What will be the wavelength of oxygen molecule in picometers moving with a velocity of `660 m s^(-1) (h = 6.6 xx 10^(-34) kg m^(2) s^(-1))`

Explanation

To find the wavelength of an oxygen molecule moving with a velocity of 660 m/s, we can use the de Broglie wavelength formula:

\[ \lambda = \frac{h}{mv} \]

Where:
- \(\lambda\) is the wavelength,
- \(h\) is the Planck constant (\(6.6 \times 10^{-34} \, \text{kg m}^2/\text{s}\)),
- \(m\) is the mass of the oxygen molecule,
- \(v\) is the velocity of the molecule (\(660 \, \text{m/s}\)).

### Step 1: Calculate the mass of an oxygen molecule
The molecular mass of oxygen (O2) is approximately 32 g/mol. To find the mass of a single oxygen molecule, we use Avogadro's number (\(6.022 \times 10^{23} \, \text{molecules/mol}\)):

\[ m = \frac{32 \, \text{g/mol}}{6.022 \times 10^{23} \, \text{molecules/mol}} = \frac{32 \times 10^{-3} \, \text{kg}}{6.022 \times 10^{23}} \approx 5.31 \times 10^{-26} \, \text{kg} \]

### Step 2: Substitute values into the de Broglie wavelength formula
Now we can substitute the values into the de Broglie formula:

\[ \lambda = \frac{6.6 \times 10^{-34} \, \text{kg m}^2/\text{s}}{(5.31 \times 10^{-26} \, \text{kg})(660 \, \text{m/s})} \]

### Step 3: Calculate the denominator
First, calculate the denominator:

\[ m \cdot v = (5.31 \times 10^{-26} \, \text{kg})(660 \, \text{m/s}) \approx 3.50 \times 10^{-23} \, \text{kg m/s} \]

### Step 4: Calculate the wavelength
Now substitute this value back into the formula for \(\lambda\):

\[ \lambda = \frac{6.6 \times 10^{-34}}{3.50 \times 10^{-23}} \approx 1.89 \times 10^{-11} \, \text{m} \]

### Step 5: Convert to picometers
Since \(1 \, \text{picometer} = 10^{-12} \, \text{m}\), we convert the wavelength into picometers:

\[ \lambda \approx 1.89 \times 10^{-11} \, \text{m} = 18.9 \, \text{pm} \]

### Final Answer
The wavelength of the oxygen molecule moving with a velocity of \(660 \, \text{m/s}\) is approximately \(18.9 \, \text{picometers}\).

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