During an experiment, an ideal gas is found to obey a condition `(p^2)/(rho) = "constant"`. (`rho` = density of the gas). The gas is initially at temperature (T), pressure (p) and density `rho`. The gas expands such that density changes to `rho//2`.

Options

A. The pressure of the gas changes to `sqrt(2) p`.
B. The temperature of the gas changes to `sqrt(2) T`
C. The graph of the above process on (p - T) diagram is parabola.
D. The graph of the above process on (p - T) diagram is hyperbola.

Explanation

To solve the problem step by step, we will use the given condition and the ideal gas law to find the relationship between the pressure and temperature of the gas after it expands.

### Step 1: Understand the given condition
We are given that \(\frac{p^2}{\rho} = \text{constant}\), where \(p\) is the pressure and \(\rho\) is the density of the gas.

### Step 2: Set up the initial conditions
Let:
- Initial pressure: \(p_1 = p\)
- Initial density: \(\rho_1 = \rho\)
- Initial temperature: \(T_1 = T\)

### Step 3: Set up the final conditions after expansion
After the gas expands, the density changes to:
\[ \rho_2 = \frac{\rho}{2} \]

### Step 4: Use the condition to find the final pressure
Using the condition \(\frac{p^2}{\rho} = \text{constant}\), we can write:
\[ \frac{p_1^2}{\rho_1} = \frac{p_2^2}{\rho_2} \]
Substituting the known values:
\[ \frac{p^2}{\rho} = \frac{p_2^2}{\frac{\rho}{2}} \]
This simplifies to:
\[ \frac{p^2}{\rho} = \frac{2p_2^2}{\rho} \]
Cancelling \(\rho\) from both sides gives:
\[ p^2 = 2p_2^2 \]
Now, solving for \(p_2\):
\[ p_2^2 = \frac{p^2}{2} \implies p_2 = \sqrt{2}p \]

### Step 5: Relate pressure and temperature using the ideal gas law
From the ideal gas law, we know:
\[ PV = nRT \quad \text{or} \quad PM = \rho RT \]
Rearranging gives:
\[ \rho = \frac{PM}{RT} \]

### Step 6: Substitute \(\rho\) into the condition
Now substituting \(\rho\) into the condition \(\frac{p^2}{\rho}\):
\[ \frac{p^2}{\frac{PM}{RT}} = \text{constant} \]
This implies:
\[ p^2 \cdot \frac{RT}{PM} = \text{constant} \]
Thus, we can express this as:
\[ p^2T = \text{constant} \cdot M/R \]

### Step 7: Establish the relationship between \(p\) and \(T\)
From the above equation, we can see that:
\[ p^2T = \text{constant} \]
This indicates a hyperbolic relationship between \(p\) and \(T\).

### Step 8: Find the final temperature
Using the relationship \(P_1T_1 = P_2T_2\):
\[ p \cdot T = \sqrt{2}p \cdot T_2 \]
Cancelling \(p\) from both sides gives:
\[ T = \sqrt{2} T_2 \implies T_2 = \frac{T}{\sqrt{2}} = \sqrt{2}T \]

### Conclusion
The final temperature of the gas after expansion is:
\[ T_2 = \sqrt{2}T \]