Which of the following is correct decreasing order of r.ms. velocity at same temperature for `H_2, N_2, CO_2` and `O_2`

Options

A. `O_2 gt CO_2 gtN_2 gt H_2`
B. `H_2 gt N_2 gt O_2 gt CO_2`
C. `H_2 gt N_2 gt CO_2 gt O_2`
D. `N_2 gt CO_2 gt H_2 gt O_2`

Answer

To determine the correct decreasing order of root mean square (r.m.s.) velocity for the gases H₂, N₂, CO₂, and O₂ at the same temperature, we can follow these steps:

### Step 1: Understand the Formula for r.m.s. Velocity
The r.m.s. velocity (v_rms) of a gas is given by the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where:
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature,
- \( M \) is the molar mass of the gas.

### Step 2: Identify the Relationship Between r.m.s. Velocity and Molar Mass
From the formula, we can see that the r.m.s. velocity is inversely proportional to the square root of the molar mass. This means that:
- A lower molar mass results in a higher r.m.s. velocity.
- A higher molar mass results in a lower r.m.s. velocity.

### Step 3: Calculate the Molar Mass of Each Gas
Now, we will calculate the molar mass of each gas:
- **H₂**: Molar mass = \( 1 \times 2 = 2 \) g/mol
- **N₂**: Molar mass = \( 14 \times 2 = 28 \) g/mol
- **CO₂**: Molar mass = \( 12 + (16 \times 2) = 44 \) g/mol
- **O₂**: Molar mass = \( 16 \times 2 = 32 \) g/mol

### Step 4: Arrange the Gases in Increasing Order of Molar Mass
Now we can arrange the gases based on their molar masses:
- H₂: 2 g/mol
- N₂: 28 g/mol
- O₂: 32 g/mol
- CO₂: 44 g/mol

The increasing order of molar mass is:
\[ H_2 < N_2 < O_2 < CO_2 \]

### Step 5: Determine the Decreasing Order of r.m.s. Velocity
Since the r.m.s. velocity is inversely proportional to the square root of the molar mass, the decreasing order of r.m.s. velocity will be the opposite of the increasing order of molar mass:
- Highest r.m.s. velocity: H₂
- Next: N₂
- Next: O₂
- Lowest r.m.s. velocity: CO₂

Thus, the correct decreasing order of r.m.s. velocity is:
\[ H_2 > N_2 > O_2 > CO_2 \]

### Conclusion
The correct decreasing order of r.m.s. velocity at the same temperature for H₂, N₂, CO₂, and O₂ is:
\[ H_2 > N_2 > O_2 > CO_2 \]