Consider the following statements for diatomic gases, the ratio `C_(p) //C_(v)` is equal to
(1) `1.40` ( lower temperature)
(2) `1.66` ( moderate temperatre)
(3) `1.29` ( higher temperature )
which of the above statements are correct

Options

A. 1,2 and 3
B. 1 and 2
C. 2 and 3
D. 1 and 3

Explanation

To solve the problem regarding the ratio of specific heats \( \frac{C_p}{C_v} \) for diatomic gases at different temperatures, we can follow these steps:

### Step 1: Understand the Definitions
- \( C_p \) is the specific heat at constant pressure.
- \( C_v \) is the specific heat at constant volume.
- For diatomic gases, the values of \( C_p \) and \( C_v \) can be expressed in terms of the gas constant \( R \).

### Step 2: Determine the Values of \( C_p \) and \( C_v \)
- For diatomic gases, the values are:
- \( C_p = \frac{7R}{2} \)
- \( C_v = \frac{5R}{2} \)

### Step 3: Calculate the Ratio \( \frac{C_p}{C_v} \)
- The ratio can be calculated as follows:
\[ \frac{C_p}{C_v} = \frac{\frac{7R}{2}}{\frac{5R}{2}} = \frac{7}{5} = 1.4 \]

### Step 4: Analyze the Effect of Temperature
- The ratio \( \frac{C_p}{C_v} \) for diatomic gases is affected by temperature. As the temperature increases, the degrees of freedom of the gas molecules increase, which can lead to a decrease in the ratio \( \frac{C_p}{C_v} \).

### Step 5: Evaluate the Given Statements
- The statements provided in the question are:
1. \( 1.40 \) (lower temperature)
2. \( 1.66 \) (moderate temperature)
3. \( 1.29 \) (higher temperature)

- From our calculations:
- At lower temperatures, \( \frac{C_p}{C_v} = 1.4 \) is correct.
- At moderate temperatures, \( \frac{C_p}{C_v} = 1.66 \) is not typically observed for diatomic gases.
- At higher temperatures, \( \frac{C_p}{C_v} \) can decrease to \( 1.29 \), which is plausible.

### Conclusion
Based on the analysis:
- The correct statements are:
- (1) \( 1.40 \) (lower temperature)
- (3) \( 1.29 \) (higher temperature)

Thus, the correct options are (1) and (3).