The root mean square speeds at STP for the gases `H_(2), N_(2),O_2` and HBr are in the order :

Options

A. `H_2 lt N_2 lt O_2 lt HBr`
B. `HBr gt O_2 gt N_2 gt H_2`
C. `H_2 gt N_2 gt O_2 gt HBr`
D. `HBr lt O_2 lt H_2 gt N_2`

Explanation

To determine the order of root mean square speeds (VRMS) for the gases H2, N2, O2, and HBr at standard temperature and pressure (STP), we can follow these steps:

### Step 1: Understand the formula for root mean square speed
The root mean square speed (VRMS) 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: Recognize that temperature is constant at STP
At standard temperature and pressure, the temperature \( T \) is the same for all gases. Therefore, the root mean square speed is inversely proportional to the square root of the molar mass:
\[ V_{RMS} \propto \frac{1}{\sqrt{M}} \]

### Step 3: Determine the molar masses of the gases
Next, we need to find the molar masses of the gases involved:
- Molar mass of \( H_2 \) (Hydrogen) = 2 g/mol
- Molar mass of \( N_2 \) (Nitrogen) = 28 g/mol
- Molar mass of \( O_2 \) (Oxygen) = 32 g/mol
- Molar mass of \( HBr \) (Hydrogen bromide) = 81 g/mol

### Step 4: Order the gases by their molar masses
Now we can order the gases from the lowest to the highest molar mass:
1. \( H_2 \) = 2 g/mol
2. \( N_2 \) = 28 g/mol
3. \( O_2 \) = 32 g/mol
4. \( HBr \) = 81 g/mol

### Step 5: Determine the order of root mean square speeds
Since \( V_{RMS} \) is inversely proportional to the square root of the molar mass, the order of root mean square speeds will be the opposite of the order of molar masses:
- Highest speed: \( H_2 \)
- Next: \( N_2 \)
- Then: \( O_2 \)
- Lowest speed: \( HBr \)

Thus, the order of root mean square speeds at STP is:
\[ V_{RMS} \text{ order: } H_2 > N_2 > O_2 > HBr \]

### Final Answer
The root mean square speeds at STP for the gases \( H_2, N_2, O_2, \) and \( HBr \) are in the order:
\[ H_2 > N_2 > O_2 > HBr \]

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