A gas may expand either adiabatically or isothermally. A number of `p-V` curves are drawn for the two processes over different ranges of pressure and volume, it will be found that
(i) Two adiabatic curves do not intersect
(ii) two isothermal curves do not intersect
(iii) an adiabatic curve and an isothermal curve may intersect.
(iv) the magnitude of the slope of an adiabatic curve is greater thanthe magnitude of the slope of an isothermal curve

Options

A. (i),(ii)
B. (ii),(iii)
C. (iii),(iv)
D. all

Answer

To analyze the statements regarding the behavior of adiabatic and isothermal processes in a gas, let's go through each statement step by step.

### Step-by-Step Solution:

1. **Statement (i): Two adiabatic curves do not intersect.**
- **Explanation**: In an adiabatic process, there is no heat exchange with the surroundings. Each adiabatic curve corresponds to a different amount of internal energy, which is a function of the temperature of the gas. If two adiabatic curves were to intersect, it would imply that the gas could exist at the same pressure and volume with two different temperatures, which is not possible. Therefore, this statement is **true**.

2. **Statement (ii): Two isothermal curves do not intersect.**
- **Explanation**: An isothermal process occurs at a constant temperature. Each isothermal curve represents a different temperature. If two isothermal curves intersected, it would mean that at the same pressure and volume, the gas could have two different temperatures, which contradicts the definition of isothermal processes. Thus, this statement is also **true**.

3. **Statement (iii): An adiabatic curve and an isothermal curve may intersect.**
- **Explanation**: It is possible for an adiabatic curve and an isothermal curve to intersect because they represent different thermodynamic processes. An intersection would imply that at a certain pressure and volume, the gas can be at a specific temperature (isothermal) while also undergoing an adiabatic change. Therefore, this statement is **true**.

4. **Statement (iv): The magnitude of the slope of an adiabatic curve is greater than the magnitude of the slope of an isothermal curve.**
- **Explanation**: To understand the slopes mathematically:
- For an adiabatic process, the relationship is given by \( PV^\gamma = k \) (where \( \gamma = C_p/C_v \)). Differentiating gives us:
\[ \frac{dP}{dV} = -\gamma \frac{P}{V} \]
- For an isothermal process, the relationship is \( PV = k \). Differentiating gives us:
\[ \frac{dP}{dV} = -\frac{P}{V} \]
- Comparing the two slopes, we see that the slope of the adiabatic curve is steeper (more negative) than that of the isothermal curve because \( \gamma > 1 \). Therefore, this statement is **true**.

### Summary of Truth Values:
- (i) True
- (ii) True
- (iii) True
- (iv) True