How much heat (in kJ) does it take to melt 28.4 grams of ice at 0 °C and raise the temperature of the resulting water to 25 °C? Heat of fusion 334 J/g, Specific heat 4.184 J/g °C ?

q = [mass ice x heat fusion] + [mass ice x specific heat H2O x (Tfinal-Tinitial)]

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To calculate the total heat required, we need to consider two steps:

1. Heat required to melt the ice at 0°C
2. Heat required to raise the temperature of the resulting water from 0°C to 25°C.

Step 1: Heat required to melt the ice
The heat required to melt the ice can be calculated using the formula:
Heat = mass × heat of fusion

Given:
Mass of ice = 28.4 grams
Heat of fusion = 334 J/g

Heat = 28.4 g × 334 J/g

Convert grams to kilograms (since the specific heat is given in joules per gram) by dividing the mass by 1000:
Heat = (28.4 g / 1000) × 334 J/g

Step 2: Heat required to raise the temperature of the resulting water
The heat required to raise the temperature of the water can be calculated using the formula:
Heat = mass × specific heat × temperature change

Given:
Mass of water = 28.4 grams
Specific heat = 4.184 J/g °C
Temperature change = 25°C - 0°C = 25°C

Heat = 28.4 g × 4.184 J/g °C × 25°C

Add the two heats from Step 1 and Step 2 to find the total heat required:
Total Heat = Heat (Step 1) + Heat (Step 2)

Total Heat = (28.4 g / 1000) × 334 J/g + 28.4 g × 4.184 J/g °C × 25°C

Convert the total heat from joules to kilojoules by dividing by 1000:
Total Heat = [(28.4 g / 1000) × 334 J/g + 28.4 g × 4.184 J/g °C × 25°C] / 1000

Now, calculate the total heat value.

To find the total heat required, we need to consider two steps:

1. Heating the ice from 0 °C to its melting point (0 °C).
2. Melting the ice at its melting point and heating the resulting water from 0 °C to 25 °C.

Let's break down the calculation for each step:

Step 1: Heating the ice from 0 °C to its melting point (0 °C)

To calculate the heat required to raise the temperature of the ice, we can use the formula:

Q = m * C * ΔT

where:
Q = heat energy (in J)
m = mass of the substance (in grams)
C = specific heat capacity (in J/g °C)
ΔT = change in temperature (in °C)

In this case:
m = 28.4 g
C = 4.184 J/g °C
ΔT = (0 °C - 0 °C) = 0 °C

Therefore, the heat required to raise the temperature of the ice is:

Q1 = (28.4 g) * (4.184 J/g °C) * (0 °C - 0 °C) = 0 J

Step 2: Melting the ice and heating the resulting water

To calculate the heat required to melt the ice and heat the resulting water, we need to consider two parts:

2.1 Heat of fusion to melt the ice:
The heat of fusion is the amount of heat required to convert a substance from a solid to a liquid state. In this case, the heat of fusion for ice is given as 334 J/g.

To calculate the heat required to melt the ice, we can use the formula:

Q2.1 = m * ΔHf

where:
Q2.1 = heat energy (in J)
m = mass of the substance (in grams)
ΔHf = heat of fusion (in J/g)

In this case:
m = 28.4 g
ΔHf = 334 J/g

Therefore, the heat required to melt the ice is:

Q2.1 = (28.4 g) * (334 J/g) = 9477.6 J

2.2 Heating the resulting water from 0 °C to 25 °C:

To calculate the heat required to raise the temperature of the resulting water, we can use the formula:

Q2.2 = m * C * ΔT

where:
Q2.2 = heat energy (in J)
m = mass of the substance (in grams)
C = specific heat capacity (in J/g °C)
ΔT = change in temperature (in °C)

In this case:
m = 28.4 g
C = 4.184 J/g °C
ΔT = (25 °C - 0 °C) = 25 °C

Therefore, the heat required to raise the temperature of the resulting water is:

Q2.2 = (28.4 g) * (4.184 J/g °C) * (25 °C - 0 °C) = 3015.64 J

Finally, we can calculate the total heat required by adding the heat from step 1 and step 2:

Total heat required = Q1 + Q2.1 + Q2.2
= 0 J + 9477.6 J + 3015.64 J
= 12493.24 J

To convert the total heat required from joules (J) to kilojoules (kJ), we divide by 1000:

Total heat required = 12493.24 J / 1000 = 12.49324 kJ

Therefore, it takes 12.49324 kJ of heat to melt 28.4 grams of ice at 0 °C and raise the temperature of the resulting water to 25 °C.