A gas expands such that its initial and final temperatures are equal. Also the process followed by the gas traces a straight line on the `P-V` diagram
(i) The temperature of the gas remains constant throughout
(ii) The temperature of the gas first increases and then decreases
(iii) The temperature of the gas first decreases and then increases
(iv) The straight line has negative slope

Options

A. The temperature of the gas remains constant throughout.
B. The temperature of the gas first increases and then decreases.
C. The temperature of the gas first decreases and then increases.
D. The straight line has a negative slope.

Explanation

To solve the problem, we need to analyze the behavior of a gas expanding along a straight line on a P-V (pressure-volume) diagram while maintaining equal initial and final temperatures. Let's break down the solution step by step:

### Step-by-Step Solution:

1. **Understanding the P-V Diagram**:
- The P-V diagram represents the relationship between pressure (P) and volume (V) of a gas.
- A straight line on this diagram indicates a linear relationship between pressure and volume during the expansion of the gas.

**Hint**: Recall that the slope of the line can indicate whether the pressure increases or decreases with volume.

2. **Identifying the Isotherm**:
- Since the initial and final temperatures of the gas are equal, the process must occur along an isotherm. An isotherm is a curve on the P-V diagram where the temperature remains constant.
- The straight line traced by the gas must be tangent to an isotherm at some point.

**Hint**: Remember that isotherms are typically hyperbolic curves on the P-V diagram, and a straight line can only be tangent to them at one point.

3. **Analyzing Temperature Changes**:
- As the gas expands, it may experience changes in temperature.
- If the process starts at a lower volume and moves to a higher volume, the gas will initially absorb heat, causing its temperature to rise until it reaches the tangent point on the isotherm. After this point, as it continues to expand, it will lose heat, causing the temperature to decrease.

**Hint**: Think about the first law of thermodynamics and how energy transfer affects temperature during expansion.

4. **Conclusion on Temperature Behavior**:
- From the analysis, we can conclude that the temperature of the gas first increases as it approaches the tangent point on the isotherm and then decreases as it continues to expand.
- Therefore, the correct option is that the temperature of the gas first increases and then decreases.

**Hint**: Consider how the gas behaves in terms of energy transfer during the expansion process.

5. **Slope of the Straight Line**:
- Since the straight line on the P-V diagram is descending (from left to right), it has a negative slope. This indicates that as the volume increases, the pressure decreases.

**Hint**: Remember that a negative slope on a graph indicates an inverse relationship between the two variables.

### Final Answer:
- The correct options are:
- (ii) The temperature of the gas first increases and then decreases.
- (iv) The straight line has a negative slope.