`C_(p)//C_(v)`, ratio for noble gases is ________.

Explanation

To find the ratio \( \frac{C_p}{C_v} \) for noble gases, we can follow these steps:

### Step 1: Understand the definitions
- \( C_v \): Molar heat capacity at constant volume.
- \( C_p \): Molar heat capacity at constant pressure.

### Step 2: Identify the degree of freedom for noble gases
- Noble gases (like helium, neon, argon, krypton, xenon, and radon) are monatomic gases.
- For monatomic gases, the degree of freedom \( F \) is equal to 3.

### Step 3: Calculate \( C_v \)
- The formula for \( C_v \) for a monatomic gas is given by:
\[ C_v = \frac{3}{2} R \]
where \( R \) is the universal gas constant.

### Step 4: Use the relationship between \( C_p \) and \( C_v \)
- The relationship between \( C_p \) and \( C_v \) is given by:
\[ C_p - C_v = R \]
- Rearranging this gives:
\[ C_p = C_v + R \]

### Step 5: Substitute \( C_v \) into the equation for \( C_p \)
- Substitute \( C_v = \frac{3}{2} R \) into the equation for \( C_p \):
\[ C_p = \frac{3}{2} R + R = \frac{3}{2} R + \frac{2}{2} R = \frac{5}{2} R \]

### Step 6: Calculate the ratio \( \frac{C_p}{C_v} \)
- Now we can find the ratio:
\[ \frac{C_p}{C_v} = \frac{\frac{5}{2} R}{\frac{3}{2} R} \]
- Simplifying this gives:
\[ \frac{C_p}{C_v} = \frac{5}{3} \]

### Step 7: Final calculation
- The ratio simplifies to:
\[ \frac{C_p}{C_v} = \frac{5}{3} \approx 1.66 \]

### Conclusion
Thus, the ratio \( \frac{C_p}{C_v} \) for noble gases is approximately **1.66**.

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