If heat energy is given to an ideal gas at constant pressure, then select the graph which best represents the variation of VT with temperature (T).

Options

A.
B.
C.
D.

Answer

To solve the problem of how the volume times temperature (VT) varies with temperature (T) for an ideal gas at constant pressure, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Ideal Gas Law**:
The ideal gas law is given by the equation:
\[ PV = nRT \]
where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is temperature.

2. **Identify the Condition of the Problem**:
The problem states that heat energy is given to the gas at constant pressure. This means that \( P \) remains constant.

3. **Rearranging the Ideal Gas Law**:
Since pressure is constant, we can rearrange the ideal gas law to express volume \( V \) in terms of temperature \( T \):
\[ V = \frac{nRT}{P} \]
Here, \( \frac{nR}{P} \) is a constant (let's call it \( k \)). Thus, we can express \( V \) as:
\[ V = kT \]

4. **Finding the Expression for VT**:
We want to find the relationship between \( VT \) and \( T \). We can multiply both sides of the equation \( V = kT \) by \( T \):
\[ VT = kT^2 \]

5. **Analyzing the Relationship**:
The equation \( VT = kT^2 \) indicates that \( VT \) is directly proportional to \( T^2 \). This means that as temperature \( T \) increases, \( VT \) increases with the square of the temperature.

6. **Graphical Representation**:
The relationship \( VT \propto T^2 \) can be represented graphically. If we plot \( VT \) on the y-axis and \( T \) on the x-axis, the graph will be a parabola opening upwards, since it is a quadratic relationship.

7. **Conclusion**:
Therefore, the graph that best represents the variation of \( VT \) with temperature \( T \) is a parabola that opens upwards.

### Final Answer:
The correct option is the graph that represents an upward parabola.

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