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

The sum of heats gained=0

Heatwater+heatsteam+heatcopper=0
.210*cw*(tf-19.5)kJ+.0210*-2.26Mj/kg+.0210*cw*(tf-100)+.0489*cc*(tf-19.5)=0
solve for tf

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

First, we need to calculate the energy transferred from the steam to the water and copper cup. This can be done by calculating the energy change for each component separately.

1. The energy transferred from the steam to the water:
- Calculate the energy change for steam:
Q_steam = m_steam * c_steam * (T_final - T_initial)
where m_steam is the mass of steam (0.0210 kg), c_steam is the specific heat capacity of steam (assumed to be the same as water, 4.19 kJ/kg.K), T_final is the final temperature of the system, and T_initial is the initial temperature of the steam (100.00°C).

2. The energy transferred from the steam to the copper cup:
- Calculate the energy change for the copper cup:
Q_copper_cup = m_copper_cup * c_copper * (T_final - T_initial)
where m_copper_cup is the mass of the copper cup (48.9 g = 0.0489 kg), c_copper is the specific heat capacity of copper (0.386 kJ/kg.K), T_final is the final temperature of the system, and T_initial is the initial temperature of the copper cup (same as the water initial temperature, 19.5°C).

3. The energy transferred from the water to the copper cup:
- Calculate the energy change for the water:
Q_water = m_water * c_water * (T_final - T_initial)
where m_water is the mass of water (0.210 kg), c_water is the specific heat capacity of water (4.19 kJ/kg.K), T_final is the final temperature of the system, and T_initial is the initial temperature of the water (19.5°C).

According to the principle of energy conservation, the total energy transferred from the steam to the system is equal to the sum of the energy changes for each component:

Q_steam = Q_water + Q_copper_cup

We can rearrange this equation to solve for the final temperature (T_final):

m_steam * c_steam * (T_final - T_initial) = m_water * c_water * (T_final - T_initial) + m_copper_cup * c_copper * (T_final - T_initial)

Substituting the given values:

0.0210 kg * 4.19 kJ/kg.K * (T_final - 100.00°C) = 0.210 kg * 4.19 kJ/kg.K * (T_final - 19.5°C) + 0.0489 kg * 0.386 kJ/kg.K * (T_final - 19.5°C)

Simplifying the equation:

(0.0210 * 4.19 * T_final - 0.0210 * 4.19 * 100.00) = (0.210 * 4.19 * T_final - 0.210 * 4.19 * 19.5) + (0.0489 * 0.386 * T_final - 0.0489 * 0.386 * 19.5)

Now you can solve this equation for T_final.