By itself no. Vegetable oil is what we call hydrophobic(fearing water). However if you add detergent it will separate each molecule of oil and completely surround them. Here it will dissolve in water.
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Most likely not.
Water is polar. That means it has a very small electrical charge, which makes it work sort of like a magnet.
Vegetable oil is not polar. This means it does not have the electrical properties that water has, and therefore wont split up bindings like water does.
No. It will become less viscous (more runny) with heat, but the long chain molecules wont mix with the water molecules. If you add detergent (soap) then this will bridge the gap between the two types of molecules, allowing the oil to be broken up. But just hot water on it own wont be much different from cold water and vegetable oil. The oil will separate ans float above the water.
Oil is a nonpolar liquid, meaning that its atoms don't attract the atoms of a polar substance, such as water.
Water (H2O) is polar because its oxygen atom pulls on the electrons of its 2 hydrogen atoms, so it has a positive side and a negative side; it and other water molecules naturally attract each other.
On the other hand, oil is composed of hydrogen and carbon atoms. Hydrogen and carbon generally have the same electronegativity, meaning that they equally share the electrons. This lack of a positive side or a negative side makes oil nonpolar, so its atoms and water molecules don't attract each other.
Thus, oil does not dissolve in water but accumulates to form a layer above the water in a container. Since water has a density of 1 g/L (gram per liter) and oil floats on top of water, oil has a density of less than 1 g/L. If a nonpolar substance has a density of greater than 1 g/L, then it would accumulate at the bottom of the water to form a dense layer in the container.
It doesn't do so to any appreciable degree. To the extent it does, it's because water is a freaking amazing solvent.