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Molten sodium chloride is to be used as a constant-temperature bath for a high-temperature chemical reactor. Two hundred kilograms of solid \(\mathrm{NaCl}\) at \(300 \mathrm{K}\) is charged into an insulated vessel, and a 3000 kW electrical heater is turned on, raising the salt to its melting point of 1073 K and melting it at a constant pressure of 1 atm. (a) The heat capacity \(\left(C_{p}\right)\) of solid \(\mathrm{NaCl}\) is \(50.41 \mathrm{J} /(\mathrm{mol} \cdot \mathrm{K})\) at \(T=300 \mathrm{K},\) and \(53.94 \mathrm{J} /(\mathrm{mol} \cdot \mathrm{K})\) at \(T=500 \mathrm{K},\) and the heat of fusion of \(\mathrm{NaCl}\) at \(1073 \mathrm{K}\) is \(30.21 \mathrm{kJ} / \mathrm{mol} .\) Use these data to determine a linear expression for \(C_{p}(T)\) and to calculate \(\Delta \hat{H}\) ( \(\mathrm{kJ} / \mathrm{mol}\) ) for the transition of \(\mathrm{NaCl}\) from a solid at 300 K to a liquid at \(1073 \mathrm{K}\). (b) Write and solve the energy balance equation for this closed system isobaric process to determine the required heat input in kilojoules. (c) If \(85 \%\) of the full power of \(3000 \mathrm{kW}\) goes into heating and melting the salt, how long does the process take?

Step by step solution

01

Developing the linear expression \(C_{p}(T)\)

Given \(C_{p}\) values at 300K and 500K, the equation of a line can be used to find the linear expression for \(C_{p}(T)\). The formula is \(y = mx + b\), where \(m\) (slope) is \(\Delta y/\Delta x\) and \(b\) (intercept) is \(y - mx\). Here, \(y\) represents \(C_{p}\), \(x\) represents \(T\), and \(m\) is \((53.94 \mathrm{J} /(\mathrm{mol} \cdot \mathrm{K}) - 50.41 \mathrm{J} /(\mathrm{mol} \cdot \mathrm{K}))/(500 \mathrm{K} - 300 \mathrm{K}) = 0.01765 \mathrm{J} / (\mathrm{mol} \cdot \mathrm{K}^{2})\). The y-intercept \(b\) can be derived with \(y - mx\) using the \(C_{p}\) and \(T\) values at 300K: \(C_{p} - mT = b => 50.41 \mathrm{J} /(\mathrm{mol} \cdot \mathrm{K}) - (0.01765 \mathrm{J} / (\mathrm{mol} \cdot \mathrm{K}^{2})) \cdot 300 \mathrm{K} = 44.605 \mathrm{J} /(\mathrm{mol} \cdot \mathrm{K})\). Therefore, the equation of \(C_{p}(T)\) is \(0.01765T + 44.605 \mathrm{J} /(\mathrm{mol} \cdot \mathrm{K})\).

02

Calculating the enthalpy change \(\Delta \hat{H}\)

Using the developed equation from step 1, \(\Delta \hat{H}\) can be calculated in two parts: the heat gained to increase the temperature (\(\Delta H_{1}\)) and the heat given at the phase change (\(\Delta H_{2}\)). \(\Delta H_{1} = \int_{300}^{1073} C_{p}(T) dT = \int_{300}^{1073} (0.01765T + 44.605) dT\) which can be evaluated as \(\Delta H_{1} = 52.94 \mathrm{kJ/mol}\). \(\Delta H_{2}\) is given as the heat of fusion at 1073K, which is 30.21 kJ/mol. \(\Delta \hat{H}\) is the sum of \(\Delta H_{1}\) and \(\Delta H_{2}\), which is 83.15 kJ/mol.

03

Determining the heat input

Based on the energy balance equation \(Q = \Delta U + W\), and accounting for the process being at constant pressure (isobaric) and in a closed system (no work done), the heat input required (\(Q\)) is equal to \(\Delta H\) (enthalpy change). Given 200kg of NaCl, we can find the molar quantity by dividing by the molar mass of NaCl (58.44 g/mol). Then, multiply it with \(\Delta \hat{H}\) (molar enthalpy change) to get \(Q\), which is \(285000 \mathrm{kJ}\).

04

Computing process time

The time required for the heating process can be calculated by using the formula \(time = Energy/Power\). Knowing that 85% of the full power is used for the process, the actual power used is 0.85 x 3000kW = 2550kW. Therefore, the time required will be \(285000 \mathrm{kJ} /2550 \mathrm{kW} = 112 \mathrm{hours}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Heat Capacity

In thermodynamics, heat capacity is a crucial concept that describes the amount of heat energy required to raise the temperature of a substance by a specific amount. It is often represented as \(C_p\) when measurements are taken at constant pressure. The heat capacity is given in units of \(\text{J/(mol}\cdot\text{K)}\), demonstrating the energy needed to heat one mole of a substance by one Kelvin.

Key factors to remember about heat capacity include:

• It varies with temperature; hence, it's important to obtain values at different temperatures, as demonstrated with NaCl at 300 K and 500 K.
• In this exercise, the linear relationship of \(C_p(T)\) was established using the provided data, allowing us to predict \(C_p\) across temperatures.

The linear expression related to heat capacity for NaCl was derived using the formula \(y = mx + b\), where the slope \(m\) captured the change of heat capacity with temperature. Understanding this provides insight into the energy needed for heating processes in practical applications.

Enthalpy Change

Enthalpy change, denoted as \(\Delta \hat{H}\), represents the total heat content change of a system under constant pressure. It's a vital concept to comprehend the energy transactions during chemical processes, particularly those involving phase transitions.

In our scenario of molten sodium chloride:

• Enthalpy change was evaluated in two stages: heating the solid to its melting point and then allowing it to transition into a liquid phase.
• The calculation combined the integrated heat capacity function over the temperature range and added the heat of fusion at 1073 K.
• Mathematically, this involved evaluating \(\int_{300}^{1073} C_p(T)\, dT\) and adding the latent heat of fusion, quantified here as \(\Delta H_2 = 30.21 \, \text{kJ/mol}\).

Gaining an understanding of enthalpy change enhances one's ability to predict how much heat energy a process will need, which is crucial for designing and operating thermal systems.

Energy Balance Equation

In order to thoroughly analyze a thermodynamic system, one must employ the energy balance equation. For an isobaric process like the melting of sodium chloride, the energy equation simplifies as the work done is zero (since the system is closed and pressure is constant).

The equation used was \(Q = \Delta U + W\), where \(Q\) is the heat added to the system, \(\Delta U\) is the change in internal energy, and \(W\) is work done. For this type of process, since work done \(W = 0\), the heat input \(Q\) is directly equal to the enthalpy change \(\Delta H\).

Steps involved include:

• Calculating the total mole quantity using NaCl's molar mass, essential for determining the entire heat input with \(\Delta \hat{H}\).
• Identifying that the heat input \(Q\) needed for the process is \(285,000 \, \text{kJ}\).
• Converting the energy needs into time, considering actual power usage (85% efficiency of the heater), to obtain process duration.

Through mastering the energy balance, engineers and chemists can precisely determine the necessary power supply and duration for thermal operation, an essential step in planning large-scale thermal systems.