A certain ideal gas undergoes a polytropic process `PV^(n)` = constant such that the molar specific heat during the process is negative. If the ratio of the specific heats of the gas be `gamma`, then the range of values of `n` will be

Options

A. (a)`0 lt n lt gamma`
B. (b)`1 lt n lt gamma`
C. (c)`n = gamma`
D. (d)`n gt gamma`

Explanation

To solve the problem regarding the range of values of \( n \) for an ideal gas undergoing a polytropic process with negative molar specific heat, we can follow these steps:

### Step 1: Understand the Polytropic Process
In a polytropic process, the relationship between pressure \( P \), volume \( V \), and the polytropic index \( n \) is given by:
\[ PV^n = \text{constant} \]

### Step 2: Molar Specific Heat in a Polytropic Process
The molar specific heat \( C \) during a polytropic process can be expressed as:
\[ C = C_v + \frac{R}{1 - n} \]
where \( C_v \) is the molar specific heat at constant volume, and \( R \) is the universal gas constant.

### Step 3: Condition for Negative Molar Specific Heat
For the molar specific heat \( C \) to be negative, we set the equation:
\[ C < 0 \]
Substituting the expression for \( C \):
\[ C_v + \frac{R}{1 - n} < 0 \]

### Step 4: Rearranging the Inequality
Rearranging the inequality gives us:
\[ \frac{R}{1 - n} < -C_v \]
This implies:
\[ 1 - n < -\frac{R}{C_v} \]
or
\[ n > 1 + \frac{R}{C_v} \]

### Step 5: Relating \( C_v \) and \( \gamma \)
Using the relationship between specific heats, we know:
\[ \gamma = \frac{C_p}{C_v} = 1 + \frac{R}{C_v} \]
From this, we can express \( C_v \) in terms of \( \gamma \):
\[ C_v = \frac{R}{\gamma - 1} \]

### Step 6: Substitute \( C_v \) in the Inequality
Substituting \( C_v \) into the inequality:
\[ n > 1 + \frac{R}{\frac{R}{\gamma - 1}} = 1 + (\gamma - 1) = \gamma \]

### Step 7: Final Range for \( n \)
Thus, we have:
\[ n > \gamma \]

### Step 8: Conclusion
Since \( n \) must also be less than \( 1 \) for the process to be polytropic with negative specific heat, we conclude:
\[ 1 < n < \gamma \]

### Final Answer
The range of values for \( n \) is:
\[ 1 < n < \gamma \]