A certain amount of ideal monoatomic gas undergoes a process given by `TV^(1//2)`= constant. The molar specific heat of the gas for the process will be given by

Options

A. R/2
B. `-3R//2`
C. `3R//2`
D. `-R//2`

Answer

To find the molar specific heat of the gas for the process defined by \( TV^{1/2} = \text{constant} \), we can follow these steps:

### Step 1: Rewrite the given equation
The process can be expressed as:
\[ TV^{1/2} = k \quad (\text{where } k \text{ is a constant}) \]

### Step 2: Express temperature in terms of pressure and volume
Using the ideal gas law, we know:
\[ PV = nRT \implies T = \frac{PV}{nR} \]
Substituting this into the equation gives:
\[ \frac{PV}{nR} V^{1/2} = k \]
This simplifies to:
\[ \frac{PV^{3/2}}{nR} = k \]
Thus, we can write:
\[ PV^{3/2} = \text{constant} \]

### Step 3: Identify the polytropic process
From the equation \( PV^{3/2} = \text{constant} \), we can identify that this is a polytropic process with:
\[ n = \frac{3}{2} \]

### Step 4: Apply the first law of thermodynamics
The first law of thermodynamics states:
\[ \Delta Q = \Delta U + \Delta W \]
For a polytropic process, we can express the heat added as:
\[ \Delta Q = nC \Delta T \]
Where \( C \) is the molar specific heat for the process.

### Step 5: Calculate the change in internal energy
The change in internal energy for an ideal gas is given by:
\[ \Delta U = nC_V \Delta T \]
For a monoatomic gas, the molar specific heat at constant volume \( C_V \) is:
\[ C_V = \frac{3}{2}R \]

### Step 6: Calculate the work done
The work done in a polytropic process is given by:
\[ \Delta W = \frac{nR}{1 - n} \Delta T \]
Substituting \( n = \frac{3}{2} \):
\[ \Delta W = \frac{nR}{1 - \frac{3}{2}} \Delta T = \frac{nR}{-\frac{1}{2}} \Delta T = -2nR \Delta T \]

### Step 7: Substitute into the first law equation
Now substituting \( \Delta U \) and \( \Delta W \) into the first law:
\[ nC \Delta T = nC_V \Delta T - 2nR \Delta T \]
Dividing through by \( n \Delta T \) (assuming \( \Delta T \neq 0 \)):
\[ C = C_V - 2R \]

### Step 8: Substitute \( C_V \)
Substituting \( C_V = \frac{3}{2}R \):
\[ C = \frac{3}{2}R - 2R = \frac{3}{2}R - \frac{4}{2}R = -\frac{1}{2}R \]

### Conclusion
Thus, the molar specific heat of the gas for the process is:
\[ C = -\frac{1}{2}R \]