A calorimeter with specific heat of 490x10^2J/kg^1K^1 has a mass of 2.00x10^-1kg that contains 3.00x10^-1kg of liquid water at 40C. If 5.00x10^-2kg of ice at 0C is dropped into the water. Find the final temperature of the mixture where the ice has melted is 23.8C. Calculate the heat of fusion of ice.

To find the final temperature of the mixture, we can use the principle of conservation of energy. The heat lost by the water and ice will equal the heat gained by the calorimeter.

Let's calculate the heat lost by the water:
q1 = m1 * c1 * ΔT1
where
m1 = mass of water
c1 = specific heat of water
ΔT1 = change in temperature of water

Given:
m1 = 0.3 kg (mass of water)
c1 = 4.90 x 10^2 J/kg°C (specific heat of water)
ΔT1 = T1 - Tfinal (change in temperature of water)

Initially, the water is at 40 °C and the final temperature is 23.8 °C.

ΔT1 = 40 - 23.8 = 16.2 °C

q1 = 0.3 kg * 4.90 x 10^2 J/kg°C * 16.2 °C

Now, let's calculate the heat gained by the calorimeter. The heat gained by the calorimeter will be in the form of heat of fusion of ice.

q2 = m2 * Lf
where
m2 = mass of ice
Lf = heat of fusion of ice

Given:
m2 = 0.05 kg (mass of ice)

Now, let's calculate the heat of fusion of ice, Lf.

q2 = 0.05 kg * Lf

As per the given information, the total heat lost by the water and gained by the calorimeter is equal to zero.

q1 = -q2

0.3 kg * 4.90 x 10^2 J/kg°C * 16.2 °C = 0.05 kg * Lf

Solving the equation:

Lf = (0.3 kg * 4.90 x 10^2 J/kg°C * 16.2 °C) / 0.05 kg

Now, let's substitute the values and calculate Lf:

Lf = (2.94 x 10^3 J/°C) / 0.05 kg

Lf = 5.88 x 10^4 J/kg

Therefore, the heat of fusion of ice is 5.88 x 10^4 J/kg.

To find the final temperature of the mixture and the heat of fusion of ice, we can apply the principle of energy conservation. Here's how we'll approach this:

Step 1: Calculate the heat transferred when the ice melts and raises the temperature of the mixture to the final temperature.
Step 2: Calculate the heat transferred from the water to the calorimeter.
Step 3: Equate the heat transferred in step 1 and step 2 to find the final temperature.
Step 4: Use the final temperature to calculate the heat of fusion of ice.

Step 1: Calculate the heat transferred when the ice melts and raises the temperature of the mixture.

To melt the ice, it requires the heat of fusion (Qf) which can be calculated using the formula:
Qf = mass of ice × heat of fusion of ice

Given: mass of ice = 5.00x10^-2 kg

But we need to determine the heat of fusion of ice.

Step 2: Calculate the heat transferred from the water to the calorimeter.

The heat transferred from the water to the calorimeter can be calculated using the formula:
Qw = mass of water × specific heat of water × change in temperature

Given:
mass of water = 3.00x10^-1 kg
specific heat of water = 490x10^2 J/kg^1K^1
change in temperature = final temperature - initial temperature
initial temperature = 40°C

Step 3: Equate the heat transferred in step 1 and step 2 to find the final temperature.

Qf = Qw

mass of ice × heat of fusion of ice = mass of water × specific heat of water × change in temperature

Substitute the given values and rearrange the equation to solve for the final temperature.

Step 4: Use the final temperature to calculate the heat of fusion of ice.

Now that we have the final temperature, we can substitute it into the equation from step 1 to calculate the heat of fusion of ice.

Using the solution obtained from step 3, substitute the final temperature into the formula:
Qf = mass of ice × heat of fusion of ice

Solving this equation will give us the value of the heat of fusion of ice.