To determine the final temperature of the frothy milk, we need to calculate the heat transferred from the steam to the milk and then use the principle of heat conservation.
The heat transferred from the steam to the milk can be calculated using the formula:
Q = m * c * ΔT
Where: Q is the heat transferred, m is the mass of the substance (milk), c is the specific heat capacity of the substance, ΔT is the change in temperature.
First, let's calculate the heat released by the steam when it cools from 110°C to 100°C and condenses into water at 100°C.
The heat released by the steam can be calculated as follows:
Q_steam = m_steam * c_steam * ΔT_steam
Given: m_steam = 20g c_steam = specific heat capacity of steam (approximately 2.03 J/g°C) ΔT_steam = 110°C - 100°C = 10°C
Q_steam = 20g * 2.03 J/g°C * 10°C = 406 J
Next, let's calculate the heat absorbed by the milk when it heats up from 15°C to its final temperature, denoted as T.
The heat absorbed by the milk can be calculated as follows:
Q_milk = m_milk * c_milk * ΔT_milk
Given: m_milk = 200g c_milk = specific heat capacity of milk (approximately 3.93 J/g°C) ΔT_milk = T - 15°C
We can assume that no heat is lost to the surroundings during this process, so the heat lost by the steam is equal to the heat gained by the milk:
Q_steam = Q_milk
406 J = 200g * 3.93 J/g°C * (T - 15°C)
Simplifying the equation:
(T - 15°C) = 406 J / (200g * 3.93 J/g°C) (T - 15°C) ≈ 0.517
T ≈ 15°C + 0.517 ≈ 15.517°C
Therefore, the approximate final temperature of the frothy milk is 15.517°C.