An ideal diatomic gas with `C_(V)=(5R)/(2)` occupies a volume `V_(1)` at a pressure `P_(1)`. The gas undergoes a process in which the pressure is proportional to the volume. At the end of the process the rms speed of the gas molecules has doubled from its initial value.
Heat supplied to the gas in the given process is

Options

A. `7P_(1)V_(1)`
B. `8P_(1)V_(1)`
C. `9P_(1)V_(1)`
D. `10P_(1)V_(1)`

Answer

To solve the problem step by step, we will analyze the given information and apply the relevant physics concepts.

### Step 1: Understand the Initial Conditions
We have an ideal diatomic gas with:
- Heat capacity at constant volume, \( C_V = \frac{5R}{2} \)
- Initial volume \( V_1 \) and initial pressure \( P_1 \).

### Step 2: Relate RMS Speed to Temperature
The RMS speed (\( v_{rms} \)) of gas molecules is given by the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where \( R \) is the universal gas constant, \( T \) is the absolute temperature, and \( M \) is the molar mass of the gas.

If the RMS speed doubles, we have:
\[ v'_{rms} = 2v_{rms} \]
This implies:
\[ \sqrt{\frac{3RT'}{M}} = 2\sqrt{\frac{3RT}{M}} \]
Squaring both sides gives:
\[ \frac{3RT'}{M} = 4 \cdot \frac{3RT}{M} \]
Cancelling \( \frac{3R}{M} \) from both sides, we find:
\[ T' = 4T \]

### Step 3: Calculate the Change in Temperature
The change in temperature (\( \Delta T \)) is:
\[ \Delta T = T' - T = 4T - T = 3T \]

### Step 4: Analyze the Process
The problem states that pressure is proportional to volume:
\[ P \propto V \quad \Rightarrow \quad PV = k \quad \text{(constant)} \]
This is characteristic of a polytropic process with \( n = -1 \).

### Step 5: Calculate the Molar Heat Capacity for the Process
The molar heat capacity \( C \) for a polytropic process is given by:
\[ C = C_V + \frac{R}{1 - n} \]
Substituting \( C_V = \frac{5R}{2} \) and \( n = -1 \):
\[ C = \frac{5R}{2} + \frac{R}{1 - (-1)} = \frac{5R}{2} + \frac{R}{2} = \frac{6R}{2} = 3R \]

### Step 6: Calculate the Heat Supplied
The heat supplied (\( \Delta Q \)) during the process is given by:
\[ \Delta Q = nC\Delta T \]
Substituting \( C = 3R \) and \( \Delta T = 3T \):
\[ \Delta Q = n(3R)(3T) = 9nRT \]

### Step 7: Relate to Ideal Gas Law
Using the ideal gas law \( PV = nRT \), we can express \( nRT \) as:
\[ nRT = P_1V_1 \]
Thus, substituting this into our equation for heat:
\[ \Delta Q = 9P_1V_1 \]

### Final Answer
The heat supplied to the gas in the given process is:
\[ \Delta Q = 9P_1V_1 \]