In a certain gas, the ratio of the velocity of sound and root mean square velocity is `sqrt(5//9)`. The molar heat capacity of the gas in a process given by `PT = constant` is.
(Take `R = 2 cal//mol K`). Treat the gas as ideal.

Options

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

Answer

To solve the problem, we need to find the molar heat capacity of a gas given the ratio of the velocity of sound to the root mean square (rms) velocity. The steps to arrive at the solution are as follows:

### Step 1: Understand the given ratio
The ratio of the velocity of sound (Vs) to the root mean square velocity (Vrms) is given as:
\[ \frac{V_s}{V_{rms}} = \sqrt{\frac{5}{9}} \]

### Step 2: Write the formulas for Vs and Vrms
The formulas for the velocity of sound and root mean square velocity for an ideal gas are:
\[ V_s = \sqrt{\frac{\gamma P}{\rho}} \]
\[ V_{rms} = \sqrt{\frac{3P}{\rho}} \]

### Step 3: Substitute the formulas into the ratio
Substituting the formulas into the given ratio:
\[ \frac{\sqrt{\frac{\gamma P}{\rho}}}{\sqrt{\frac{3P}{\rho}}} = \sqrt{\frac{5}{9}} \]

### Step 4: Simplify the ratio
This simplifies to:
\[ \sqrt{\frac{\gamma}{3}} = \sqrt{\frac{5}{9}} \]

### Step 5: Square both sides
Squaring both sides gives:
\[ \frac{\gamma}{3} = \frac{5}{9} \]

### Step 6: Solve for γ
Multiplying both sides by 3:
\[ \gamma = \frac{5}{3} \]

### Step 7: Use the relationship between molar heat capacities
For an ideal gas, the relationship between the heat capacities is given by:
\[ C_p - C_v = R \]
and
\[ \gamma = \frac{C_p}{C_v} \]

From this, we can express \(C_v\) in terms of \(R\) and \(\gamma\):
\[ C_v = \frac{R}{\gamma - 1} \]

### Step 8: Substitute the value of γ
Substituting \(\gamma = \frac{5}{3}\):
\[ C_v = \frac{R}{\frac{5}{3} - 1} = \frac{R}{\frac{2}{3}} = \frac{3R}{2} \]

### Step 9: Calculate \(C_p\)
Using the relationship \(C_p = C_v + R\):
\[ C_p = \frac{3R}{2} + R = \frac{3R}{2} + \frac{2R}{2} = \frac{5R}{2} \]

### Step 10: Determine the molar heat capacity for the given process
In the process where \(PT = \text{constant}\), the molar heat capacity \(C\) can be expressed as:
\[ C = C_v + 2R \]
Substituting \(C_v\):
\[ C = \frac{3R}{2} + 2R = \frac{3R}{2} + \frac{4R}{2} = \frac{7R}{2} \]

### Final Answer
Thus, the molar heat capacity of the gas in the given process is:
\[ C = \frac{7R}{2} \]