The addition of soap changes the surface tension of water to σ and that of sugar changes it to σ . Then

The correct option is ${\sigma }_1<{\sigma }_2$.

Let us analyze the effect of different types of impurities on the surface tension of water:
1. Effect of Soap (Surfactant):
Soap is a surfactant, which consists of hydrophobic and hydrophilic parts. When soap is added to water, its molecules accumulate at the surface and disrupt the cohesive hydrogen bonding between water molecules. This significantly decreases the surface tension of water. If we denote the surface tension of pure water as ${\sigma }_0$, then:
${\sigma }_1<{\sigma }_0$
2. Effect of Sugar (Soluble Impurity):
Sugar is a highly soluble substance. When highly soluble substances are dissolved in water, the strong forces of attraction between the solute and solvent molecules increase the intermolecular forces at the surface. This results in an increase in the surface tension of the solution. Thus:
${\sigma }_2>{\sigma }_0$

Comparing both effects relative to pure water, we obtain the inequality:
${\sigma }_1<{\sigma }_0<{\sigma }_2$
Therefore, we conclude that:
${\sigma }_1<{\sigma }_2$