Calculate the number of degrees of freedom in `10 cm^(3)` of `O_(2)` at N.T.P.

Answer

To calculate the number of degrees of freedom in 10 cm³ of O₂ at N.T.P (Normal Temperature and Pressure), we can follow these steps:

### Step 1: Determine the number of molecules in 10 cm³ of O₂
At N.T.P, 1 mole of any gas occupies a volume of 22,400 cm³ and contains Avogadro's number of molecules, which is approximately \(6.022 \times 10^{23}\) molecules.

To find the number of molecules in 10 cm³ of O₂, we can use the following formula:

\[ \text{Number of molecules} = \left(\frac{\text{Volume of gas}}{\text{Molar volume}}\right) \times \text{Avogadro's number} \]

Substituting the values:

\[ \text{Number of molecules} = \left(\frac{10 \, \text{cm}^3}{22,400 \, \text{cm}^3}\right) \times 6.022 \times 10^{23} \]

### Step 2: Calculate the number of molecules
Now, we calculate the number of molecules:

\[ \text{Number of molecules} = \left(\frac{10}{22,400}\right) \times 6.022 \times 10^{23} \]

Calculating the fraction:

\[ \frac{10}{22,400} \approx 0.0004464 \]

Now, multiplying by Avogadro's number:

\[ \text{Number of molecules} \approx 0.0004464 \times 6.022 \times 10^{23} \approx 2.688 \times 10^{20} \text{ molecules} \]

### Step 3: Determine the degrees of freedom per molecule
For diatomic gases like O₂, each molecule has 5 degrees of freedom (3 translational and 2 rotational).

### Step 4: Calculate the total degrees of freedom
To find the total degrees of freedom for all the molecules, we multiply the number of molecules by the degrees of freedom per molecule:

\[ \text{Total degrees of freedom} = \text{Number of molecules} \times \text{Degrees of freedom per molecule} \]

Substituting the values:

\[ \text{Total degrees of freedom} = (2.688 \times 10^{20}) \times 5 \]

Calculating this gives:

\[ \text{Total degrees of freedom} \approx 1.344 \times 10^{21} \]

### Final Answer
The total number of degrees of freedom in 10 cm³ of O₂ at N.T.P is approximately \(1.34 \times 10^{21}\).

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