An unknown material has a normal melting/freezing point of -20.4 °C, and the liquid phase has a specific heat capacity of 179 J/(kg C°). One-tenth of a kilogram of the solid at -20.4 °C is put into a 0.130-kg aluminum calorimeter cup that contains 0.198 kg of glycerin. The temperature of the cup and the glycerin is initially 26.4 °C. All the unknown material melts, and the final temperature at equilibrium is 19.1 °C. The calorimeter neither loses energy to nor gains energy from the external environment. What is the latent heat of fusion of the unknown material?

19.4

19.4

To find the latent heat of fusion of the unknown material, we can use the concept of energy conservation.

Let's break down the problem step by step:

Step 1: Calculate the heat absorbed by the unknown material to raise its temperature from -20.4 °C to 0 °C.

We can use the specific heat capacity formula:

Q = mcΔT

Where:
Q = Heat absorbed or released
m = Mass of the substance
c = Specific heat capacity
ΔT = Change in temperature

In this case, m = 0.1 kg (mass of the solid), c = 179 J/(kg°C), and ΔT = (0 °C) - (-20.4 °C) = 20.4 °C.

Q1 = (0.1 kg) * (179 J/(kg°C)) * (20.4 °C) = 368.76 J

Step 2: Calculate the heat absorbed by the unknown material during the phase change from solid to liquid.

The heat absorbed during phase change is given by:

Q2 = mL

Where:
m = Mass of the substance
L = Latent heat of fusion

In this case, m = 0.1 kg (mass of the solid) and Q2 = Q1 (from step 1) since all the heat absorbed in step 1 is used for the phase change.

So, Q2 = 368.76 J

Step 3: Calculate the heat absorbed by the glycerin and the calorimeter to decrease their temperature from 26.4 °C to 19.1 °C.

We can use the specific heat capacity formula:

Q3 = mcΔT

Where:
Q3 = Heat absorbed or released
m = Mass of the substance (glycerin + calorimeter)
c = Specific heat capacity
ΔT = Change in temperature

In this case, m = 0.198 kg + 0.13 kg = 0.328 kg (mass of glycerin + calorimeter), c = specific heat capacity of glycerin (unknown value), and ΔT = (19.1 °C) - (26.4 °C) = -7.3 °C (negative value because the temperature is decreasing).

Q3 = (0.328 kg) * (c J/(kg°C)) * (-7.3 °C)

Step 4: Set up the energy conservation equation.

Since no energy is lost or gained by the calorimeter system, the total heat absorbed by the system is equal to the total heat released by the system.

Q1 + Q2 = Q3

(368.76 J) + (Q2) = Q3

Substitute the values of Q2 and Q3:

368.76 J + 368.76 J = (0.328 kg) * (c J/(kg°C)) * (-7.3 °C)

733.52 J = -2.3936 c

Step 5: Solve for the specific heat capacity of glycerin (c).

c = (733.52 J) / (-2.3936 * (-7.3 °C))

c = 31.92 J/(kg°C)

The specific heat capacity of glycerin is approximately 31.92 J/(kg°C).

Step 6: Calculate the latent heat of fusion (L) of the unknown material.

We know that Q2 = mL

Q2 = (0.1 kg) * L

368.76 J = (0.1 kg) * L

L = (368.76 J) / (0.1 kg)

L = 3687.6 J/kg

Therefore, the latent heat of fusion of the unknown material is approximately 3687.6 J/kg.

To find the latent heat of fusion of the unknown material, we can use the principle of conservation of energy. The energy gained by the solid material as it heats up and melts must be equal to the energy lost by the aluminum calorimeter cup and the glycerin as they cool down.

First, let's calculate the energy gained by the solid material:

1. Calculate the energy required to heat the solid material from -20.4 °C to the melting point of the material (0 °C):

Energy1 = mass1 * specific heat capacity1 * temperature change1
where mass1 = 0.1 kg (mass of solid material)
specific heat capacity1 = 179 J/(kg C°) (specific heat capacity of the liquid phase of the material)
temperature change1 = (-20.4 °C) - (0 °C) = -20.4 °C

Energy1 = 0.1 kg * 179 J/(kg C°) * (-20.4 °C) = -367.56 J (negative because energy is being lost)

2. Calculate the energy required to melt the solid material:

Energy2 = mass1 * latent heat of fusion
where latent heat of fusion is what we are trying to find

Now, let's calculate the energy lost by the aluminum calorimeter cup and the glycerin:

3. Calculate the energy lost by the aluminum calorimeter cup:

Energy3 = mass3 * specific heat capacity3 * temperature change3
where mass3 = 0.130 kg (mass of aluminum calorimeter cup)
specific heat capacity3 = 900 J/(kg C°) (specific heat capacity of aluminum)
temperature change3 = (19.1 °C) - (26.4 °C) = -7.3 °C (negative because temperature is decreasing)

Energy3 = 0.130 kg * 900 J/(kg C°) * (-7.3 °C) = -842.7 J (negative because energy is being lost)

4. Calculate the energy lost by the glycerin:

Energy4 = mass4 * specific heat capacity4 * temperature change4
where mass4 = 0.198 kg (mass of glycerin)
specific heat capacity4 = 2400 J/(kg C°) (specific heat capacity of glycerin)
temperature change4 = (19.1 °C) - (26.4 °C) = -7.3 °C (negative because temperature is decreasing)

Energy4 = 0.198 kg * 2400 J/(kg C°) * (-7.3 °C) = -3234.96 J (negative because energy is being lost)

Now, according to the principle of conservation of energy, the energy gained by the solid material (Energy1 + Energy2) must be equal to the energy lost by the aluminum calorimeter cup and the glycerin (Energy3 + Energy4):

Energy1 + Energy2 = Energy3 + Energy4

-367.56 J + Energy2 = -842.7 J - 3234.96 J

Simplifying the equation:

Energy2 = -842.7 J - 3234.96 J + 367.56 J

Energy2 = -4710.1 J

Therefore, the latent heat of fusion of the unknown material is 4710.1 J.