Answer :
To determine the total energy absorbed by the 383.9 gram sample of the substance as it is heated from [tex]\(-23.1^{\circ} C\)[/tex] to [tex]\(51.8^{\circ} C\)[/tex], we must consider the different stages: heating the solid phase to its melting point, melting the substance (phase change), and heating the liquid phase to the final temperature. Here’s a step-by-step solution to calculate the energy absorbed:
1. Heating the solid to its melting point:
The substance starts at [tex]\(-23.1^{\circ} C\)[/tex] and must be heated to its melting point at [tex]\(17.6^{\circ} C\)[/tex]. The specific heat capacity of the solid phase is [tex]\(2.96 J/(g\cdot^{\circ} C)\)[/tex].
The energy required for this stage can be calculated with the formula:
[tex]\[ q_1 = \text{mass} \times \text{specific heat capacity of solid} \times \Delta T \][/tex]
[tex]\[ q_1 = 383.9 \, \text{g} \times 2.96 \, \text{J/(g\cdot^{\circ} C)} \times (17.6^{\circ} C - (-23.1^{\circ} C)) \][/tex]
[tex]\[ q_1 = 383.9 \, \text{g} \times 2.96 \, \text{J/(g\cdot^{\circ} C)} \times 40.7^{\circ} C \][/tex]
[tex]\[ q_1 = 46249.2008 \, \text{J} \][/tex]
2. Melting the solid at its melting point:
The substance undergoes a phase change at its melting point. The enthalpy of fusion ([tex]\(\Delta H_{fus}\)[/tex]) is given as [tex]\(8.04 \, \text{kJ/mol}\)[/tex]. First, we need to convert this to [tex]\(\text{J/g}\)[/tex]:
[tex]\[ \Delta H_{fus} = \frac{8.04 \, \text{kJ}}{92.41 \, \text{g/mol}} \times 1000 \, \text{J/kJ} = 86.98 \, \text{J/g} \][/tex]
The energy required for this stage is:
[tex]\[ q_2 = \text{mass} \times \Delta H_{fus} \][/tex]
[tex]\[ q_2 = 383.9 \, \text{g} \times 86.98 \, \text{J/g} \][/tex]
[tex]\[ q_2 = 33400.67092306027 \, \text{J} \][/tex]
3. Heating the liquid from its melting point to the final temperature:
Once the substance is in the liquid phase, it must be heated from [tex]\(17.6^{\circ} C\)[/tex] to [tex]\(51.8^{\circ} C\)[/tex]. The specific heat capacity of the liquid phase is [tex]\(1.75 \, \text{J/(g\cdot^{\circ} C)}\)[/tex].
The energy required for this stage can be calculated with the formula:
[tex]\[ q_3 = \text{mass} \times \text{specific heat capacity of liquid} \times \Delta T \][/tex]
[tex]\[ q_3 = 383.9 \, \text{g} \times 1.75 \, \text{J/(g\cdot^{\circ} C)} \times (51.8^{\circ} C - 17.6^{\circ} C) \][/tex]
[tex]\[ q_3 = 383.9 \, \text{g} \times 1.75 \, \text{J/(g\cdot^{\circ} C)} \times 34.2^{\circ} C \][/tex]
[tex]\[ q_3 = 22976.414999999994 \, \text{J} \][/tex]
4. Total energy absorbed:
We sum the energies calculated for each stage to get the total energy absorbed:
[tex]\[ q_{\text{total}} = q_1 + q_2 + q_3 \][/tex]
[tex]\[ q_{\text{total}} = 46249.2008 \, \text{J} + 33400.67092306027 \, \text{J} + 22976.414999999994 \, \text{J} \][/tex]
[tex]\[ q_{\text{total}} = 102626.28672306026 \, \text{J} \][/tex]
5. Convert the total energy to kJ:
[tex]\[ q_{\text{total, kJ}} = \frac{102626.28672306026 \, \text{J}}{1000} \][/tex]
[tex]\[ q_{\text{total, kJ}} = 102.62628672306026 \, \text{kJ} \][/tex]
Therefore, the total energy absorbed to heat the solid sample from [tex]\(-23.1^{\circ} C\)[/tex] to [tex]\(51.8^{\circ} C\)[/tex] is [tex]\(102.62628672306026 \, \text{kJ}\)[/tex].
1. Heating the solid to its melting point:
The substance starts at [tex]\(-23.1^{\circ} C\)[/tex] and must be heated to its melting point at [tex]\(17.6^{\circ} C\)[/tex]. The specific heat capacity of the solid phase is [tex]\(2.96 J/(g\cdot^{\circ} C)\)[/tex].
The energy required for this stage can be calculated with the formula:
[tex]\[ q_1 = \text{mass} \times \text{specific heat capacity of solid} \times \Delta T \][/tex]
[tex]\[ q_1 = 383.9 \, \text{g} \times 2.96 \, \text{J/(g\cdot^{\circ} C)} \times (17.6^{\circ} C - (-23.1^{\circ} C)) \][/tex]
[tex]\[ q_1 = 383.9 \, \text{g} \times 2.96 \, \text{J/(g\cdot^{\circ} C)} \times 40.7^{\circ} C \][/tex]
[tex]\[ q_1 = 46249.2008 \, \text{J} \][/tex]
2. Melting the solid at its melting point:
The substance undergoes a phase change at its melting point. The enthalpy of fusion ([tex]\(\Delta H_{fus}\)[/tex]) is given as [tex]\(8.04 \, \text{kJ/mol}\)[/tex]. First, we need to convert this to [tex]\(\text{J/g}\)[/tex]:
[tex]\[ \Delta H_{fus} = \frac{8.04 \, \text{kJ}}{92.41 \, \text{g/mol}} \times 1000 \, \text{J/kJ} = 86.98 \, \text{J/g} \][/tex]
The energy required for this stage is:
[tex]\[ q_2 = \text{mass} \times \Delta H_{fus} \][/tex]
[tex]\[ q_2 = 383.9 \, \text{g} \times 86.98 \, \text{J/g} \][/tex]
[tex]\[ q_2 = 33400.67092306027 \, \text{J} \][/tex]
3. Heating the liquid from its melting point to the final temperature:
Once the substance is in the liquid phase, it must be heated from [tex]\(17.6^{\circ} C\)[/tex] to [tex]\(51.8^{\circ} C\)[/tex]. The specific heat capacity of the liquid phase is [tex]\(1.75 \, \text{J/(g\cdot^{\circ} C)}\)[/tex].
The energy required for this stage can be calculated with the formula:
[tex]\[ q_3 = \text{mass} \times \text{specific heat capacity of liquid} \times \Delta T \][/tex]
[tex]\[ q_3 = 383.9 \, \text{g} \times 1.75 \, \text{J/(g\cdot^{\circ} C)} \times (51.8^{\circ} C - 17.6^{\circ} C) \][/tex]
[tex]\[ q_3 = 383.9 \, \text{g} \times 1.75 \, \text{J/(g\cdot^{\circ} C)} \times 34.2^{\circ} C \][/tex]
[tex]\[ q_3 = 22976.414999999994 \, \text{J} \][/tex]
4. Total energy absorbed:
We sum the energies calculated for each stage to get the total energy absorbed:
[tex]\[ q_{\text{total}} = q_1 + q_2 + q_3 \][/tex]
[tex]\[ q_{\text{total}} = 46249.2008 \, \text{J} + 33400.67092306027 \, \text{J} + 22976.414999999994 \, \text{J} \][/tex]
[tex]\[ q_{\text{total}} = 102626.28672306026 \, \text{J} \][/tex]
5. Convert the total energy to kJ:
[tex]\[ q_{\text{total, kJ}} = \frac{102626.28672306026 \, \text{J}}{1000} \][/tex]
[tex]\[ q_{\text{total, kJ}} = 102.62628672306026 \, \text{kJ} \][/tex]
Therefore, the total energy absorbed to heat the solid sample from [tex]\(-23.1^{\circ} C\)[/tex] to [tex]\(51.8^{\circ} C\)[/tex] is [tex]\(102.62628672306026 \, \text{kJ}\)[/tex].
