Find de-Broglie wavelength corresponding to the root-mean square velocity of hydrogen molecules at room temperature `(20^@C)`.

Explanation

To find the de-Broglie wavelength corresponding to the root-mean square (RMS) velocity of hydrogen molecules at room temperature (20°C), we will follow these steps:

### Step 1: Convert Temperature to Kelvin
The temperature in Celsius needs to be converted to Kelvin using the formula:
\[ T(K) = T(°C) + 273 \]
For 20°C:
\[ T = 20 + 273 = 293 \, K \]

### Step 2: Calculate the Root Mean Square Velocity (Vrms)
The formula for the root mean square velocity of gas molecules is given by:
\[ V_{rms} = \sqrt{\frac{3RT}{M}} \]
Where:
- \( R \) is the gas constant, \( R = 8.314 \, J/(mol \cdot K) \)
- \( T \) is the temperature in Kelvin (which we found to be 293 K)
- \( M \) is the molar mass of hydrogen. The molar mass of hydrogen (H₂) is approximately \( 2 \, g/mol = 2 \times 10^{-3} \, kg/mol \).

### Step 3: Calculate the Molar Mass in kg
The mass of one hydrogen molecule can be calculated as:
\[ M = \frac{2 \times 10^{-3} \, kg/mol}{N_A} \]
Where \( N_A \) (Avogadro's number) is approximately \( 6.022 \times 10^{23} \, molecules/mol \):
\[ M = \frac{2 \times 10^{-3}}{6.022 \times 10^{23}} \approx 3.32 \times 10^{-27} \, kg \]

### Step 4: Substitute Values into the Vrms Formula
Now we can substitute the values into the Vrms formula:
\[ V_{rms} = \sqrt{\frac{3 \times 8.314 \times 293}{3.32 \times 10^{-27}}} \]

### Step 5: Calculate Vrms
Calculating the above expression:
\[ V_{rms} \approx \sqrt{\frac{7300.74}{3.32 \times 10^{-27}}} \approx \sqrt{2.20 \times 10^{30}} \approx 1.48 \times 10^{15} \, m/s \]

### Step 6: Calculate the de-Broglie Wavelength
The de-Broglie wavelength \( \lambda \) is given by the formula:
\[ \lambda = \frac{h}{mv} \]
Where:
- \( h \) is Planck's constant, \( h \approx 6.626 \times 10^{-34} \, J \cdot s \)
- \( m \) is the mass of one hydrogen molecule (which we calculated)
- \( v \) is the RMS velocity \( V_{rms} \)

Substituting the values:
\[ \lambda = \frac{6.626 \times 10^{-34}}{3.32 \times 10^{-27} \times 1.48 \times 10^{15}} \]

### Step 7: Calculate λ
Calculating the above expression:
\[ \lambda \approx \frac{6.626 \times 10^{-34}}{4.91 \times 10^{-12}} \approx 1.35 \times 10^{-22} \, m \]
Converting to Angstroms (1 Angstrom = \( 10^{-10} \, m \)):
\[ \lambda \approx 1.35 \times 10^{-12} \, m = 1.35 \, \text{Å} \]

### Final Answer
The de-Broglie wavelength corresponding to the root-mean square velocity of hydrogen molecules at room temperature (20°C) is approximately:
\[ \lambda \approx 1.35 \, \text{Å} \]