Question

Carbon at \(25^{\circ} \mathrm{C}, 1\) atm enters a reactor operating at steady state and burns with oxygen entering at \(127^{\circ} \mathrm{C}, 1 \mathrm{~atm}\). The entering streams have equal molar flow rates. An equilibrium mixture of \(\mathrm{CO}_{2}, \mathrm{CO}\), and \(\mathrm{O}_{2}\) exits at \(2727^{\circ} \mathrm{C}, 1\) atm. Determine, per kmol of carbon, (a) the composition of the exiting mixture. (b) the heat transfer between the reactor and its surroundings, in \(\mathrm{kJ}\). Neglect kinetic and potential energy effects.

Short answer

The composition of the exiting mixture can be found using the equilibrium constant at 2727°C. Heat transfer requires energy balance based on combustion enthalpies.

Step-by-step solution

- Specify the chemical reaction

Consider the combustion reaction of carbon with oxygen: \[ C + \frac{1}{2}O_2 \rightarrow CO \ C + O_2 \rightarrow CO_2 \] Since we have CO and CO2 in the product, the actual reaction will be a combination of these two.

- Write the equilibrium expression

Given the exiting products are in equilibrium, we need to write the equilibrium expression for the reaction. Using the reaction: \[ C + \frac{1}{2}O_2 \rightarrow CO \ C + O_2 \rightarrow CO_2 \] the equilibrium constant expression for this can be written based on the partial pressures: \[ K = \frac{P_{CO}^2}{P_{CO_2} \times P_{O_2}} \]

- Use the specified conditions

The incoming streams have equal molar flow rates, implying for 1 kmol of carbon there is 1 kmol of oxygen. With these as inputs, the equilibrium mole fractions will need to be found given the exit temperature of 2727°C.

- Apply mass balance

For every 1 kmol of carbon entering, the output must match the molar flow rates of the product. Define the exits as fractions x for CO2, y for CO, and z for O2 where: \[ x + y + z = 1 \]

- Solve for the equilibrium composition

Use the given constant K and the temperature to solve for x, y, and z such that: \[ K = \frac{(y)^2}{x \times z} \]. This will require using the known equilibrium constant value at 2727°C.

- Apply energy balance

Determine the heat transfer (𝑄) between the reactor and surroundings using enthalpy changes. The enthalpy change for each mole of reactant to product can be found using standard enthalpies.

- Calculate the overall enthalpy change

Considering standard enthalpies of combustion/formation: \[ \text{Heat Energy } Q = \text{nΔH}_{comb}\text{(CO)} + \text{nΔH}_{comb}\text{(CO2)} - \text{nΔH}_{form}\text{(C,O2)} \]. Calculate the sum using specific enthalpies and given temperature.

Key concepts

Equilibrium Composition

Understanding equilibrium composition is crucial for reactions like combustion. In chemical reactions, equilibrium is achieved when the rate of the forward reaction equals the rate of the reverse reaction. Let's consider the combustion of carbon with oxygen. The reaction leads to the formation of carbon dioxide (CO₂) and carbon monoxide (CO). Given the problem, the equilibrium mixture includes CO₂, CO, and O₂.

To find the equilibrium composition, we use the equilibrium constant (K). This constant depends on temperature and reflects the ratio of product concentrations to reactant concentrations. For our combustion reaction, at the specified temperature of 2727°C, the equilibrium expression based on partial pressures is:

\( K = \frac{P_{CO}^2}{P_{CO_2} \times P_{O_2}} \).

We need to determine the ratios of CO₂, CO, and O₂ in the equilibrium mixture. Using a balance equation where x, y, and z represent the mole fractions of CO₂, CO, and O₂ respectively, we get:

\( x + y + z = 1 \).

Solving these equations with the equilibrium constant will give us the composition of the exiting gases.

Mass Balance

Mass balance ensures that the mass entering the reactor equals the mass exiting. This principle is fundamental in chemical engineering. For our combustion reaction, we have 1 kmol of carbon and 1 kmol of oxygen entering the reactor.

The reactants and products are linked through stoichiometric relationships. Carbon and oxygen combine in certain fixed ratios to form CO₂ and CO. Our task is to establish a mass balance for the reaction products.

Given the problem: for every 1 kmol of carbon, the products must equal the molar flow rates of carbon derivatives. Let's denote the exit products by:

- x kmol of CO₂
- y kmol of CO
- z kmol of O₂

Using the relationship:

\( x + y + z = 1 \),

and the stoichiometry from the equilibrium constant equation, we solve for x, y, and z. This ensures that the total mass entering the reactor is the same as the mass exiting.

Energy Balance

Energy balance in a reactor shows that the energy entering equals the energy exiting, plus any energy stored or lost. In steady-state operations, where storage is not considered, we ensure the total energy input equals the energy output.

In this combustion problem, we need to assess the heat transfer between the reactor and its surroundings. The contributions to energy input include the enthalpy of the reactants: carbon and oxygen at their inlet temperatures. Meanwhile, the energy output includes the enthalpy of the product gases at the exit temperature.

For each reactant and product, we calculate the enthalpic energy change based on their standard enthalpies of formation (ΔH_f) and specific heat capacities (C_p). Next, we sum these enthalpic changes over all reactants and products based on their molar flow rates and temperature differences.

The energy balance equation becomes:

\( Q_{in} + H_{reactants} = Q_{out} + H_{products} \).

Solving for Q, we determine the net heat transfer in or out of the reactor.

Enthalpy Change

Enthalpy change (ΔH) represents the heat absorbed or released during a reaction at constant pressure. For our combustion reaction, we determine the enthalpy change of the system using the standard enthalpies of formation for carbon (C), oxygen (O₂), carbon monoxide (CO), and carbon dioxide (CO₂).

We start with the combustion equations:

1. \( C + \frac{1}{2}O_2 \rightarrow CO \)
2. \( C + O_2 \rightarrow CO_2 \)

The enthalpy change for each equation is calculated. Summing these changes, considering the molar flow rates of each product (CO, CO₂, O₂), gives the overall enthalpy change for the reaction.

The net change is computed as:

\( ΔH_{total} = (n_{CO} ΔH_{comb, CO}) + (n_{CO2} ΔH_{comb, CO2}) - (n_{C} ΔH_{form, C}) \).

This combined enthalpy change informs us of the heat released or absorbed, thus helping in the energy balance calculations.
Understanding enthalpy changes helps in designing efficient reactors and determining heat transfer requirements.