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The equilibrium constant for the ethane dehydrogenation reaction, $$\mathrm{C}_{2} \mathrm{H}_{6}(\mathrm{g}) \rightleftharpoons \mathrm{C}_{2} \mathrm{H}_{4}(\mathrm{g})+\mathrm{H}_{2}(\mathrm{g})$$ is defined as $$K_{p}(\mathrm{atm})=\frac{y_{\mathrm{C}_{2} \mathrm{H}_{4}} y_{\mathrm{H}_{2}}}{y_{\mathrm{C}_{2} \mathrm{H}_{6}}} P$$ where \(P(\text { atm })\) is the total pressure and \(y_{i}\) is the mole fraction of the ith substance in an equilibrium mixture. The equilibrium constant has been found experimentally to vary with temperature according to the formula $$K_{p}(T)=7.28 \times 10^{6} \exp [-17,000 / T(\mathrm{K})]$$ The heat of reaction at \(1273 \mathrm{K}\) is \(+145.6 \mathrm{kJ}\), and the heat capacities of the reactive species may be approximated by the formulas $$\left.\begin{array}{rl}\left(C_{p}\right)_{\mathrm{C}_{2} \mathrm{H}_{4}} & =9.419+0.1147 T(\mathrm{K}) \\\\\left(C_{p}\right)_{\mathrm{H}_{2}} & =26.90+4.167 \times 10^{-3} T(\mathrm{K}) \\ \left(C_{p}\right)_{\mathrm{C}_{2} \mathrm{H}_{6}} & =11.35+0.1392 T(\mathrm{K}) \end{array}\right\\}[\mathrm{J} /(\mathrm{mol} \cdot \mathrm{K})]$$ Suppose pure cthane is fed to a continuous constant-pressure adiabatic reactor at \(1273 \mathrm{K}\) and pressure \(P(\text { atm }),\) the products emerge at \(T_{\mathrm{f}}(\mathrm{K})\) and \(P(\mathrm{atm}),\) and the residence time of the reaction mixture in the reactor is large enough for the outlet stream to be considered an equilibrium mixture of ethane, ethylene, and hydrogen. (a) Prove that the fractional conversion of ethane in the reactor is $$f=\left(\frac{K_{p}}{P+K_{p}}\right)^{1 / 2}$$ (b) Write an energy balance on the reactor, and use it to prove that $$f=\frac{1}{1+\phi\left(T_{\mathrm{f}}\right)}$$ where Finally, substitute for \(\Delta H_{\mathrm{r}}\) and the heat capacities in Equation 4 to derive an explicit expression for \(\phi\left(T_{\mathrm{f}}\right)\) (c) We now have two expressions for the fractional conversion \(f\) : Equation 2 and Equation 3 . If these expressions are equated, \(K_{p}\) is replaced by the expression of Equation \(1,\) and \(\phi\left(T_{\mathrm{f}}\right)\) is replaced by the expression derived in Part (b), the result is one equation in one unknown, \(T_{\mathrm{f}}\). Derive this equation, and transpose the right side to obtain an expression of the form $$\psi\left(T_{\mathrm{f}}\right)=0$$ (d) Prepare a spreadsheet to take \(P\) as input, solve Equation 5 for \(T_{\mathrm{f}}\) (use Goal Seek or Solver), and determine the final fractional conversion, \(\left.f \text { . (Suggestion: Set up columns for } P, T_{\mathrm{f}}, f, K_{p}, \phi, \text { and } \psi .\right)\) Run the program for \(P(\text { atm })=0.01,0.05,0.10,0.50,1.0,5.0,\) and \(10.0 .\) Plot \(T_{\mathrm{f}}\) versus \(P\) and \(f\) versus \(P,\) using a logarithmic coordinate scale for \(P\).

Step by step solution

01

Proving the Fractional Conversion of Ethane

The mole fraction of ethane \(y_{C_{2}H_{6}}\) at any instance is given by (1- f) and those of ethylene and hydrogen are equal and given by f / 2 due to stoichiometric relationships. Since the reaction mixture is assumed to reach equilibrium, plug in these values into the equilibrium constant expression and simplify to obtain the equation \(f=\left(\frac{K_{p}}{P+K_{P}}\right)^{1 / 2}\).

02

Writing an Energy Balance Equation

The energy balance for an adiabatic reactor can be expressed as -\(\Delta H_{r} . f = \int_{T_{0}}^{T_{f}} C_{p,m}\) dT, where the left side represents the heat of reaction and the right side corresponds to the heat capacity of the mixture integrated over temperature range. \(\Delta H_{r}\) denotes the enthalpy change of reaction and \(C_{p,m}\) is the mixture heat capacity. Expressing \(C_{p,m}\) as the weighted sum of individual heat capacities, we can rearrange the energy balance equation and solve for f, leading to the equation \(f=\frac{1}{1+\phi\left(T_{f}\right)}\), where \( \phi\left(T_{f}\right)=\frac{\int_{T_{0}}^{T_{f}} C_{p,m}\, dT}{-\Delta H_{r}}\)

03

Deriving the Heat Capacity Expression

Substituting for the heat capacity expressions into the above equation and integrating, we obtain an explicit expression for \( \phi\left(T_{f}\right)\).

04

Equating Fractional Conversion Expressions

Equating the initial \(f\) derived from equilibrium analysis with the \(f\) derived from energy balance, we substitute the expressions for \(K_{p}\) and \( \phi\left(T_{f}\right)\) to obtain the explicit equation for \( T_{f}\) as the unknown, then rearranging to the form \( \psi\left(T_{f}\right)=0\).

05

Performing Numerical Calculations

Prepare a spreadsheet with columns for \(P, T_{f}, f, K_{p}\), \( \phi, \text { and } \psi.\) Use Goal Seek or Solver to solve the equation \( \psi\left(T_{f}\right)=0\) for \( T_{f}\). Record the final fractional conversions for each pressure input. Plot \( T_{f}\) and \(f\) against \(P\) on a logarithmic scale.

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Key Concepts

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

Chemical Equilibrium Constant

Understanding the chemical equilibrium constant is essential when studying reactions like ethane dehydrogenation. This constant, denoted as Kp for a gas-phase reaction, relates the partial pressures of the products and reactants at equilibrium. It is crucial to note that Kp is temperature-dependent, which is evident in the provided formula $$K_{p}(T)=7.28 \times 10^{6} \exp [-17,000 / T(\mathrm{K})]$$where T is the temperature in Kelvin. The exponential relationship indicates that as temperature changes, the equilibrium position also shifts—a concept known as Le Chatelier's Principle. In practical terms, if you increase the temperature, the equilibrium favors the endothermic direction of the reaction, adjusting Kp accordingly.

Understanding Kp is not just academic; it has real-world applications in industrial chemical processes, such as the optimization of conditions for maximum yield. In our exercise example, Kp is used to calculate the fractional conversion of ethane, which is a measure of how much ethane has been converted to ethylene and hydrogen in the adiabatic reactor.

• Kp indicates the extent of reaction at equilibrium.
• It is temperature sensitive and follows the van't Hoff equation.
• It is a critical factor in determining reaction yields in industrial applications.

Enthalpy Change of Reaction

The enthalpy change of reaction, denoted as g(H_r), represents the heat absorbed or released during a chemical reaction at constant pressure. It's a measure of the difference in enthalpy between reactants and products. The sign of H_r indicates whether a reaction is exothermic (negative H_r) or endothermic (positive H_r). In our exercise, the enthalpy change for the dehydrogenation of ethane at $$1273 \mathrm{K}$$ is $$+145.6 \mathrm{kJ}$$, suggesting that the reaction absorbs heat from its surroundings, thus being endothermic.

In an adiabatic reactor, where no heat is exchanged with the environment, the temperature inside the reactor changes as a direct consequence of H_r. If H_r is positive, the temperature will decrease unless the reaction is coupled with other reactions or processes that can compensate for the loss of heat—making the understanding of H_r vital for reactor design and operation.

• Positive H_r indicates heat is absorbed (endothermic reaction).
• Negative H_r indicates heat is released (exothermic reaction).
• It helps in designing energy balances for chemical reactors.

Heat Capacity

Heat capacity, often denoted by C_p, is a property that determines how much heat is needed to raise the temperature of a substance by a given amount. Specifically, for each of the substances in our reaction—ethane, ethylene, and hydrogen—the heat capacity is expressed as a function of temperature. These expressions take into account the fact that heat capacity is not constant but varies with temperature: $$\left(C_{p}\right)_{\mathrm{C}_{2} \mathrm{H}_{4}} = 9.419 + 0.1147 T(\mathrm{K})$$$$\left(C_{p}\right)_{\mathrm{H}_{2}} = 26.90 + 4.167 \times 10^{-3} T(\mathrm{K})$$$$\left(C_{p}\right)_{\mathrm{C}_{2} \mathrm{H}_{6}} = 11.35 + 0.1392 T(\mathrm{K})$$

In the context of the adiabatic reactor, these capacities allow us to calculate the overall heat capacity of the reaction mixture, which is necessary for the energy balance equation and, ultimately, for determining the reactor's outlet temperature and the conversion of ethane.

• Heat capacities vary with temperature and are provided in temperature-dependent formulas.
• They are integral to calculating the energy balance in chemical reactors.
• Understanding heat capacities is critical when designing processes for thermal efficiency.

Fractional Conversion of Ethane

The fractional conversion of ethane is a measure of the extent to which ethane is converted into other products, such as ethylene and hydrogen, in a chemical process. In the scenario of the adiabatic reactor, where pure ethane is fed into the system, the fractional conversion f is a crucial performance metric. It tells us how efficiently the reactant is being utilized under the set conditions. Respecting the equation from the exercise $$f=\left(\frac{K_{p}}{P+K_{p}}\right)^{1 / 2}$$we interpret it as the ratio of ethane that has reacted to form the products at equilibrium. Moreover, it highlights the relationship between conversion, equilibrium constant, and pressure. A high equilibrium constant relative to the pressure suggests a high conversion; conversely, high pressure relative to Kp suggests lower conversion.

Proficiency in determining fractional conversion is key for engineers and chemists in process controls and optimization, thus a foundational topic for students.

• It is a direct indicator of process efficiency.
• Dependent on both the equilibrium constant and system pressure.
• Essential for process optimization and reactor design.