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Methane and \(30 \%\) excess air are to be fed to a combustion reactor. An inexperienced technician mistakes his instructions and charges the gases together in the required proportion into an evacuated closed tank. (The gases were supposed to be fed directly into the reactor.) The contents of the charged tank are at \(25^{\circ} \mathrm{C}\) and 4.00 atm absolute. (a) Calculate the standard internal energy of combustion of the methane combustion reaction. \(\Delta \hat{U}_{c}^{\circ}(\mathrm{kJ} / \mathrm{mol}),\) taking \(\mathrm{CO}_{2}(\mathrm{g})\) and \(\mathrm{H}_{2} \mathrm{O}(\mathrm{v})\) as the presumed products. Then prove that if the constant-pressure heat capacity of an ideal-gas species is independent of temperature, the specific internal energy of that species at temperature \(T\left(^{\circ} \mathrm{C}\right)\) relative to the same species at \(25^{\circ} \mathrm{C}\) is given by the expression $$\hat{U}=\left(C_{p}-R\right)\left(T-25^{\circ} \mathrm{C}\right)$$ where \(R\) is the gas constant. Use this formula in the next part of the problem. (b) You wish to calculate the maximum temperature, \(T_{\max }\left(^{\circ} \mathrm{C}\right),\) and corresponding pressure, \(P_{\max }(\text { atm }),\) that the tank would have to withstand if the mixture it contains were to be accidentally ignited. Taking molecular species at \(25^{\circ} \mathrm{C}\) as references and treating all species as ideal gases, prepare an inlet-outlet internal energy table for the closed system combustion process. In deriving expressions for each \(\dot{U}_{i}\) at the final reactor condition \(\left(T_{\max }, P_{\max }\right),\) use the following approximate values for \(C_{p_{i}}\left[\mathrm{k} J /\left(\mathrm{mol} \cdot^{\circ} \mathrm{C}\right)\right]: 0.033 \mathrm{for} \mathrm{O}_{2}, 0.032\) for \(\mathrm{N}_{2}, 0.052 \mathrm{for} \mathrm{CO}_{2},\) and \(0.040 \mathrm{for} \mathrm{H}_{2} \mathrm{O}(\mathrm{v}) .\) Then use an energy balance and the ideal-gas equation of state to perform the required calculations. (c) Why would the actual temperature and pressure attained in a real tank be less than the values calculated in Part (a)? (State several reasons.) (d) Think of ways that the tank contents might be accidentally ignited. The list should suggest why accepted plant safety regulations prohibit the storage of combustible vapor mixtures.

Step by step solution

01

Calculate the combustion reaction

Determine the balanced chemical equation for methane combustion with \(30 \%\) excess air. The combustion equation is \(CH_{4} + 2O_{2} \rightarrow CO_{2} + 2H_{2}O\). Considering \(30 \%\) excess air, the reaction becomes: \(CH_{4} + 2.6O_{2} \rightarrow CO_{2} + 2H_{2}O + 0.6O_{2}\).

02

Calculate the standard internal energy of combustion

Use standard enthalpies of formation to calculate the standard internal energy change for the reaction using the formula \( \Delta U = \Delta H - T \Delta S\) and data from literature or textbooks. Enthalpies and entropies of formation are usually given at standard temperature and pressure (STP), i.e., \(25^{\circ} \mathrm{C}\) and 1 atm.

03

Derive the specific internal energy formula

Demonstrate that the specific internal energy of an ideal gas species at a certain temperature relative to \(25^{\circ} \mathrm{C}\) equals \(\hat{U} = (C_{p}-R) (T-25^{\circ} \mathrm{C})\). This comes from the definition of specific heat capacity at constant pressure \(C_{p}\), and the fact that for an ideal gas, the change in enthalpy equals the change in internal energy plus the product of pressure and change in volume, which leads to a difference in the constant volume and constant pressure heat capacities equal to the gas constant \(R\).

04

Prepare energy table

Calculate the vortex quantities for the combustion process using the formula \(U = n (C_{p}-R) (T-25^{\circ} \mathrm{C})\). Use given values for molar heat capacities and the vortex quantities from the balanced chemical equation. Solve the expressions for each molecule at the final reactor condition \(\left(T_{\max }, P_{\max }\right)\) using the provided values for each species’ specific heat capacities.

05

Perform energy balance calculation

Apply energy conservation to calculate the max temperature and pressure. The energy released by combustion equals the increase in internal energy of the combustion products. Then use ideal-gas equation of state \(PV=nRT\) to solve for pressure.

06

Explaining fluctuation in realistic values

Explain why the real temperature and pressure might be less than the calculated values, such as heat loss to the environment, non-ideality of gases, incomplete combustion, and others.

07

Identify potential ignition sources

List sources of ignition which include static electricity, external heat, sparks or open flames, among other things.

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Key Concepts

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

Combustion Reactions

Combustion reactions are chemical processes in which a fuel reacts with an oxidant, typically oxygen, to release energy. This energy is often in the form of heat and light. In the case of methane combustion, the balanced chemical equation is:

• CHâ‚„ + 2Oâ‚‚ → COâ‚‚ + 2Hâ‚‚O.

When there is a 30% excess of air, extra oxygen molecules are present, making the equation:

• CHâ‚„ + 2.6Oâ‚‚ → COâ‚‚ + 2Hâ‚‚O + 0.6Oâ‚‚.

This reaction is exothermic, meaning it releases energy. The energy from combustion can be harnessed in engines and power plants.
To find out how much energy a reaction releases or absorbs, chemists calculate the reaction's internal energy change (∆U). This involves using the energy stored in chemical bonds and determining how much changes as new bonds form.
Understanding combustion reactions is crucial in designing safe and efficient systems for energy production.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation that describes the behavior of an ideal gas. It is expressed as:

• PV = nRT

Here:

• P represents pressure,
• V is volume,
• n denotes the number of moles,
• R is the universal gas constant, and
• T stands for temperature in Kelvin.

This equation helps predict how a gas will respond to changes in pressure, volume, or temperature when other conditions remain constant.
In our scenario, calculating the maximum pressure ( P_{ ext{max}}) involved using the Ideal Gas Law after combustion in a closed tank, starting with conditions like 25°C and 4.00 atm, is critical.
Understanding these calculations is essential in determining whether a system can withstand certain pressures and temperatures during chemical reactions.

Internal Energy

Internal energy is the total energy contained within a system. It includes kinetic energy of molecules and the energy stored in chemical bonds.
For a reaction, such as methane combustion, internal energy change (∆U) can be calculated using the formula:

• ∆U = ∆H - T∆S.

Here:

• ∆H is the enthalpy change, and
• T∆S accounts for energy affecting entropy at temperature T.

Through combustion, the internal energy of reactants decreases as energy is released.
When analyzing ideal gases, specific internal energy changes can be calculated using \[\hat{U} = (C_p - R)(T - 25^{\circ}C)\]This equation indicates how internal energy varies with temperature, factoring in heat capacity (C_p) and the constant R. Understanding changes in internal energy helps predict how energy is transferred during reactions and the resulting temperature variations.

Heat Capacity

Heat capacity is the amount of heat energy required to raise the temperature of a substance by one degree Celsius. It is an intrinsic property of matter that determines how substances respond to added or removed heat. In the context of ideal gases, the heat capacity at constant pressure ( C_p) is critical.
For our calculation, specific heat capacities ( C_p) for gases involved in the reaction are necessary. These values are typically provided in scientific problems and can vary for each gas. For example:

• Oâ‚‚ has a C_p of 0.033 kJ/(mol•°C),
• Nâ‚‚ has a C_p of 0.032 kJ/(mol•°C),
• COâ‚‚ has a C_p of 0.052 kJ/(mol•°C), and
• Hâ‚‚O(v) has a C_p of 0.040 kJ/(mol•°C).

Using these values helps calculate temperature changes in the system and the energy required or released during reactions. Having a deeper understanding of heat capacity allows engineers and scientists to design systems that effectively manage thermal energy.

Safety Regulations

Safety regulations are critical in preventing accidents in industrial and laboratory settings. They provide guidelines to mitigate risks associated with processes like combustion, where flammable gases are involved.
In our scenario, storing combustible mixtures in tanks can be hazardous. Ignition sources, such as static electricity or open flames, can accidentally ignite combustible vapors, causing explosions.
Following safety regulations means proper handling and storing of materials, using equipment designed to withstand foreseeable pressures and temperatures, and employing sensors to detect and manage leaks or ignition risks.
These protocols aim to protect workers and the environment from dangerous chemical reactions and their potential consequences. Understanding and adhering to safety regulations is as crucial as understanding the chemistry of reactions themselves.