On the `P-T` graph of an ideal gas, choose the correct option(s):

Options

A. Adiabatic process will be a straight line.
B. Isochoric process will be straight line passing through the origin.
C. adiabatic curve will have a positive slope.
D. the slope of adiabatic curve is proportional T.

Explanation

To solve the question regarding the `P-T` graph of an ideal gas, we will analyze the different processes (adiabatic and isochoric) and their characteristics on the graph.

### Step-by-Step Solution:

1. **Understanding the Adiabatic Process:**
- The adiabatic process for an ideal gas is described by the equation \( PV^{\gamma} = \text{constant} \), where \( \gamma \) is the adiabatic exponent.
- From the ideal gas law, we know \( PV = nRT \). We can express pressure \( P \) in terms of temperature \( T \) and volume \( V \):
\[ P = \frac{nRT}{V} \]
- Substituting this into the adiabatic equation gives:
\[ \left(\frac{nRT}{V}\right)V^{\gamma} = \text{constant} \]
- Rearranging, we find:
\[ P \propto T^{\frac{1}{\gamma - 1}} \]
- This indicates that the relationship between \( P \) and \( T \) is not linear, hence the adiabatic process will not be a straight line on the `P-T` graph.

2. **Analyzing the Isochoric Process:**
- In an isochoric process, the volume \( V \) is constant. Thus, the relationship simplifies to:
\[ P \propto T \]
- This means that for a constant volume, pressure is directly proportional to temperature, which results in a straight line on the `P-T` graph.

3. **Slope of the Adiabatic Curve:**
- The slope of the adiabatic curve can be derived from the relationship established earlier. By differentiating the logarithmic form of the relationship, we find:
\[ \frac{dP}{dT} = \frac{\gamma P}{T(\gamma - 1)} \]
- This shows that the slope depends on both pressure \( P \) and temperature \( T \), and is positive as both \( P \) and \( T \) are positive in the context of ideal gases.

4. **Conclusion:**
- The adiabatic process does not represent a straight line on the `P-T` graph.
- The isochoric process does represent a straight line on the `P-T` graph.
- The slope of the adiabatic curve is not directly proportional to temperature but rather depends on both pressure and temperature.

### Final Answer:
- The correct options are:
- **B**: Isochoric process will be a straight line.
- **C**: The slope of the adiabatic curve is positive.