An ideal monatomic gas expands in such a way that `TV^((1)/(2))`=` constant, where T and V are the temperature & volume of the gas, choose the correct option(s)

Options

A. The molar heat capacity is `C=C_(V)-2R`
B. The internal energy of gas decreases.
C. Temperature of gas decreases.
D. Pressure of gas increases.

Answer

To solve the problem, we need to analyze the relationship given by the equation \( TV^{1/2} = \text{constant} \) for an ideal monatomic gas. Let's go through the steps systematically.

### Step 1: Understand the given relationship
The equation \( TV^{1/2} = \text{constant} \) indicates a specific type of process involving the temperature \( T \) and volume \( V \) of the gas. We can denote this constant as \( C \):
\[ T V^{1/2} = C \]

### Step 2: Express temperature in terms of volume
From the equation, we can express temperature \( T \) in terms of volume \( V \):
\[ T = \frac{C}{V^{1/2}} \]

### Step 3: Analyze the internal energy of the gas
For a monatomic ideal gas, the internal energy \( U \) is given by:
\[ U = n C_v T \]
where \( C_v = \frac{3}{2} R \) for a monatomic gas. Substituting the expression for \( T \):
\[ U = n C_v \left(\frac{C}{V^{1/2}}\right) = \frac{n C_v C}{V^{1/2}} \]

### Step 4: Determine the change in internal energy with volume
As the volume \( V \) increases, \( V^{1/2} \) also increases, which implies that \( U \) will decrease because it is inversely proportional to \( V^{1/2} \):
\[ U \propto \frac{1}{V^{1/2}} \]
Thus, as \( V \) increases, \( U \) decreases.

### Step 5: Analyze the pressure of the gas
Using the ideal gas law \( PV = nRT \) and substituting for \( T \):
\[ P = \frac{nR}{V} \cdot \frac{C}{V^{1/2}} = \frac{nRC}{V^{3/2}} \]
From this equation, we can see that as \( V \) increases, \( P \) decreases because \( P \) is inversely proportional to \( V^{3/2} \).

### Conclusion
From our analysis, we can summarize the findings:
1. The molar heat capacity \( C \) is related to \( C_v \) and can be calculated.
2. The internal energy of the gas decreases as the volume increases.
3. The pressure of the gas decreases as the volume increases.

### Final Answer
The correct options are:
- Molar heat capacity is correctly defined.
- Internal energy of the gas decreases.
- Pressure of the gas decreases.