The ratio of the specific heats of a gas is `(C_(p))/(C_(v))=1.66` then the gas may be

Options

A. `CO_(2)`
B. `He`
C. `H_(2)`
D. `NO_(2)`

Answer

To determine the type of gas based on the given ratio of specific heats \( \frac{C_p}{C_v} = 1.66 \), we can follow these steps:

### Step 1: Understand the relationship between specific heats
The ratio of specific heats \( \gamma \) (gamma) is defined as:
\[ \gamma = \frac{C_p}{C_v} \]
where \( C_p \) is the specific heat at constant pressure and \( C_v \) is the specific heat at constant volume.

### Step 2: Use the relationship with degrees of freedom
For an ideal gas, the relationship between \( \gamma \) and the degrees of freedom \( n \) is given by:
\[ \gamma = 1 + \frac{2}{n} \]
where \( n \) is the number of degrees of freedom of the gas molecules.

### Step 3: Substitute the given value of \( \gamma \)
Substituting the given value of \( \gamma \):
\[ 1.66 = 1 + \frac{2}{n} \]

### Step 4: Rearrange the equation to solve for \( n \)
Subtract 1 from both sides:
\[ 0.66 = \frac{2}{n} \]
Now, cross-multiply to solve for \( n \):
\[ n \cdot 0.66 = 2 \]
\[ n = \frac{2}{0.66} \approx 3.03 \]

### Step 5: Interpret the value of \( n \)
The calculated value of \( n \approx 3.03 \) suggests that the gas has approximately 3 degrees of freedom.

### Step 6: Identify the type of gas
For a monatomic gas, \( n = 3 \) (3 translational degrees of freedom). For a diatomic gas, \( n = 5 \) (3 translational + 2 rotational). Since \( n \approx 3.03 \), it indicates that the gas is likely to be a monatomic gas.

### Conclusion
Thus, the gas with a ratio of specific heats \( \frac{C_p}{C_v} = 1.66 \) is likely a monatomic gas.

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