Consider the four gases hydrogen, oxygen, nitrogen and helium at the same temperature. Arrange them in the increasing order of the de Broglie wavelengths of their molecules.

Options

A. Hydrogen, helium, nitrogen, oxygen
B. Oxygen, netrogen, hydrogen, helium
C. Oxygen, nitrogen, helium, hydrogen
D. Nitrogen, oxygen, helium, hydrogen

Explanation

To solve the problem of arranging the gases hydrogen, oxygen, nitrogen, and helium in increasing order of their de Broglie wavelengths, we can follow these steps:

### Step 1: Understand the de Broglie wavelength formula
The de Broglie wavelength (λ) of a gas molecule is given by the formula:
\[ \lambda = \frac{H}{\sqrt{3m k_B T}} \]
where:
- \(H\) is Planck's constant,
- \(m\) is the mass of the gas molecule,
- \(k_B\) is the Boltzmann constant,
- \(T\) is the temperature.

### Step 2: Analyze the dependencies
Since all gases are at the same temperature (T), and both \(H\) and \(k_B\) are constants, the de Broglie wavelength is inversely proportional to the square root of the mass of the gas:
\[ \lambda \propto \frac{1}{\sqrt{m}} \]
This means that as the mass of the gas increases, the de Broglie wavelength decreases.

### Step 3: Determine the molar masses of the gases
Next, we need to find the molar masses of the gases:
- Hydrogen (H₂): 2 g/mol (1 g/mol per atom)
- Helium (He): 4 g/mol
- Nitrogen (N₂): 28 g/mol (14 g/mol per atom)
- Oxygen (O₂): 32 g/mol (16 g/mol per atom)

### Step 4: Arrange the gases by mass
Now we can arrange the gases in order of increasing mass:
1. Hydrogen (2 g/mol)
2. Helium (4 g/mol)
3. Nitrogen (28 g/mol)
4. Oxygen (32 g/mol)

### Step 5: Determine the order of de Broglie wavelengths
Since the de Broglie wavelength is inversely proportional to the square root of the mass, we can now arrange the gases in increasing order of their de Broglie wavelengths:
- Highest mass (Oxygen) → Lowest wavelength
- Next highest mass (Nitrogen) → Next lowest wavelength
- Next (Helium) → Higher wavelength than nitrogen
- Lowest mass (Hydrogen) → Highest wavelength

Thus, the increasing order of de Broglie wavelengths is:
1. Oxygen (lowest wavelength)
2. Nitrogen
3. Helium
4. Hydrogen (highest wavelength)

### Final Answer
The increasing order of the de Broglie wavelengths of the gases is:
**Oxygen < Nitrogen < Helium < Hydrogen**