Question

If \(3.02 \times 10^{19}\) molecules are removed from \(98 \mathrm{mg}\) of \(\mathrm{H}_{2} \mathrm{SO}_{4}\), then the number of moles of \(\mathrm{H}_{2} \mathrm{SO}_{4}\) left are (a) \(0.1 \times 10^{-3}\) (b) \(5 \times 10^{-4}\) (c) \(1.2 \times 10^{-4}\) (d) \(1.5 \times 10^{-3}\)

Short answer

Approximately \(9.5 \times 10^{-4}\) moles of \(H_2SO_4\) remain.

Step-by-step solution

Calculate Molar Mass of H2SO4

First, find the molar mass of \(H_2SO_4\). The atomic masses are approximately: Hydrogen (H) = 1 g/mol, Sulfur (S) = 32 g/mol, and Oxygen (O) = 16 g/mol. Therefore, the molar mass of \(H_2SO_4\) is \( (2 \times 1) + 32 + (4 \times 16) = 2 + 32 + 64 = 98 \text{ g/mol}. \)

Determine Initial Moles of H2SO4

Next, calculate the initial moles of \(H_2SO_4\) in 98 mg. Since 98 mg is equal to 0.098 g, we have:\(\text{Moles of } H_2SO_4 = \frac{0.098}{98} = 0.001 \text{ mol}\).This is the initial amount before any molecules are removed.

Determine Molecules Removed to Moles

Now, convert the number of molecules removed to moles using Avogadro's number \( (6.022 \times 10^{23} \text{ molecules/mol}) \). The moles removed are:\(\text{Moles removed} = \frac{3.02 \times 10^{19}}{6.022 \times 10^{23}} \approx 5.02 \times 10^{-5} \text{ mol}.\)

Calculate Moles of H2SO4 Left

Subtract the moles removed from the initial moles to find the remaining moles:\(\text{Moles left} = 0.001 - 5.02 \times 10^{-5} = 0.0009498. \approx 9.5 \times 10^{-4} \text{ moles}.\)

Match to Closest Option

Examine the answer choices and determine the one closest to \(9.5 \times 10^{-4}\). The closest option is \(1 imes 10^{-3}\), but all choices are slightly different. As a more accurate solution is not provided, reassess intermediate rounding or approach for precise correspondence.

Key concepts

Avogadro's Number

Avogadro's Number is an extraordinary concept in chemistry that acts as a bridge between the very tiny molecular world and the more tangible world of moles and grams we deal with. It's a fundamental constant used to convert between the number of atoms, ions, or molecules, and their corresponding amount in moles. The value of Avogadro's Number is approximately \(6.022 \times 10^{23}\) entities per mole. This means that one mole of any substance contains about \(6.022 \times 10^{23}\) of its constituent particles. This vast number makes it feasible to count entities that are otherwise exceptionally small and numerous.
Think of Avogadro's Number as a sort of 'chemist's dozen', except rather than a dozen representing 12, it represents \(6.022 \times 10^{23}\). This helps translate our calculations from small-scale events—involving individual atoms and molecules—into quantities we can measure and use in the lab. Anytime we need to convert a given number of molecules to moles, Avogadro's Number becomes our go-to tool, such as in this exercise where we needed to find how many moles \(3.02 \times 10^{19}\) molecules corresponded to.

Molar Mass Calculation

To solve problems in chemistry involving moles, calculating the molar mass is crucial. The molar mass is essentially the weight of one mole of a given substance, expressed in grams per mole (g/mol). For instance, when calculating the molar mass of a compound like sulfuric acid \( \text{H}_2\text{SO}_4 \), we use the atomic masses of its constituent elements.
In this specific calculation, you add up the following:

• Hydrogen (H): Each atom has an atomic mass of about 1 g/mol, so for 2 atoms, it's \( 2 \times 1 = 2 \text{ g/mol} \).
• Sulfur (S): With one atom at 32 g/mol, contributing \(32 \text{ g/mol} \).
• Oxygen (O): 16 g/mol per atom, \(4 \times 16 = 64 \text{ g/mol} \).

Totaling these gives the molar mass of sulfuric acid, \( \text{H}_2\text{SO}_4 \), equal to \( 98 \text{ g/mol} \). Understanding how to perform this calculation is fundamental for progressing to more complex mole conversion problems that require precise values, as demonstrated in the step-by-step solution for the problem.

Molecule to Mole Conversion

Converting molecules to moles is an essential skill in chemistry, especially when dealing with reactions and amounts of substances. It involves using Avogadro's Number to express how many molecules correspond to a substance's amount in moles. For example, if you have \(3.02 \times 10^{19}\) molecules of a substance, to find how many moles this represents, divide by \(6.022 \times 10^{23}\).
Through this division:

• We set up the equation: \( \text{Moles} = \frac{3.02 \times 10^{19}}{6.022 \times 10^{23}} \).
• The calculation results in \(5.02 \times 10^{-5} \text{ moles} \).

This conversion effectively tells us the amount of the substance present in moles, which is indispensable for having a consistent method to measure and predict the behavior of substances in a given context. This is why molecule to mole conversion is frequently employed in practical scenarios, such as in this exercise to determine how moles of \( \text{H}_2\text{SO}_4 \) remained after removing some molecules.