The correct equation that gives the relationship between the specific gravity of soil grains (G) and the hydraulic gradient (i) for initiating quick condition in sand with a void ratio of 0.5 is given by option 'C': G = 1.5i.

Explanation:

1. Understanding Specific Gravity:
- Specific gravity is defined as the ratio of the density of a substance to the density of a reference substance, usually water.
- In the case of soil, the specific gravity of soil grains (G) refers to the ratio of the density of soil grains to the density of water.

2. Understanding Hydraulic Gradient:
- Hydraulic gradient (i) is a measure of the variation in hydraulic head per unit distance along the flow path.
- It represents the slope of the water table or the pressure head in the flow direction.

3. Quick Condition in Soil:
- Quick condition refers to the state in which the soil loses its strength and behaves as a viscous liquid when subjected to an increase in pore water pressure.
- This phenomenon is commonly observed in saturated loose sands.

4. Relationship between Specific Gravity and Hydraulic Gradient:
- The specific gravity of soil grains does not directly influence the hydraulic gradient.
- However, the specific gravity of soil grains affects the critical hydraulic gradient for initiating quick condition in sand.

5. Void Ratio:
- Void ratio is defined as the ratio of the volume of voids to the volume of solids in a soil mass.
- In this case, the given sand has a void ratio of 0.5.

6. Critical Hydraulic Gradient for Quick Condition in Sand:
- The critical hydraulic gradient for initiating quick condition in sand is influenced by several factors, including the void ratio and the specific gravity of soil grains.
- Empirical relationships have been established to estimate the critical hydraulic gradient for different soil conditions.

7. Option 'C' Explanation:
- Option 'C' states that G = 1.5i, which means the specific gravity of soil grains is equal to 1.5 times the hydraulic gradient.
- This equation suggests that the specific gravity of soil grains has a direct relationship with the hydraulic gradient.
- The specific gravity of soil grains is expected to increase with increasing hydraulic gradient.

In conclusion, the correct equation that gives the relationship between the specific gravity of soil grains (G) and the hydraulic gradient (i) for initiating quick condition in sand with a void ratio of 0.5 is G = 1.5i. This equation indicates that the specific gravity of soil grains increases with increasing hydraulic gradient.