There is a small amount of water at the bottom of a sealed container of volume 5.2 liters which is otherwise full of an ideal gas. A thin tube open to the atmosphere extends down into the water. The system is initially at atmospheric pressure and temperature 113 Celsius.

If we increase the temperature of the gas until water rises in the tube to a height of 120 cm, then what is the temperature at that instant?

If the system is initially at 1 atm, then there can be no liquid water at the bottom when the temperature is 113 C. It would all have evaporated.

Maybe they are expecting you to assume that the water has not yet heated up to the gas temperature. This would be a nonequilibrium state. You could use the water column height to get the new water pressure, and use that to compute the new temperature.

This is a poorly thought out question, in my opinion.

how would i do that using the water coumn height?

To determine the temperature at the instant when the water rises to a height of 120 cm in the tube, we can use the principles of gas laws and hydrostatic pressure.

Step 1: Convert the height of the water column to meters
The height of the water column is given as 120 cm. Since we are working with SI units, we need to convert this to meters by dividing by 100.
120 cm ÷ 100 = 1.2 m

Step 2: Calculate the pressure exerted by the column of water
The pressure exerted by the column of water can be calculated using the formula P = ρgh, where P is the pressure, ρ is the density of the liquid, g is the acceleration due to gravity, and h is the height of the liquid column.
Consider water to be incompressible and have a density of 1000 kg/m^3.
Using the acceleration due to gravity, g = 9.8 m/s^2, the pressure can be calculated as
P = (1000 kg/m^3) × (9.8 m/s^2) × (1.2 m)
P = 11760 Pa

Step 3: Calculate the new pressure of the gas
Since the system is initially at atmospheric pressure, we can calculate the new pressure of the gas by adding the pressure exerted by the water column to the atmospheric pressure.
Atmospheric pressure at sea level is approximately 101,325 Pa.
New pressure = Atmospheric pressure + Pressure from water column
New pressure = 101,325 Pa + 11,760 Pa
New pressure = 113,085 Pa

Step 4: Use the Ideal Gas Law to find the temperature
The Ideal Gas Law states that PV = nRT, where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature in Kelvin.
Since the system is sealed, the volume remains constant at 5.2 liters (converted to cubic meters by dividing by 1000):
V = 5.2 L ÷ 1000 = 0.0052 m^3

We can rearrange the Ideal Gas Law equation to solve for temperature:
T = (PV) / (nR)
Since the number of moles of gas is not given, we can assume it remains constant and cancel it out.

T = (New pressure * Volume) / (R)
T = (113,085 Pa * 0.0052 m^3) / (8.314 J/(mol·K))

The ideal gas constant R is approximately 8.314 J/(mol·K).

Calculating this expression will give you the temperature at the instant when the water rises to a height of 120 cm in the tube.