The relation between internal energy `U`, pressure `P` and volume `V` of a gas in an adiabatic process is
`U = a + bPV` where a and b are constant . What is the effective value of adiabatic constant `gamma`.

Options

A. `(a)/(b)`
B. `(b + 1)/(b)`
C. `(a + 1)/(a)`
D. `(b)/(a)`

Explanation

To find the effective value of the adiabatic constant \(\gamma\) for the given relation \(U = a + bPV\), we can follow these steps:

### Step 1: Understand the Adiabatic Process
In an adiabatic process, the relationship between pressure \(P\) and volume \(V\) is given by:
\[ PV^\gamma = \text{constant} \]
where \(\gamma\) is the adiabatic constant.

### Step 2: Use the First Law of Thermodynamics
According to the first law of thermodynamics:
\[ Q = \Delta U + W \]
For an adiabatic process, the heat exchange \(Q\) is zero. Therefore:
\[ 0 = \Delta U + W \]
This implies:
\[ \Delta U = -W \]

### Step 3: Express Work Done \(W\)
The work done \(W\) in an adiabatic process can be expressed as:
\[ W = P dV \]
Thus, we can rewrite the change in internal energy as:
\[ \Delta U = -P dV \]

### Step 4: Differentiate the Internal Energy Expression
Given the internal energy relation:
\[ U = a + bPV \]
Taking the differential, we have:
\[ dU = b(P dV + V dP) \]

### Step 5: Set the Two Expressions for \(\Delta U\) Equal
From the first law, we have:
\[ dU = -P dV \]
Equating the two expressions for \(dU\):
\[ b(P dV + V dP) = -P dV \]

### Step 6: Rearrange the Equation
Rearranging gives:
\[ bP dV + bV dP + P dV = 0 \]
Factoring out \(dV\):
\[ (bP + P) dV + bV dP = 0 \]
This simplifies to:
\[ (b + 1)P dV + bV dP = 0 \]

### Step 7: Separate Variables
Rearranging gives:
\[ (b + 1)P dV = -bV dP \]
Dividing both sides by \(PV\):
\[ \frac{dV}{V} = -\frac{b}{b + 1} \frac{dP}{P} \]

### Step 8: Integrate Both Sides
Integrating both sides:
\[ \int \frac{dV}{V} = -\frac{b}{b + 1} \int \frac{dP}{P} \]
This results in:
\[ \ln V = -\frac{b}{b + 1} \ln P + C \]
Exponentiating gives:
\[ V = C' P^{-\frac{b}{b + 1}} \]

### Step 9: Relate to the Adiabatic Process
The relation \(PV^\gamma = \text{constant}\) can be rewritten as:
\[ P V^{\frac{b + 1}{b}} = \text{constant} \]
Thus, we can identify:
\[ \gamma = \frac{b + 1}{b} \]

### Step 10: Final Expression for \(\gamma\)
The effective value of the adiabatic constant \(\gamma\) is:
\[ \gamma = 1 + \frac{1}{b} \]

### Conclusion
Thus, the effective value of the adiabatic constant \(\gamma\) is:
\[ \gamma = 1 + \frac{1}{b} \]

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