The pressure and density of a diatomic gas `(gamma=7//5)` change adiabatically from (p,d) to `(p^(1),d^(2))`. If `(d^(1))/(d)=32`, then `(P^(1))/(P)` should be

Options

A. 1/128
B. 32
C. 128
D. none of the above

Explanation

To solve the problem, we need to use the relationship between pressure and density for an adiabatic process involving a diatomic gas. The relevant equations are derived from the principles of thermodynamics and the kinetic theory of gases.

### Step-by-Step Solution:

1. **Understand the relationship between pressure and density in an adiabatic process:**
For an adiabatic process, the relationship between pressure (P) and density (d) for a gas is given by:
\[ P \propto d^{\gamma} \]
where \(\gamma\) is the adiabatic index. For a diatomic gas, \(\gamma = \frac{7}{5}\).

2. **Set up the initial and final conditions:**
Let the initial pressure and density be \(P\) and \(d\), respectively. The final pressure and density after the adiabatic change are \(P^1\) and \(d^2\). We are given that:
\[ \frac{d^1}{d} = 32 \]
This means that the final density \(d^1\) is 32 times the initial density \(d\):
\[ d^1 = 32d \]

3. **Apply the adiabatic relation:**
Using the relationship \(P \propto d^{\gamma}\), we can express the initial and final states as:
\[ P \propto d^{\frac{7}{5}} \quad \text{and} \quad P^1 \propto (d^1)^{\frac{7}{5}} \]

4. **Relate the pressures using the densities:**
From the proportionality, we can write:
\[ \frac{P^1}{P} = \left(\frac{d^1}{d}\right)^{\frac{7}{5}} \]
Substituting \(d^1 = 32d\):
\[ \frac{P^1}{P} = \left(32\right)^{\frac{7}{5}} \]

5. **Calculate \(32^{\frac{7}{5}}\):**
We can simplify this expression:
\[ 32 = 2^5 \Rightarrow 32^{\frac{7}{5}} = (2^5)^{\frac{7}{5}} = 2^7 = 128 \]

6. **Final result:**
Therefore, the ratio of the final pressure to the initial pressure is:
\[ \frac{P^1}{P} = 128 \]

### Conclusion:
The final answer is:
\[ \frac{P^1}{P} = 128 \]