What amount of energy is required to melt a 27.4 g piece of ice at 0oC?

The heat of fusion of ice = 333 Jg-1

Heat required to melt the ice 9.12e3 = J

What amount of energy must be removed from 123 g of water to cool it from 71.6oC to 9.41oC?
Specific heat of water = 4.184 JK-1g-1

Enter the heat as a negative amount (heat removed from a system is by convention always negative).
Heat removed to cool the water =-3.2e4 J

What is the final temperature after a 27.4 g piece of ice is placed into a styrofoam cup containing 123 g of hot water at 71.6oC. The mass of the ice, and the mass and initial temperature of the hot water are the same as in the two previous parts of this question.

Final temperature = oC

I got the first two answers I just don't know how to find the answer to the last questions.

[mass ice x heat fusion]+[mass melted ice x specific heat ice x (Tfinal-Tinitial)] + [mass warm water x specific heat H2O x (Tfinal-Tinitial)] = 0

Substitute the numbers and solve for Tf.

To find the final temperature after the ice is placed into the cup of hot water, we need to use the principle of conservation of energy. The heat lost by the hot water will be equal to the heat gained by the ice.

The heat lost by the hot water can be calculated using the equation:

Q_water = m_water * c_water * (T_initial_water - T_final)

Where:
Q_water = heat lost by the water (negative value)
m_water = mass of water (123 g)
c_water = specific heat of water (4.184 J/Kg°C)
T_initial_water = initial temperature of the water (71.6°C)
T_final = final temperature of the water (to be determined)

Substituting the given values into the equation:

Q_water = 123 g * 4.184 J/g°C * (71.6°C - T_final)

The heat gained by the ice can be calculated using the equation:

Q_ice = m_ice * H_fusion

Where:
Q_ice = heat gained by the ice
m_ice = mass of ice (27.4 g)
H_fusion = heat of fusion of ice (333 J/g)

Substituting the given values into the equation:

Q_ice = 27.4 g * 333 J/g

Since the total heat lost by the water is equal to the total heat gained by the ice (according to the principle of conservation of energy), we can set up the equation:

123 g * 4.184 J/g°C * (71.6°C - T_final) = 27.4 g * 333 J/g

Now we can solve the equation to find the final temperature (T_final).

123 * 4.184 * (71.6 - T_final) = 27.4 * 333

First, multiply the numbers on the left side of the equation:

515.0528 * (71.6 - T_final) = 9064.2

Next, distribute the multiplication:

36917.11968 - 515.0528 * T_final = 9064.2

Now, isolate the term with T_final on one side:

-515.0528 * T_final = 9064.2 - 36917.11968

Simplify both sides of the equation:

-515.0528 * T_final = -27852.91968

Finally, divide both sides of the equation by -515.0528 to solve for T_final:

T_final = (-27852.91968) / (-515.0528)

T_final ≈ 54.03°C

Therefore, the final temperature after placing the ice into the cup of hot water is approximately 54.03°C.