Suppose 0.0210 kg of steam (at 100.00°C) is added to 0.210 kg of water (initially at 19.5°C.). The water is inside a copper cup of mass 48.9 g. The cup is inside a perfectly insulated calorimetry container that prevents heat flow with the outside environment. Find the final temperature (in °C) of the water after equilibrium is reached.

use the following values:
c_copper=0.386 kJ/kg.K
c_water=4.19 kJ/kg.K
L_fusion=2.26 MJ/kg

To find the final temperature of the water after equilibrium is reached, we can use the principle of conservation of energy.

Let's break down the steps involved in solving this problem:

1. Calculate the heat gained or lost by each component:

- Heat gained or lost by the copper cup (Q_cup):
Q_cup = m_cup * c_copper * ΔT_cup
where m_cup is the mass of the copper cup (48.9 g), c_copper is the specific heat capacity of copper (0.386 kJ/kg.K), and ΔT_cup is the change in temperature of the cup (final temperature - initial temperature).

- Heat gained or lost by the water (Q_water):
Q_water = m_water * c_water * ΔT_water
where m_water is the mass of the water (0.210 kg), c_water is the specific heat capacity of water (4.19 kJ/kg.K), and ΔT_water is the change in temperature of the water (final temperature - initial temperature).

- Heat gained or lost during the phase change of steam (Q_phase_change):
Q_phase_change = m_steam * L_fusion
where m_steam is the mass of the steam (0.0210 kg) and L_fusion is the latent heat of fusion for steam (2.26 MJ/kg).

2. Since the calorimeter is perfectly insulated and prevents heat flow with the outside environment, the heat lost by one component must be equal to the heat gained by the other component, as well as the heat gained during the phase change:

Q_cup + Q_water + Q_phase_change = 0

3. Rearrange the equation to solve for the final temperature:

Q_cup + Q_water + Q_phase_change = 0
m_cup * c_copper * ΔT_cup + m_water * c_water * ΔT_water + m_steam * L_fusion = 0

ΔT_water can be expressed as (final temperature - initial temperature), and ΔT_cup can be expressed as (final temperature - 100.00°C). Substitute these values into the equation:

m_cup * c_copper * (final temperature - 100.00°C) + m_water * c_water * (final temperature - 19.5°C) + m_steam * L_fusion = 0

4. Solve the equation for the final temperature:

Plug in the given values:
(0.0489 kg) * (0.386 kJ/kg.K) * (final temperature - 100.00°C) + (0.210 kg) * (4.19 kJ/kg.K) * (final temperature - 19.5°C) + (0.0210 kg) * (2.26 MJ/kg) = 0

Simplify and convert units (1 MJ = 1000 kJ):
(0.0489 kg) * (0.386 kJ/kg.K) * (final temperature - 100.00°C) + (0.210 kg) * (4.19 kJ/kg.K) * (final temperature - 19.5°C) + (0.0210 kg) * (2260 kJ/kg) = 0

Rearrange the equation to isolate the final temperature:
(0.0489 kg * 0.386 kJ/kg.K + 0.210 kg * 4.19 kJ/kg.K) * final temperature - (0.0489 kg * 0.386 kJ/kg.K * 100.00°C + 0.210 kg * 4.19 kJ/kg.K * 19.5°C + 0.0210 kg * 2260 kJ/kg) = 0

Simplify the equation further and solve for the final temperature using algebraic techniques or a scientific calculator.

The final temperature obtained from this calculation will be the equilibrium temperature reached by the water after adding the steam.