At room temperature `(300K)`, the rms speed of the molecules of a certain diatomic gas is found to be `1930 m//s`. Can you gusess name of the gas? Find the temperature at which the rms speed is double of the speed in part one `(R = 25//3 J// mol -K)`

Explanation

To solve the problem, we will follow these steps:

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

### Step 2: Rearrange the formula to find the molar mass
Rearranging the formula to solve for \( M \):
\[ M = \frac{3RT}{v_{rms}^2} \]

### Step 3: Substitute the known values
Given:
- \( R = \frac{25}{3} \, \text{J/mol-K} \)
- \( T = 300 \, \text{K} \)
- \( v_{rms} = 1930 \, \text{m/s} \)

Substituting these values into the equation:
\[ M = \frac{3 \cdot \frac{25}{3} \cdot 300}{(1930)^2} \]

### Step 4: Calculate the molar mass
Calculating the numerator:
\[ 3 \cdot \frac{25}{3} \cdot 300 = 25 \cdot 300 = 7500 \]
Calculating the denominator:
\[ (1930)^2 = 3724900 \]
Now substituting back:
\[ M = \frac{7500}{3724900} \approx 0.002013 \, \text{kg/mol} = 2.013 \, \text{g/mol} \]

### Step 5: Identify the gas
The molar mass of approximately \( 2.013 \, \text{g/mol} \) corresponds to hydrogen gas (H₂), which has a molar mass of about \( 2 \, \text{g/mol} \).

### Step 6: Find the temperature for double the RMS speed
We need to find the temperature at which the RMS speed is double:
\[ v_{rms, final} = 2 \cdot v_{rms, initial} \]
From the relationship:
\[ v_{rms} \propto \sqrt{T} \]
This implies:
\[ \frac{v_{rms, final}}{v_{rms, initial}} = \sqrt{\frac{T_{final}}{T_{initial}}} \]
Substituting the known values:
\[ 2 = \sqrt{\frac{T_{final}}{300}} \]
Squaring both sides:
\[ 4 = \frac{T_{final}}{300} \]
Thus:
\[ T_{final} = 4 \cdot 300 = 1200 \, \text{K} \]

### Final Answer
The gas is hydrogen (H₂), and the temperature at which the RMS speed is double the initial speed is \( 1200 \, \text{K} \).

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