The poisson's ratio for `O_(2)` is `1.4`. Which of the following are correct for `O_(2)`?

Options

A. `C_(v) = 5 cal//mol K`
B. `c_(v) or s_(v) = 0.45 cal//mol K`
C. `C_(p)=(R gamma)/(gamma-1)`
D. `C_(v) = (R )/((gamma-1))`

Explanation

To solve the problem regarding the Poisson's ratio for \( O_2 \) (oxygen), which is given as \( \gamma = 1.4 \), we need to analyze the implications of this value on the specific heats \( C_p \) and \( C_v \) of the gas.

### Step-by-Step Solution:

1. **Understanding Poisson's Ratio**:
The Poisson's ratio (\( \gamma \)) is defined as the ratio of the specific heat at constant pressure (\( C_p \)) to the specific heat at constant volume (\( C_v \)):
\[ \gamma = \frac{C_p}{C_v} \]

2. **Using the Relation Between \( C_p \) and \( C_v \)**:
We also know the relationship between \( C_p \) and \( C_v \):
\[ C_p - C_v = R \]
where \( R \) is the ideal gas constant.

3. **Expressing \( C_p \) in Terms of \( C_v \)**:
Rearranging the equation gives us:
\[ C_p = C_v + R \]
Substituting this into the equation for \( \gamma \):
\[ \gamma = \frac{C_v + R}{C_v} \]

4. **Simplifying the Equation**:
This can be simplified to:
\[ \gamma = 1 + \frac{R}{C_v} \]
Rearranging gives:
\[ \frac{R}{C_v} = \gamma - 1 \]
Therefore:
\[ C_v = \frac{R}{\gamma - 1} \]

5. **Calculating \( C_v \)**:
Given \( \gamma = 1.4 \) and using \( R = 2 \) calorie, we can calculate \( C_v \):
\[ C_v = \frac{2}{1.4 - 1} = \frac{2}{0.4} = 5 \text{ calorie per mole Kelvin} \]

6. **Finding \( C_p \)**:
Now, we can find \( C_p \):
\[ C_p = C_v + R = 5 + 2 = 7 \text{ calorie per mole Kelvin} \]

7. **Verifying the Options**:
- **Option A**: \( C_v = 5 \text{ calorie per mole Kelvin} \) - **Correct**
- **Option B**: \( C_v = 0.45 \text{ calorie per mole Kelvin} \) - **Incorrect**
- **Option C**: \( C_p = \frac{R \gamma}{\gamma - 1} \) - **Correct** (since \( C_p = 7 \text{ calorie per mole Kelvin} \))
- **Option D**: \( C_v = R(\gamma - 1) \) - **Incorrect** (should be \( C_v = \frac{R}{\gamma - 1} \))

### Conclusion:
The correct options are A and C.