An ideal gas `(C_p / C_v = gamma)` is taken through a process in which the pressure and volume vary as `(p = aV^(b))`. Find the value of b for which the specific heat capacity in the process is zero.

Explanation

To solve the problem, we need to find the value of \( b \) for which the specific heat capacity in the process is zero. We start with the given relationship between pressure \( p \) and volume \( V \):

\[ p = a V^b \]

### Step 1: Understand the condition for zero specific heat capacity
The specific heat capacity \( C \) in a process is defined as:

\[ C = \frac{dq}{dT} \]

where \( dq \) is the heat added to the system and \( dT \) is the change in temperature. If \( C = 0 \), it implies that no heat is exchanged during the process, which characterizes an adiabatic process.

### Step 2: Identify the relationship for an adiabatic process
For an adiabatic process involving an ideal gas, the relationship between pressure and volume is given by:

\[ PV^\gamma = \text{constant} \]

where \( \gamma = \frac{C_p}{C_v} \).

### Step 3: Rewrite the adiabatic condition
We can express this relationship in terms of pressure \( p \):

\[ p = \frac{k}{V^\gamma} \]

where \( k \) is a constant.

### Step 4: Compare the two expressions for pressure
We have two expressions for pressure:

1. From the problem: \( p = a V^b \)
2. From the adiabatic condition: \( p = \frac{k}{V^\gamma} \)

### Step 5: Set the two expressions equal to each other
Setting the two expressions for \( p \) equal gives:

\[ a V^b = \frac{k}{V^\gamma} \]

### Step 6: Rearranging the equation
Rearranging this equation, we get:

\[ a V^{b + \gamma} = k \]

### Step 7: Analyze the relationship
For this equation to hold for all \( V \), the exponent of \( V \) must be zero (since \( k \) is a constant). Therefore, we set:

\[ b + \gamma = 0 \]

### Step 8: Solve for \( b \)
From the equation \( b + \gamma = 0 \), we can solve for \( b \):

\[ b = -\gamma \]

### Final Answer
Thus, the value of \( b \) for which the specific heat capacity in the process is zero is:

\[ \boxed{-\gamma} \]