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 find the value of \( b \) for which the specific heat capacity in the process is zero, we can follow these steps:

### Step 1: Understand the relationship between heat transfer, internal energy, and work done
From the first law of thermodynamics, we know:
\[ dQ = dU + dW \]
where \( dQ \) is the heat added to the system, \( dU \) is the change in internal energy, and \( dW \) is the work done by the system.

### Step 2: Express heat transfer and internal energy for an ideal gas
For an ideal gas, the heat added can be expressed as:
\[ dQ = n C_p dT \]
and the change in internal energy is given by:
\[ dU = n C_v dT \]

### Step 3: Set the condition for specific heat capacity being zero
If the specific heat capacity \( C_B \) in the process is zero, then:
\[ dQ = 0 \]
This implies:
\[ n C_p dT = 0 \]
For this to hold true, either \( n = 0 \) (which is not the case) or \( dT = 0 \). If \( dT = 0 \), it indicates that the temperature does not change, suggesting an isothermal process.

### Step 4: Consider the implications of \( dQ = 0 \)
If \( dQ = 0 \), it indicates that the process is adiabatic. For an adiabatic process, we have the relation:
\[ PV^{\gamma} = \text{constant} \]

### Step 5: Relate the given pressure-volume relationship to the adiabatic condition
Given the relationship \( p = aV^b \), we can express it in terms of the adiabatic condition:
\[ PV^{\gamma} = k \quad \text{(constant)} \]
This means:
\[ pV^{\gamma} = k \]
Comparing this with the given equation \( p = aV^b \), we can rewrite it as:
\[ p = \frac{k}{V^{\gamma}} \]
This leads us to:
\[ aV^b = \frac{k}{V^{\gamma}} \]

### Step 6: Equate the powers of \( V \)
From the equation \( aV^b = \frac{k}{V^{\gamma}} \), we can equate the powers of \( V \):
\[ b = -\gamma \]

### Conclusion
Thus, the value of \( b \) for which the specific heat capacity in the process is zero is:
\[ \boxed{-\gamma} \]