Introduction:
This problem is related to the phase behavior of binary liquid mixtures. We need to determine the pressure at which half of the liquid is converted into vapor for a binary liquid mixture containing 10 moles each of liquid A and B. The given information is the initial pressures of the two components.

Given Data:
- Number of moles of liquid A and B = 10 each
- Initial pressure of A (P0A) = 200 mm Hg
- Initial pressure of B (P0B) = 100 mm Hg

Solution:
To determine the pressure at which half of the liquid is converted into vapor, we need to first calculate the vapor pressure of the mixture at the given temperature. This can be done using Raoult's law, which states that the vapor pressure of a component in a mixture is proportional to its mole fraction in the mixture.

- Vapor pressure of A (PA) = mole fraction of A (XA) * vapor pressure of pure A (P0A)
- Vapor pressure of B (PB) = mole fraction of B (XB) * vapor pressure of pure B (P0B)

The mole fraction of A and B in the mixture can be calculated as follows:
- Mole fraction of A (XA) = number of moles of A / total number of moles = 10 / 20 = 0.5
- Mole fraction of B (XB) = number of moles of B / total number of moles = 10 / 20 = 0.5

Using the above equations, we get:
- Vapor pressure of A (PA) = 0.5 * 200 = 100 mm Hg
- Vapor pressure of B (PB) = 0.5 * 100 = 50 mm Hg

The total vapor pressure of the mixture (Ptotal) can be calculated as the sum of the vapor pressures of A and B:
- Ptotal = PA + PB = 100 + 50 = 150 mm Hg

To determine the pressure at which half of the liquid is converted into vapor, we need to compare the total pressure of the mixture with the vapor pressure of the liquid at that pressure. At the pressure where the vapor pressure equals the total pressure, half of the liquid will have vaporized.

Therefore, the pressure at which half of the liquid is converted into vapor is 150 mm Hg, which is the total vapor pressure of the mixture.

Conclusion:
The pressure at which half of the liquid is converted into vapor for a binary liquid mixture containing 10 moles each of liquid A and B, with initial pressures of 200 mm Hg and 100 mm Hg respectively, is 150 mm Hg. This can be calculated using Raoult's law and comparing the vapor pressure of the liquid with the total pressure of the mixture.