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 constants. What is the effective value of adiabatic constant `gamma` ?

Options

A. 1. `a/b`
B. 2. `(b+1)/b`
C. 3. `(a+1)/a`
D. 4. `b/a`

Answer

To find the effective value of the adiabatic constant \( \gamma \) given the relation between internal energy \( U \), pressure \( P \), and volume \( V \) of a gas in an adiabatic process as \( U = a + bPV \), where \( a \) and \( b \) are constants, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Adiabatic Process**:
In an adiabatic process, there is no heat exchange, which means \( \delta Q = 0 \). According to the first law of thermodynamics, this implies:
\[ dU = -dW \]

2. **Express Change in Internal Energy**:
The change in internal energy \( dU \) for an ideal gas can be expressed as:
\[ dU = nC_v dT \]
where \( C_v \) is the molar heat capacity at constant volume.

3. **Relate \( C_v \) to \( \gamma \)**:
We know that:
\[ C_v = \frac{R}{\gamma - 1} \]
where \( R \) is the universal gas constant.

4. **Substitute \( C_v \) into \( dU \)**:
Substituting \( C_v \) into the expression for \( dU \):
\[ dU = n \frac{R}{\gamma - 1} dT \]

5. **Integrate to Find Internal Energy**:
Integrating \( dU \) gives us:
\[ U = n \frac{R}{\gamma - 1} T + C \]
where \( C \) is the integration constant.

6. **Use the Ideal Gas Law**:
From the ideal gas law, we have:
\[ PV = nRT \implies nRT = PV \]
Substituting this into the expression for \( U \):
\[ U = \frac{PV}{\gamma - 1} + C \]

7. **Compare with Given Relation**:
The problem states that:
\[ U = a + bPV \]
By comparing both expressions for \( U \):
\[ \frac{PV}{\gamma - 1} + C = a + bPV \]
This implies:
- The coefficient of \( PV \) gives us:
\[ \frac{1}{\gamma - 1} = b \]
- The constant terms give us:
\[ C = a \]

8. **Solve for \( \gamma \)**:
Rearranging \( \frac{1}{\gamma - 1} = b \):
\[ \gamma - 1 = \frac{1}{b} \implies \gamma = \frac{1}{b} + 1 \]
Thus, we can express \( \gamma \) as:
\[ \gamma = \frac{b + 1}{b} \]

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