The root mean square speed of hydrogen molecules of an ideal hydrogen gas kept in a gas chamber at `0^(@)C` is 3180 m/s. The pressure on the hydrogen gas is ………..
(Density of hydrogen gas is `8.99xx10^(-2)kg//m^(3)`, 1 atmosphere= `1.01xx10^(5)N/m^(2)`

Answer

To find the pressure of the hydrogen gas, we can use the relationship between the root mean square (RMS) speed, density, and pressure of the gas. Here are the steps to solve the problem:

### Step 1: Write down the formula for RMS speed
The root mean square speed (v_rms) of gas molecules is given by the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where:
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature in Kelvin,
- \( M \) is the molar mass of the gas.

### Step 2: Relate pressure, density, and temperature
We can also relate pressure (P), density (D), and temperature (T) using the equation:
\[ P = D \cdot \frac{RT}{M} \]
From this, we can express \( \frac{RT}{M} \) as:
\[ \frac{RT}{M} = \frac{P}{D} \]

### Step 3: Substitute into the RMS speed equation
Substituting \( \frac{RT}{M} \) into the RMS speed formula, we get:
\[ v_{rms} = \sqrt{\frac{3P}{D}} \]

### Step 4: Rearrange the equation to solve for pressure
Squaring both sides gives:
\[ v_{rms}^2 = \frac{3P}{D} \]
Rearranging for pressure (P), we have:
\[ P = \frac{D \cdot v_{rms}^2}{3} \]

### Step 5: Substitute the given values
Given:
- \( v_{rms} = 3180 \, \text{m/s} \)
- \( D = 8.99 \times 10^{-2} \, \text{kg/m}^3 \)

Substituting these values into the equation:
\[ P = \frac{(8.99 \times 10^{-2}) \cdot (3180)^2}{3} \]

### Step 6: Calculate \( v_{rms}^2 \)
Calculating \( v_{rms}^2 \):
\[ v_{rms}^2 = 3180^2 = 10112400 \, \text{m}^2/\text{s}^2 \]

### Step 7: Calculate pressure
Now substituting back:
\[ P = \frac{(8.99 \times 10^{-2}) \cdot (10112400)}{3} \]
Calculating this gives:
\[ P = \frac{908,901.6}{3} \approx 302967.2 \, \text{N/m}^2 \]

### Step 8: Convert pressure to atmospheres
To convert the pressure from N/m² to atmospheres:
\[ P_{atm} = \frac{P}{1.01 \times 10^5} = \frac{302967.2}{1.01 \times 10^5} \approx 3 \, \text{atm} \]

### Final Answer
The pressure on the hydrogen gas is approximately **3 atm**.

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