What is the mass in grams of one molecule of caffeine `(C_(8)H_(10)N_(4)O_(2))`?

Explanation

To find the mass in grams of one molecule of caffeine (C₈H₁₀N₄O₂), we will follow these steps:

### Step 1: Calculate the Molar Mass of Caffeine
First, we need to determine the molar mass of caffeine using the atomic masses of its constituent elements:

- Carbon (C): 12 g/mol
- Hydrogen (H): 1 g/mol
- Nitrogen (N): 14 g/mol
- Oxygen (O): 16 g/mol

The molecular formula of caffeine is C₈H₁₀N₄O₂. We can calculate the molar mass as follows:

\[ \text{Molar mass of caffeine} = (8 \times \text{atomic mass of C}) + (10 \times \text{atomic mass of H}) + (4 \times \text{atomic mass of N}) + (2 \times \text{atomic mass of O}) \]

Substituting the values:

\[ \text{Molar mass of caffeine} = (8 \times 12) + (10 \times 1) + (4 \times 14) + (2 \times 16) \]

Calculating each term:

- \(8 \times 12 = 96\)
- \(10 \times 1 = 10\)
- \(4 \times 14 = 56\)
- \(2 \times 16 = 32\)

Now, adding these values together:

\[ \text{Molar mass of caffeine} = 96 + 10 + 56 + 32 = 194 \text{ g/mol} \]

### Step 2: Use Avogadro's Number
Next, we use Avogadro's number, which is \(6.022 \times 10^{23}\) molecules/mol. This tells us how many molecules are in one mole of a substance.

### Step 3: Calculate the Mass of One Molecule
Now, we can find the mass of one molecule of caffeine by dividing the molar mass by Avogadro's number:

\[ \text{Mass of one molecule} = \frac{\text{Molar mass}}{\text{Avogadro's number}} = \frac{194 \text{ g/mol}}{6.022 \times 10^{23} \text{ molecules/mol}} \]

Calculating this gives:

\[ \text{Mass of one molecule} = \frac{194}{6.022 \times 10^{23}} \approx 3.22 \times 10^{-22} \text{ grams} \]

### Final Answer
The mass of one molecule of caffeine is approximately \(3.22 \times 10^{-22}\) grams.

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