A monatomic ideal gas sample is given heat Q. One half of this heat is used as work done by the gas and rest is used for increasing its internal energy. The equation of process in terms of volume and temperature is

Options

A. `(v^(2))/(T^(3))=" constant"`
B. `(v^(2))/(sqrtT)=" constant"`
C. `VT^(3)=" constant"`
D. `V^(2)sqrtT=" constant"`

Answer

To solve the problem step by step, we will analyze the relationship between the heat added to a monatomic ideal gas, the work done by the gas, and the change in internal energy.

### Step 1: Understand the given information
We have a monatomic ideal gas that receives heat \( Q \). According to the problem, half of this heat is used for work done \( W \), and the other half is used to increase the internal energy \( \Delta U \).

### Step 2: Write the equations
From the first law of thermodynamics, we know:
\[ Q = \Delta U + W \]
Given that:
\[ W = \frac{1}{2} Q \quad \text{and} \quad \Delta U = \frac{1}{2} Q \]

### Step 3: Relate internal energy to temperature
For a monatomic ideal gas, the change in internal energy can be expressed as:
\[ \Delta U = n C_V \Delta T \]
where \( C_V = \frac{3R}{2} \) for a monatomic gas.

### Step 4: Substitute into the equation
Substituting \( \Delta U \) into the first law equation:
\[ Q = n C_V \Delta T + W \]
Since \( W = P \Delta V \) and \( P = \frac{nRT}{V} \), we can write:
\[ W = \frac{nRT}{V} \Delta V \]

### Step 5: Set up the equation
Now substituting \( W \) into the equation:
\[ Q = n C_V \Delta T + \frac{nRT}{V} \Delta V \]
Since \( \Delta U = \frac{1}{2} Q \) and \( W = \frac{1}{2} Q \), we can equate:
\[ \frac{1}{2} Q = n C_V \Delta T \quad \text{and} \quad \frac{1}{2} Q = \frac{nRT}{V} \Delta V \]

### Step 6: Express \( \Delta T \) and \( \Delta V \)
From the equation for \( \Delta U \):
\[ \Delta T = \frac{Q}{n C_V} = \frac{Q}{n \cdot \frac{3R}{2}} = \frac{2Q}{3nR} \]
From the equation for \( W \):
\[ \Delta V = \frac{Q V}{nRT} \]

### Step 7: Relate \( T \) and \( V \)
Now we can relate \( T \) and \( V \):
Using the ideal gas law \( PV = nRT \), we can express \( P \) in terms of \( T \) and \( V \):
\[ P = \frac{nRT}{V} \]

### Step 8: Combine equations
Substituting \( \Delta T \) and \( \Delta V \) into the equations, we get:
\[ \frac{2Q}{3nR} = \frac{Q V}{nRT} \]
Canceling \( Q \) and rearranging gives:
\[ T \propto V^{\frac{3}{2}} \]

### Final Step: Write the final equation
Thus, we can express the relationship as:
\[ \frac{T^{\frac{3}{2}}}{V} = \text{constant} \]
or equivalently,
\[ V^2 \propto T^3 \]
This implies:
\[ \frac{V^2}{T^3} = \text{constant} \]

### Conclusion
The equation of the process in terms of volume and temperature is:
\[ \frac{V^2}{T^3} = \text{constant} \]