Explanation:

Convection coefficient:
The convection coefficient, also known as the film coefficient, is a measure of the effectiveness of heat transfer between a solid surface and a fluid medium. It represents the rate of heat transfer per unit area per unit temperature difference between the surface and the fluid.

Condensation over a vertical surface:
Condensation is the process by which a vapor or gas changes into a liquid state. When condensation occurs over a vertical surface, such as a wall or a tube, it is referred to as vertical condensation. This process is commonly observed in applications such as heat exchangers, air conditioning systems, and refrigeration systems.

Variation of convection coefficient:
The convection coefficient for condensation over a vertical surface is influenced by various factors such as the properties of the fluid, the surface characteristics, and the flow conditions. However, one of the key factors that determine the variation of the convection coefficient is the thermal conductivity of the fluid.

Effect of thermal conductivity:
The thermal conductivity of a fluid is a measure of its ability to conduct heat. It is denoted by the symbol 'k'. When the thermal conductivity of the fluid is high, it means that the fluid can conduct heat more effectively. On the other hand, when the thermal conductivity is low, the heat transfer is less efficient.

Explanation of the correct answer:
The correct answer is option 'A' - k^0.25. This means that the convection coefficient varies with the square root of the thermal conductivity of the fluid. In other words, as the thermal conductivity of the fluid increases, the convection coefficient also increases, but at a slower rate. This is because the square root function is a mathematical relationship that increases at a slower rate than the original value.

Other options:
Option 'B' - k^0.33: This option suggests that the convection coefficient varies with the cube root of the thermal conductivity. However, this relationship does not accurately represent the variation observed in condensation over a vertical surface.

Option 'C' - k^0.75: This option suggests that the convection coefficient varies with the 3/4th power of the thermal conductivity. Again, this relationship does not accurately represent the variation observed in condensation.

Option 'D' - k^-0.5: This option suggests that the convection coefficient varies inversely with the square root of the thermal conductivity. However, this is not an accurate representation of the variation observed in condensation over a vertical surface.

Conclusion:
In condensation over a vertical surface, the convection coefficient varies with the square root of the thermal conductivity of the fluid. This relationship is represented by option 'A' - k^0.25.