \times 10^{22}\,atoms}{6.022 \times 10^{23} \, atoms/mol} = 0.0465 \, mol\]. To determine the molar mass, we can now divide the mass by the number of moles. \[ Molar\;mass = \frac{Mass}{n} = \frac{2.09 \, g}{0.0465 \, mol} = 45 \, g/mol\]. The element with a molar mass near to 45 g/mol is Scandium (Sc).

Step 3: c) Calculate the mass of a phosphorus sample

We know that the molar mass of Mg is 24.3 g/mol. Let’s first calculate the number of moles in 44.75 g of Mg: \[ n_{Mg} = \frac{44.75 g}{24.3 g/mol} = 1.84 mol \]. Since the number of atoms in a given mass is the same for all elements, the number of phosphorus atoms equals the number of magnesium atoms. Thus, the number of moles of Phosphorus, \(n_{P} = n_{Mg}= 1.84 mol\). Consequently, the mass of phosphorus required can be found using its molar mass of 30.97 g/mol: \[ Mass_{P} = n_{P} \times (\text{Molar mass of Phosphorus}) = 1.84 mol \times 30.97 g/mol\].

Key Concepts

Molar Mass

Molar mass is the mass of one mole of a substance (usually in grams per mole, g/mol). It plays a crucial role in chemistry when converting between the mass of a substance and the moles of it. To calculate the molar mass of an element, you'll need to refer to the periodic table, where each element has a unique molar mass. For instance, krypton ( Kr ) has a molar mass of approximately 83.8 g/mol. This value allows you to determine how heavy a mole of krypton atoms would be.
To find the molar mass of a compound, sum the individual molar masses of all atoms in its molecular formula. This means if you have a complex molecule, each atom's molar mass must be taken into account and added together to obtain the total molar mass.
Being able to calculate and understand molar mass is vital for any chemist as it allows for conversion between mass and moles, essential for any kind of quantitative chemical analysis.

Avogadro's Number

One of the more fundamental constants in chemistry is Avogadro's number, which is approximately \(6.022 \times 10^{23}\). This number tells us how many atoms or molecules are in one mole of a substance. It serves as the bridge between the macroscopic and microscopic worlds, converting measurable amounts of material to actual particle counts.
When you have a known quantity of substance in moles, multiplying by Avogadro's number gives you the total number of atoms or molecules you're dealing with. For example, if you calculate that you have 0.0465 moles of an element, you can find the total number of particles by calculating \(0.0465 \times 6.022 \times 10^{23}\).
This constant is essential for all chemical calculations involving large numbers of entities, such as finding the number of atoms in a given mass of material or determining the atoms in reactions.

Element Identification

Identifying an element involves determining its unique properties such as molar mass. During exercises, you may be given a sample's weight in grams and the number of atoms. Using these values, you can calculate the molar mass and potentially identify the element by matching it to known values from the periodic table.
In the example from the exercise, a sample of 2.80\( \times 10^{22}\) atoms weighs 2.09 grams. By calculating the moles with Avogadro's number and using the mass provided, the molar mass is found to be around 45 g/mol. Referring to the periodic table, Scandium (Sc) closely matches this molar mass, enabling its identification.
Identification can sometimes be straightforward with the right data and tools, making it crucial for effectively analyzing and understanding chemical compositions in various practical applications.