One mole of an ideal gas at standard temperature and pressure occupies 22.4 L (molar volume). IF the size of the hydrogent molecule is about 1A. What is the ratio of molar volume to the atomic volume of a mole of hydrogen?

Options

A. `10^4`
B. `10^2`
C. `10^3`
D. `10^(-4)`

Answer

To solve the problem of finding the ratio of molar volume to the atomic volume of a mole of hydrogen, we can follow these steps:

### Step 1: Understand the Molar Volume
The molar volume of an ideal gas at standard temperature and pressure (STP) is given as 22.4 L. We need to convert this volume into cubic meters for our calculations:
\[ \text{Molar Volume (Vm)} = 22.4 \, \text{L} = 22.4 \times 10^{-3} \, \text{m}^3 \]

### Step 2: Determine the Size of a Hydrogen Molecule
The size of a hydrogen molecule is given as 1 Ångström (1 Å = \(10^{-10}\) m). Therefore, the radius \(r\) of a hydrogen molecule is:
\[ r = \frac{1 \, \text{Å}}{2} = 0.5 \, \text{Å} = 0.5 \times 10^{-10} \, \text{m} \]

### Step 3: Calculate the Volume of a Single Hydrogen Atom
The volume \(V\) of a single hydrogen atom can be calculated using the formula for the volume of a sphere:
\[ V = \frac{4}{3} \pi r^3 \]
Substituting the radius:
\[ V = \frac{4}{3} \pi (0.5 \times 10^{-10})^3 \]
Calculating this:
\[ V = \frac{4}{3} \times \frac{22}{7} \times (0.5^3) \times (10^{-30}) = 0.524 \times 10^{-30} \, \text{m}^3 \]

### Step 4: Calculate the Atomic Volume for One Mole of Hydrogen
One mole of hydrogen contains Avogadro's number of molecules, which is approximately \(6.023 \times 10^{23}\). Thus, the atomic volume \(V_a\) for one mole of hydrogen is:
\[ V_a = 6.023 \times 10^{23} \times 0.524 \times 10^{-30} \]
Calculating this:
\[ V_a = 3.16 \times 10^{-7} \, \text{m}^3 \]

### Step 5: Calculate the Ratio of Molar Volume to Atomic Volume
Now that we have both the molar volume and the atomic volume, we can find the ratio:
\[ \text{Ratio} = \frac{V_m}{V_a} = \frac{22.4 \times 10^{-3}}{3.16 \times 10^{-7}} \]
Calculating this gives:
\[ \text{Ratio} \approx 7.08 \times 10^{4} \]

### Final Answer
The ratio of the molar volume to the atomic volume of a mole of hydrogen is approximately \(7.08 \times 10^{4}\).

---