During an adiabatic process, the pressure of a gas is found to be proportional to cube of its absolute temperature. The poision's ratio of gas is:

Options

A. `(3)/(2)`
B. `(5)/(3)`
C. `(7)/(2)`
D. `(4)/(3)`

Explanation

To solve the problem, we need to find the Poisson's ratio (γ) of a gas during an adiabatic process where the pressure (P) is proportional to the cube of its absolute temperature (T). Let's go through the steps systematically:

### Step 1: Understand the relationship given in the problem
We know that during an adiabatic process, the pressure of the gas is proportional to the cube of its absolute temperature. This can be expressed mathematically as:
\[ P \propto T^3 \]
This implies:
\[ P = kT^3 \]
where \( k \) is a constant of proportionality.

### Step 2: Use the ideal gas law
From the ideal gas law, we have:
\[ PV = nRT \]
where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles
- \( R \) = universal gas constant
- \( T \) = absolute temperature

We can rearrange this to express \( T \):
\[ T = \frac{PV}{nR} \]

### Step 3: Substitute \( T \) into the pressure equation
Substituting \( T \) from the ideal gas law into the equation \( P = kT^3 \):
\[ P = k\left(\frac{PV}{nR}\right)^3 \]
This expands to:
\[ P = k \frac{P^3 V^3}{(nR)^3} \]

### Step 4: Rearranging the equation
Rearranging this equation gives:
\[ P^3 = \frac{n^3 R^3 P}{k V^3} \]
This simplifies to:
\[ P^2 V^3 = \frac{n^3 R^3}{k} \]
Let’s denote \( \frac{n^3 R^3}{k} \) as a new constant \( K \):
\[ P^2 V^3 = K \]

### Step 5: Relate to the adiabatic process
For an adiabatic process, we have the relation:
\[ PV^\gamma = C \]
where \( C \) is a constant and \( \gamma \) is the Poisson's ratio.

### Step 6: Equate the two expressions
From the previous steps, we have:
- \( P^2 V^3 = K \)
- \( PV^\gamma = C \)

We can equate the two constants since they both represent the same relationship:
\[ P^2 V^3 = PV^\gamma \]

### Step 7: Solve for \( \gamma \)
Dividing both sides by \( P \):
\[ PV^3 = V^\gamma \]
Now, dividing both sides by \( V^3 \):
\[ P = V^{\gamma - 3} \]

### Step 8: Identify the value of \( \gamma \)
For the equation to hold true for all values of \( P \) and \( V \), the exponents must be equal. Therefore:
\[ \gamma - 3 = 0 \]
This leads to:
\[ \gamma = \frac{3}{2} \]

### Conclusion
Thus, the Poisson's ratio (γ) of the gas is:
\[ \gamma = \frac{3}{2} \]