Calculate the rms speed of an ideal diatomic gas having molecular weight 32 gm/mol at `0^@ C`. If the specific heats at constant pressure and volume are respectively 8.3 J `mol^(-1) K^(-1) and 6.34` cal `mol^(-1) K^(-1)` respectively.

Answer

To calculate the root mean square (RMS) speed of an ideal diatomic gas, we can use 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 molecular weight in kg/mol.

### Step 1: Convert the molecular weight from grams to kilograms
Given the molecular weight \( M = 32 \, \text{g/mol} \), we need to convert it to kg/mol:

\[ M = \frac{32 \, \text{g/mol}}{1000} = 0.032 \, \text{kg/mol} \]

**Hint**: Remember to convert grams to kilograms by dividing by 1000.

### Step 2: Convert the temperature from Celsius to Kelvin
The temperature is given as \( 0^\circ C \). To convert this to Kelvin:

\[ T = 0 + 273 = 273 \, \text{K} \]

**Hint**: To convert Celsius to Kelvin, add 273.15 (for simplicity, we can use 273).

### Step 3: Use the value of the universal gas constant \( R \)
The value of the universal gas constant \( R \) in SI units is:

\[ R = 8.314 \, \text{J/(mol K)} \]

**Hint**: Ensure you are using the correct units for \( R \) (Joules per mole per Kelvin).

### Step 4: Substitute the values into the RMS speed formula
Now we can substitute the values of \( R \), \( T \), and \( M \) into the RMS speed formula:

\[ v_{rms} = \sqrt{\frac{3 \times 8.314 \, \text{J/(mol K)} \times 273 \, \text{K}}{0.032 \, \text{kg/mol}}} \]

### Step 5: Calculate the numerator
Calculating the numerator:

\[ 3 \times 8.314 \times 273 = 6816.342 \, \text{J} \]

### Step 6: Divide by the molecular weight
Now divide by the molecular weight:

\[ \frac{6816.342}{0.032} = 213,011.31 \, \text{m}^2/\text{s}^2 \]

### Step 7: Take the square root
Finally, take the square root to find \( v_{rms} \):

\[ v_{rms} = \sqrt{213,011.31} \approx 461.28 \, \text{m/s} \]

### Final Answer
The RMS speed of the ideal diatomic gas is approximately:

\[ v_{rms} \approx 461.28 \, \text{m/s} \]

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