A 45.62-gram sample of calcium hydrogen sulfate [Ca(HSO4)2] would contain how many total atoms?
Is it: 1.525E24, 1.173E23, 5.135, or 0.1948
The correct answer and explanation is :
To find how many total atoms are in a 45.62-gram sample of calcium hydrogen sulfate $\text{Ca(HSO}_4)_2$, we must go through a few steps involving molar mass, moles, and Avogadro’s number.
Step 1: Find the molar mass of Ca(HSO₄)₂
Break down the atomic masses (approximate values from periodic table):
- Ca = 40.08 g/mol
- H = 1.01 g/mol
- S = 32.07 g/mol
- O = 16.00 g/mol
Each molecule of $\text{Ca(HSO}_4)_2$ contains:
- 1 Ca
- 2 H
- 2 S
- 8 O
So, the molar mass is:
$$
40.08 + 2(1.01) + 2(32.07) + 8(16.00) = 40.08 + 2.02 + 64.14 + 128.00 = 234.24 \text{ g/mol}
$$
Step 2: Convert grams to moles
$$
\text{Moles} = \frac{45.62 \text{ g}}{234.24 \text{ g/mol}} \approx 0.1948 \text{ mol}
$$
Step 3: Determine number of molecules
Use Avogadro’s number:
$$
1 \text{ mol} = 6.022 \times 10^{23} \text{ molecules}
$$
$$
\text{Molecules} = 0.1948 \times 6.022 \times 10^{23} \approx 1.173 \times 10^{23} \text{ molecules}
$$
Step 4: Count atoms per molecule
Each $\text{Ca(HSO}_4)_2$ has:
- 1 Ca
- 2 H
- 2 S
- 8 O
= 13 atoms per molecule
Step 5: Total atoms = molecules × atoms per molecule
$$
\text{Total atoms} = 1.173 \times 10^{23} \times 13 \approx 1.525 \times 10^{24}
$$
✅ Correct Answer: 1.525E24
🔍 Explanation Summary (300 words)
To calculate the total number of atoms in a 45.62-gram sample of calcium hydrogen sulfate $\text{Ca(HSO}_4)_2$, we first determine its molar mass. This compound consists of calcium, hydrogen, sulfur, and oxygen. By adding the atomic masses of its components—1 calcium (40.08 g/mol), 2 hydrogens (2.02 g/mol), 2 sulfurs (64.14 g/mol), and 8 oxygens (128.00 g/mol)—we find the molar mass to be 234.24 g/mol.
Next, we convert the given mass to moles:
$$
\frac{45.62}{234.24} \approx 0.1948 \text{ mol}
$$
This tells us how many moles of calcium hydrogen sulfate are present. Then, we use Avogadro’s number ($6.022 \times 10^{23}$) to convert moles to molecules:
$$
0.1948 \times 6.022 \times 10^{23} \approx 1.173 \times 10^{23} \text{ molecules}
$$
Each molecule of $\text{Ca(HSO}_4)_2$ contains 13 atoms: 1 Ca, 2 H, 2 S, and 8 O.
Multiplying the number of molecules by the number of atoms per molecule:
$$
1.173 \times 10^{23} \times 13 = 1.525 \times 10^{24} \text{ atoms}
$$
So, the total number of atoms in the sample is approximately 1.525E24, making that the correct choice. This problem illustrates how molar mass and Avogadro’s number are essential for translating between mass and atomic-level quantities in chemistry.