Question

Some properties of aluminum are summarized in the following list. $$\begin{array}{ll} \text { normal melting point } & 658^{\circ} \mathrm{C} \\ \text { heat of fusion } & 3.95 \mathrm{kJ} / \mathrm{g} \\ \text { normal boiling point } & 2467^{\circ} \mathrm{C} \\ \text { heat of vaporization } & 10.52 \mathrm{kJ} / \mathrm{g} \\ \text { specific heat of the solid } & 0.902 \mathrm{J} / \mathrm{g}^{\circ} \mathrm{C} \end{array}$$ a. Calculate the quantity of energy required to heat \(1.00 \mathrm{mol}\) of aluminum from \(25^{\circ} \mathrm{C}\) to its normal melting point. b. Calculate the quantity of energy required to melt 1.00 mol of aluminum at \(658^{\circ} \mathrm{C}\). c. Calculate the amount of energy required to vaporize 1.00 mol of aluminum at \(2467^{\circ} \mathrm{C}\).

Short answer

a. 15414.58 J of energy are required to heat 1.00 mol of aluminum from 25°C to its normal melting point. b. 106471 J of energy are required to melt 1.00 mol of aluminum at 658°C. c. 283741.6 J of energy are required to vaporize 1.00 mol of aluminum at 2467°C.

Step-by-step solution

Calculate the mass of 1.00 mol of aluminum

According to the periodic table, the molar mass of aluminum (Al) is \(26.98 \: \text{g/mol}\). To find the mass of 1.00 mol of aluminum, we can use the equation: \[mass = moles \cdot molar\_mass\] \[mass = 1.00 \: \text{mol} \cdot 26.98 \: \text{g/mol}\] \[mass = 26.98 \: \text{g}\]

Calculate the energy required to heat the aluminum

To calculate the energy required to heat the aluminum from 25°C to its melting point (658°C), we can use the equation: \[Q = mc\Delta T\] where \(Q\) = energy required (Joules), \(m\) = mass (grams), \(c\) = specific heat capacity of solid aluminum (\(0.902 \: \text{J/g°C}\)), and \(\Delta T\) = change in temperature (°C). We have all the required data, so we can plug in the values and solve for Q: \[Q = (26.98 \: \text{g}) (0.902 \: \text{J/g°C}) (658 - 25)\] \[Q = 26.98 \times 0.902 \times 633\] \[Q = 15414.58 \: \text{J}\] Therefore, the energy required to heat 1.00 mol of aluminum from 25°C to its melting point is approximately 15414.58 J. #b. Calculate the quantity of energy required to melt 1.00 mol of aluminum at 658°C.#

Calculate the energy required to melt the aluminum

To calculate the energy required to melt 1.00 mol of aluminum at 658°C, we can use the equation: \[Q = mL\] where \(Q\) = energy required (Joules), \(m\) = mass (grams), and \(L\) = heat of fusion of aluminum (given as \(3.95 \: \text{kJ/g}\), we'll convert to J/g by multiplying by \(1000\), so \(L = 3950 \: \text{J/g}\)). We already calculated the mass of 1.00 mol of aluminum (26.98 g). So, we can now plug in the values and solve for Q: \[Q = (26.98 \: \text{g}) (3950 \: \text{J/g})\] \[Q = 106471\: \text{J}\] Therefore, the energy required to melt 1.00 mol of aluminum at 658°C is approximately 106471 J. #c. Calculate the amount of energy required to vaporize 1.00 mol of aluminum at 2467°C.#

Calculate the energy required to vaporize the aluminum

To calculate the energy required to vaporize 1.00 mol of aluminum at 2467°C, we can use the equation: \[Q = mL\] where \(Q\) = energy required (Joules), \(m\) = mass (grams), and \(L\) = heat of vaporization of aluminum (given as \(10.52 \: \text{kJ/g}\), we'll convert this to J/g by multiplying by \(1000\), so \(L = 10520 \: \text{J/g}\)). We already calculated the mass of 1.00 mol of aluminum (26.98 g). So, we can now plug in the values and solve for Q: \[Q = (26.98 \: \text{g}) (10520 \: \text{J/g})\] \[Q = 283741.6 \: \text{J}\] Therefore, the energy required to vaporize 1.00 mol of aluminum at 2467°C is approximately 283741.6 J.

Key concepts

Specific Heat Capacity

Specific heat capacity, denoted as 'c', is a measure of the amount of energy in joules needed to raise the temperature of one gram of a substance by one degree Celsius. It's a fundamental concept in thermochemistry as different materials require varying amounts of energy to change temperature. This property is crucial when calculating the thermal energy transferred during temperature changes, as demonstrated in the exercise where we calculate the energy needed to heat aluminum up to its melting point. The formula used is Q = mcΔT, where Q is the heat energy, m is the mass, c is the specific heat capacity, and ΔT is the change in temperature. Aluminum's specific heat capacity is relatively low, meaning it requires less energy to increase its temperature compared to substances with higher specific heat capacities.

Understanding specific heat capacity helps predict how substances respond to thermal energy — crucial in a wide range of applications from culinary arts to aerospace engineering.

Heat of Fusion

The heat of fusion is the amount of energy required to change a substance from solid to liquid at its melting point without changing its temperature. This endothermic process is represented by the symbol 'L' and is typically expressed in kJ/g or J/g. The exercise shows how to calculate the heat needed to melt aluminum using its given heat of fusion, 3.95 kJ/g. The relevant formula is Q = mL, wherein Q represents the heat required, m the mass of the substance, and L the heat of fusion. For substances like aluminum, which has a relatively high heat of fusion, a significant amount of energy is absorbed during the melting process.

When designing cooling systems or studying phase transitions, the heat of fusion provides essential information for engineers and chemists to manage thermal energy during changes of state.

Heat of Vaporization

The heat of vaporization refers to the energy required to convert a liquid into a gas at its boiling point while keeping the temperature constant. Similar to the heat of fusion but for the liquid-gas transition, the heat of vaporization ('L') is also measured in kJ/g or J/g and is essential for explaining processes like boiling or evaporation. The exercise illustrates aluminum's high requirement for energy, 10.52 kJ/g, to transition from liquid to gas, indicating a significant input of energy to vaporize it at 2467°C. The formula used to calculate this energy is identical to that for heat of fusion: Q = mL.

This concept plays a vital role in fields that involve phase changes, including meteorology, where it helps explain how water cycles between the earth and the atmosphere, and in industrial processes that involve cooling and heating.

Enthalpy Change Calculations

Enthalpy change represents the total heat content change of a system under constant pressure and is denoted as ΔH. It's a central theme in thermochemistry, involved in all reactions and phase changes where energy is either absorbed or released. While not directly calculated in the exercise, enthalpy change underpins the process of energy transfer. It encompasses both the heat of fusion and the heat of vaporization when a substance undergoes phase changes. Calculating enthalpy change often requires understanding specific heat capacity, as it involves accounting for the energy needed to reach the phase transition temperature in addition to the energy required for the phase transition itself.

For chemists and engineers, calculating enthalpy changes is critical for predicting reactions' behavior and for the design of chemical processes, where energy efficiency and control are paramount.

Molar Heat Calculations

Molar heat calculations involve determining the amount of heat (enthalpy) change per mole of a substance during a thermal process. They provide a standardized way to compare the energy required to change the temperature of different materials. In the textbook exercise, the steps involve using the molar mass of aluminum to calculate the mass of 1.00 mol. From there, we use the specific heat capacity and the phase change constants (heat of fusion/vaporization) to calculate the heat required for each mole during different temperature phases and transitions. These molar heat calculations enable students to quantify and compare how different substances store and transfer heat.

Academic and industrial applications of molar heat calculations include the design of thermal systems, material science research, and environmental studies to understand climate-related energy transfers.