A certain amount of an ideal monoatomic gas undergoes a thermodynamic process such that `VT^(2)=` constat where `V=` volume of gas, `T=` temperature of gas. Then under process

Options

A. When heat is supplied to gas its temperature will increase
B. the coefficient of volume extension of gas equals `-(2)/(T)`
C. The molar heat capacity of gas is 2R
D. When heat is supplied to the gas its temperature decreases.

Explanation

To solve the problem, we need to analyze the thermodynamic process described by the equation \( V T^2 = \text{constant} \), where \( V \) is the volume and \( T \) is the temperature of an ideal monoatomic gas. We will derive the molar heat capacity and other properties step by step.

### Step-by-Step Solution

1. **Understanding the Given Relation**:
We start with the relation \( V T^2 = C \), where \( C \) is a constant. This implies that as the volume \( V \) changes, the temperature \( T \) must change in such a way that the product \( V T^2 \) remains constant.

2. **Expressing Pressure**:
For an ideal gas, we can use the ideal gas law:
\[ PV = nRT \]
Rearranging gives us:
\[ P = \frac{nRT}{V} \]

3. **Substituting for Temperature**:
From the relation \( V T^2 = C \), we can express \( T \) as:
\[ T = \sqrt{\frac{C}{V}} \]

4. **Substituting \( T \) in the Pressure Equation**:
Now substituting \( T \) into the pressure equation, we have:
\[ P = \frac{nR \sqrt{\frac{C}{V}}}{V} = \frac{nR \sqrt{C}}{V^{3/2}} \]
This shows that \( PV^{3/2} = \text{constant} \).

5. **Identifying the Process**:
The equation \( PV^{n} = \text{constant} \) indicates that this is a polytropic process where \( n = \frac{3}{2} \).

6. **Calculating Molar Heat Capacity**:
The molar heat capacity \( C \) for a polytropic process is given by:
\[ C = C_V + \frac{R}{n - 1} \]
For a monoatomic ideal gas, \( C_V = \frac{3R}{2} \). Substituting \( n = \frac{3}{2} \):
\[ C = \frac{3R}{2} + \frac{R}{\frac{3}{2} - 1} = \frac{3R}{2} + \frac{R}{\frac{1}{2}} = \frac{3R}{2} + 2R = \frac{7R}{2} \]

7. **Interpreting the Result**:
The calculated molar heat capacity \( C = \frac{7R}{2} \) indicates that when heat is supplied to the gas, the temperature will increase, which contradicts the earlier assumption of a negative heat capacity.

8. **Coefficient of Volume Expansion**:
To find the coefficient of volume expansion \( \beta \), we use the relation:
\[ \beta = \frac{\Delta V/V}{\Delta T/T} \]
From the relation \( V T^2 = C \), we differentiate:
\[ dV \cdot T^2 + V \cdot 2T \cdot dT = 0 \]
Rearranging gives:
\[ \frac{dV}{V} = -2 \frac{dT}{T} \]
Thus, the coefficient of volume expansion is:
\[ \beta = -2 \frac{1}{T} \]

### Final Answers
- The molar heat capacity \( C \) is \( \frac{7R}{2} \).
- The coefficient of volume expansion \( \beta \) is \( -\frac{2}{T} \).