Condensation on a Glass and the Conservation of Heat

Problem Statement

In a room, a piece of ice at an initial temperature of $T_1$T1​ was placed into a glass containing water of mass $m_w$mw​. After leaving it undisturbed for a while, the ice completely melted into water, and the temperature of all the water in the glass stabilized at $T_2$T2​.

From this state, the glass was left in the room for a further period of time, during which water vapor from the air condensed on the outside of the glass, forming water droplets with a mass of $m_d$md​. After a sufficient amount of time had elapsed, the water inside the glass and the water droplets on the outside combined to reach thermal equilibrium, and the overall temperature became $T_3$T3​. In this process, the following conditions are assumed:

・Heat transfer is considered only in terms of the latent heat released when water vapor condenses into liquid water on the outside of the glass; direct heat conduction from the surrounding air, the heat capacity of the glass itself, and the evaporation of water can be neglected. ・Let $L$L be the amount of heat of condensation released when water vapor undergoes a phase transition to liquid water. ・Let $c_w$cw​ be the specific heat of water.

Derive the temperature $T_3$T3​ at thermal equilibrium using an algebraic expression, then substitute the given constraint values to find the value of $T_3$T3​.

Constraints

• Mass of water in the glass: $m_w = 0.158 \text{ kg}$mw​=0.158 kg
• Mass of ice added: $m_i = 0.040 \text{ kg}$mi​=0.040 kg
• Initial temperature of the system: $T_2 = 278 \text{ K}$T2​=278 K
• Mass of condensed water droplets: $m_d = 0.002 \text{ kg}$md​=0.002 kg
• Latent heat of condensation of water: $L = 2.52 \times 10^6 \text{ J/kg}$L=2.52×106 J/kg
• Specific heat of water: $c_w = 4200 \mathrm{ J/(kg\cdot K)}$cw​=4200J/(kg⋅K)

Input Format

Enter the value of the temperature $T_3$T3​ at thermal equilibrium as a natural number.