when an ideal gas with pressure p and volume V is compressed Isothermally to one - fourth of its volume, is pressure is `P_1` when the same gas is compressed polytropically according to the equation `PV^(1.5)` contents to one - fourth of its initial volume, the pressure is `P_2` the ratio `P_1/P_2` is

Options

A. `1/2`
B. `1/(2^(1.5))`
C. 2
D. `2^(1.5)`

Answer

To solve the problem, we need to find the ratio \( \frac{P_1}{P_2} \) where \( P_1 \) is the pressure after isothermal compression to one-fourth of the volume and \( P_2 \) is the pressure after polytropic compression according to the equation \( PV^{1.5} = \text{constant} \).

### Step 1: Calculate \( P_1 \) for Isothermal Compression

In an isothermal process, the relationship between pressure and volume is given by:

\[ PV = \text{constant} \]

Let the initial pressure be \( P \) and the initial volume be \( V \). After isothermal compression to one-fourth of the volume, the final volume \( V_1 \) becomes:

\[ V_1 = \frac{V}{4} \]

Using the isothermal condition:

\[ P \cdot V = P_1 \cdot V_1 \]

Substituting the values:

\[ P \cdot V = P_1 \cdot \frac{V}{4} \]

Now, we can cancel \( V \) from both sides:

\[ P = P_1 \cdot \frac{1}{4} \]

Rearranging gives:

\[ P_1 = 4P \]

### Step 2: Calculate \( P_2 \) for Polytropic Compression

For the polytropic process, we have the relationship:

\[ PV^{1.5} = \text{constant} \]

Using the initial conditions, we have:

\[ P \cdot V^{1.5} = P_2 \cdot V_2^{1.5} \]

After compression, the final volume \( V_2 \) is:

\[ V_2 = \frac{V}{4} \]

Substituting this into the equation gives:

\[ P \cdot V^{1.5} = P_2 \cdot \left(\frac{V}{4}\right)^{1.5} \]

Calculating \( \left(\frac{V}{4}\right)^{1.5} \):

\[ \left(\frac{V}{4}\right)^{1.5} = \frac{V^{1.5}}{4^{1.5}} = \frac{V^{1.5}}{8} \]

Now substituting back into the equation:

\[ P \cdot V^{1.5} = P_2 \cdot \frac{V^{1.5}}{8} \]

Canceling \( V^{1.5} \) from both sides:

\[ P = P_2 \cdot \frac{1}{8} \]

Rearranging gives:

\[ P_2 = 8P \]

### Step 3: Calculate the Ratio \( \frac{P_1}{P_2} \)

Now we can find the ratio:

\[ \frac{P_1}{P_2} = \frac{4P}{8P} = \frac{4}{8} = \frac{1}{2} \]

### Final Answer

Thus, the ratio \( \frac{P_1}{P_2} \) is:

\[ \frac{P_1}{P_2} = \frac{1}{2} \]

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