Calculate the rms velocity of `CO_(2)` molecules at N.T.P. Given `R = 8.31 K "mole"^(-1)K^(-1)`.

Answer

To calculate the RMS velocity of CO₂ molecules at Normal Temperature and Pressure (NTP), we will follow these steps:

### Step 1: Understand the formula for RMS velocity
The formula for the root mean square (RMS) velocity (\(V_{rms}\)) of gas molecules 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 kg/mol.

### Step 2: Identify the values needed
- The value of \(R\) is given as \(8.31 \, \text{J/(mol K)}\).
- At NTP, the temperature \(T\) is \(20^\circ C\). To convert this to Kelvin:
\[ T = 20 + 273 = 293 \, \text{K} \]

### Step 3: Calculate the molar mass of CO₂
The molar mass of CO₂ can be calculated as follows:
- Carbon (C) has an atomic mass of \(12 \, \text{g/mol}\).
- Oxygen (O) has an atomic mass of \(16 \, \text{g/mol}\), and there are 2 oxygen atoms in CO₂.
\[ \text{Molar mass of CO}_2 = 12 + (2 \times 16) = 12 + 32 = 44 \, \text{g/mol} \]
Now, convert this to kg/mol:
\[ M = 44 \, \text{g/mol} = 0.044 \, \text{kg/mol} \]

### Step 4: Substitute the values into the RMS velocity formula
Now we can substitute \(R\), \(T\), and \(M\) into the RMS velocity formula:
\[ V_{rms} = \sqrt{\frac{3 \times 8.31 \times 293}{0.044}} \]

### Step 5: Calculate the value
First, calculate the numerator:
\[ 3 \times 8.31 \times 293 = 7305.03 \]
Now, divide by \(0.044\):
\[ \frac{7305.03}{0.044} = 166,250.68 \]
Now take the square root:
\[ V_{rms} = \sqrt{166,250.68} \approx 407.65 \, \text{m/s} \]

### Final Answer
The RMS velocity of CO₂ molecules at NTP is approximately:
\[ V_{rms} \approx 407.65 \, \text{m/s} \]

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