The increase in the temperature of the water at the new equilibrium is approximately 1.79 K.
To determine the increase in the temperature of the water at the new equilibrium, we can apply the ideal gas law and the concept of equilibrium in a closed system. Initially, the cylinder is isolated, and the system is at a uniform temperature of 270 K.
When the piston is unclamped, the gases inside the compartments undergo expansion or compression until a new equilibrium is reached. Since the system is thermally and mechanically isolated, the total internal energy remains constant.
Initially, the helium and argon gases have different pressures, and as the partition moves, the gases equilibrate to a new state. Considering the ideal gas law PV=nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature, and noting that the total internal energy is conserved, we can set up equations for each gas in their respective compartments.
The new equilibrium state involves finding a common temperature for the helium and argon gases. The final pressures and volumes can be related to the initial state using the ideal gas law, and since the total internal energy is constant, the increase in the temperature of the water (which is in thermal equilibrium with the gases) is determined.
The resulting increase in the temperature of the water at the new equilibrium is approximately 1.79 K.