Heating Curve Calculation - Free Printable

To solve the problem of determining the change in energy when 25.0 g of water is heated from \(-35^\circ \text{C}\) to \(65^\circ \text{C}\), we need to consider the following steps:

Step 1: Understand the Process

The process involves three distinct stages:
1. Heating ice from \(-35^\circ \text{C}\) to \(0^\circ \text{C}\): This requires calculating the heat needed to raise the temperature of ice.
2. Melting ice at \(0^\circ \text{C}\): This requires the latent heat of fusion to convert ice into liquid water.
3. Heating water from \(0^\circ \text{C}\) to \(65^\circ \text{C}\): This requires calculating the heat needed to raise the temperature of liquid water.

Step 2: Known Values

- Mass of water, \( m = 25.0 \, \text{g} \)
- Initial temperature, \( T_{\text{initial}} = -35^\circ \text{C} \)
- Final temperature, \( T_{\text{final}} = 65^\circ \text{C} \)
- Specific heat capacity of ice, \( c_{\text{ice}} = 2.09 \, \text{J/g}^\circ \text{C} \)
- Latent heat of fusion of ice, \( \Delta H_{\text{fus}} = 6.02 \, \text{kJ/mol} \)
- Molar mass of water, \( M_{\text{H}_2\text{O}} = 18.01 \, \text{g/mol} \)
- Specific heat capacity of water, \( c_{\text{water}} = 4.18 \, \text{J/g}^\circ \text{C} \)

Step 3: Calculate the Energy for Each Stage

#### Stage 1: Heating Ice from \(-35^\circ \text{C}\) to \(0^\circ \text{C}\)
The formula for heat required to change the temperature of a substance is:
\[
q = m \cdot c \cdot \Delta T
\]
where:
- \( m = 25.0 \, \text{g} \)
- \( c = 2.09 \, \text{J/g}^\circ \text{C} \)
- \( \Delta T = 0^\circ \text{C} - (-35^\circ \text{C}) = 35^\circ \text{C} \)

Substitute the values:
\[
q_1 = 25.0 \, \text{g} \cdot 2.09 \, \text{J/g}^\circ \text{C} \cdot 35^\circ \text{C}
\]
\[
q_1 = 25.0 \cdot 2.09 \cdot 35 = 1829 \, \text{J}
\]

#### Stage 2: Melting Ice at \(0^\circ \text{C}\)
The formula for heat required to melt a substance is:
\[
q = n \cdot \Delta H_{\text{fus}}
\]
where:
- \( n \) is the number of moles of ice
- \( \Delta H_{\text{fus}} = 6.02 \, \text{kJ/mol} = 6020 \, \text{J/mol} \)

First, calculate the number of moles of ice:
\[
n = \frac{m}{M_{\text{H}_2\text{O}}} = \frac{25.0 \, \text{g}}{18.01 \, \text{g/mol}}
\]
\[
n \approx 1.388 \, \text{mol}
\]

Now, calculate the heat required for melting:
\[
q_2 = n \cdot \Delta H_{\text{fus}} = 1.388 \, \text{mol} \cdot 6020 \, \text{J/mol}
\]
\[
q_2 \approx 8357 \, \text{J}
\]

#### Stage 3: Heating Water from \(0^\circ \text{C}\) to \(65^\circ \text{C}\)
The formula for heat required to change the temperature of a substance is:
\[
q = m \cdot c \cdot \Delta T
\]
where:
- \( m = 25.0 \, \text{g} \)
- \( c = 4.18 \, \text{J/g}^\circ \text{C} \)
- \( \Delta T = 65^\circ \text{C} - 0^\circ \text{C} = 65^\circ \text{C} \)

Substitute the values:
\[
q_3 = 25.0 \, \text{g} \cdot 4.18 \, \text{J/g}^\circ \text{C} \cdot 65^\circ \text{C}
\]
\[
q_3 = 25.0 \cdot 4.18 \cdot 65 = 6857.5 \, \text{J}
\]

Step 4: Total Energy Change

The total change in energy is the sum of the energies for each stage:
\[
q_{\text{total}} = q_1 + q_2 + q_3
\]
\[
q_{\text{total}} = 1829 \, \text{J} + 8357 \, \text{J} + 6857.5 \, \text{J}
\]
\[
q_{\text{total}} = 17043.5 \, \text{J}
\]

Final Answer:

\[
\boxed{17043.5 \, \text{J}}
\]