The root mean square velocity of hydrogen at S.T.P.is

Options

A. `1.8 xx 10^5 cm sec^(-1)`
B. `3.6 xx 10^5 cm sec^(-1)`
C. `1.8 xx 10^10 cm sec^(-1)`
D. `1.8 xx 10^8 cm sec^(-1)`

Answer

To find the root mean square (RMS) velocity of hydrogen at standard temperature and pressure (STP), we can follow these steps:

### Step 1: Understand the Formula
The formula for the root mean square velocity (RMS velocity) is given by:

\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]

where:
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature in Kelvin,
- \( M \) is the molar mass of the gas in kilograms.

### Step 2: Identify the Values
At STP, the temperature is:
- \( T = 0^\circ C = 273 \, K \)

The molar mass of hydrogen gas (H₂) is:
- \( M = 2 \, \text{g/mol} = 2 \times 10^{-3} \, \text{kg/mol} \)

The universal gas constant is:
- \( R = 8.314 \, \text{J/(mol K)} \)

### Step 3: Substitute the Values into the Formula
Now, we can substitute the values into the RMS velocity formula:

\[ v_{rms} = \sqrt{\frac{3 \times 8.314 \, \text{J/(mol K)} \times 273 \, K}{2 \times 10^{-3} \, \text{kg/mol}}} \]

### Step 4: Calculate the Numerator
First, calculate the numerator:

\[ 3 \times 8.314 \times 273 = 6818.382 \, \text{J} \]

### Step 5: Calculate the Full Expression
Now, substitute this back into the equation:

\[ v_{rms} = \sqrt{\frac{6818.382}{2 \times 10^{-3}}} \]

Calculating the division:

\[ \frac{6818.382}{2 \times 10^{-3}} = 3409191 \, \text{m}^2/\text{s}^2 \]

### Step 6: Take the Square Root
Now take the square root:

\[ v_{rms} = \sqrt{3409191} \approx 1848.4 \, \text{m/s} \]

### Step 7: Convert to Centimeters per Second
Since the final answer is required in centimeters per second, we convert meters per second to centimeters per second:

\[ v_{rms} = 1848.4 \, \text{m/s} \times 100 = 184840 \, \text{cm/s} \approx 1.85 \times 10^5 \, \text{cm/s} \]

### Final Answer
Thus, the root mean square velocity of hydrogen at STP is approximately:

\[ v_{rms} \approx 1.8 \times 10^5 \, \text{cm/s} \]