An empty 4.00-L steel vessel is filled with 1.00 atm of CH4(g) and 4.00 atm of O2(g) at 300 °C. A spark causes the CH4 to burn completely, according to the equation CH4(g) + 2 O2(g) → CO2(g) + 2 H2O(g) ΔH° = -802 kJ (b) What is the final temperature inside the vessel after combustion, assuming that the steel vessel has a mass of 14.500 kg, the mixture of gases has an average molar heat capacity of 21 J/(mol·°C), and the heat capacity of steel is 0.449 J/(g·°C)?
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Step 1: Calculate the initial moles of CH4 and O2 using the ideal gas law, PV = nRT. Here, P is the pressure, V is the volume, R is the ideal gas constant (0.0821 L atm/mol K), and T is the temperature in Kelvin.
Step 2: Use the stoichiometry of the balanced chemical equation CH4(g) + 2 O2(g) → CO2(g) + 2 H2O(g) to determine the moles of O2 required for complete combustion of CH4 and to find the limiting reactant.
Step 3: Calculate the total energy released during the combustion using the given ΔH° value and the moles of CH4 that reacted. Remember that the energy released will be absorbed by the steel vessel and the gases inside it.
Step 4: Calculate the heat absorbed by the steel vessel using its mass, specific heat capacity, and the change in temperature. Use the formula Q = mcΔT, where m is mass, c is specific heat capacity, and ΔT is the change in temperature.
Step 5: Calculate the final temperature of the gases using the total heat absorbed by the gases, their average molar heat capacity, and the moles of gases present after the reaction. Use the formula Q = nCΔT, where n is the number of moles, C is the molar heat capacity, and ΔT is the change in temperature.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Stoichiometry of Combustion Reactions
Stoichiometry involves the calculation of reactants and products in chemical reactions. In combustion reactions, such as the one between methane (CH4) and oxygen (O2), the balanced equation indicates the molar ratios of reactants consumed and products formed. Understanding these ratios is crucial for determining how much energy is released and how it affects the temperature of the system.
Heat transfer refers to the movement of thermal energy from one object to another, which can change the temperature of the objects involved. Specific heat capacity is the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius. In this scenario, the specific heat capacities of both the gas mixture and the steel vessel are essential for calculating the final temperature after the combustion reaction.
Thermodynamics is the study of energy transformations, particularly heat and work. The enthalpy change (ΔH) of a reaction indicates the heat released or absorbed during the reaction at constant pressure. In this case, the negative ΔH value of -802 kJ signifies that the combustion of methane releases energy, which will increase the temperature of the gases and the steel vessel, necessitating calculations to find the final temperature.
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