An empty 4.00-L steel vessel is filled with 1.00 atm of CH41g2 and 4.00 atm of O21g2 at 300 °C. A spark causes the CH4 to burn completely, according to the equation CH41g2 + 2 O21g2¡CO21g2 + 2 H2O1g2 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 J1mol # °C2, and the heat capacity of steel is 0.449 J1g # °C2?

Question

An empty 4.00-L steel vessel is filled with 1.00 atm of CH41g2 and 4.00 atm of O21g2 at 300 °C. A spark causes the CH4 to burn completely, according to the equation CH41g2 + 2 O21g2¡CO21g2 + 2 H2O1g2 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>1mol # °C2, and the heat capacity of steel is 0.449 J>1g # °C2?

Accepted Answer

Hi everyone for this problem, it reads an empty 3.5 liter steel container was filled with a mixture consisting of 1.20 atmospheres of propane and 7.20 atmospheres of oxygen gas at 350 degrees Celsius according to the equation. The standard entropy change equals negative 2220 kg jewels. The probe was combusted using a spark given that the mass of the steel container is 15.700 g. The average molar heat capacity of the mixture is 14.5 jewels per mole degrees Celsius and the heat capacity of steel is 0.449 jules per gram degree Celsius, determine the final temperature inside the container after the combustion. So what we want to do here is answer the question of what is the final temperature inside the container after the combustion? So let's go ahead and calculate the heat of the reaction. Okay, in order for us to calculate the heat of the reaction, we're going to need to know the moles of the reaction and this wasn't given. So because we're dealing with gasses, let's go ahead and write out our ideal gas law PV equals N R. T. Pressure, times volume equals the number of moles, times gas constant r times temperature in kelvin. So we want to calculate the moles or we need the moles in order to calculate the heat of the reaction. So let's rearrange this equation so that we're solving for moles. Okay, we'll divide both sides by R. T. And we're going to get the number of moles is equal to P. V. Over R. T. So we need to calculate the moles of propane and the moles of oxygen gas. Alright, so let's go ahead and first calculate our moles of propane. Okay, so our most, let's write this in a different color. So our moles of propane is going to equal. So we need P. V. Over R. T. We know the pressure, we're told the pressure is 1. atmospheres. We know the volume were given its 3. leaders are is a gas constant that we should have memorized. This is 0.08-0 six leaders atmosphere over mole Calvin. And temperature were given is 350°C. So we need to convert this degrees C to Calvin. So we're going to take degrees Celsius and add .15. Okay, so once we do this calculation, we're going to get the moles of propane is equal to 0. moles. So let's go ahead and do the same thing. But now we're going to calculate the moles of oxygen gas. So it's going to be the same thing P. V over R. T. So let's go ahead and plug in. The values are pressure for oxygen gas is 7.20 atmospheres. The volume is 3.50 liters. This is over gas constant. R 0. leaders atmosphere over mole kelvin times temperature. So the temperature is the same. So we're going to take 2 350 degrees Celsius and add 273.15. So once we do this calculation, we're going to get the moles for oxygen gas is going to equal 0.49 to eight moles of oxygen gas. Okay, so now that we know the moles, we can now calculate the heat of the reaction. Okay, so our heat of the reaction is going to equal the moles of propane times the Anthony change of the reaction. So this is going to be times negative. 2220. And this was given in the problem. So, negative 2220 kg joules per mole of propane. Okay, So once we do this calculation, we're going to get the heat of the reaction is equal to negative 182.3286. Kill it jules. All right, so now that we know the heat of the reaction. We're going to rewrite our reaction out so that we can figure out the total moles of gas in order to figure out the total moles of gas, we need to create an ice table because we need to look at what is going to be in the equilibrium row of our ice table. So, to create an ice table, we first need to write out the reaction. So our reaction is up at the top. Our reaction was given or the equation was given. And that equation is propane gas Plus five moles of oxygen gas yields three moles of carbon dioxide gas plus four moles of water as a gas. So we're going to go ahead and write out ice Okay, where I stand for our initial concentration, C is our change in concentration and E. is equilibrium. So we just calculated our initial concentration of propane and we said the initial concentration of propane is 0.08213. And we also calculated our initial concentration or moles of oxygen gas and it is 0.49-8. And we have no products. So that means our reaction is moving in the four direction, which means our reactant are getting consumed and our products and we have products forming. So for our change row for our reactant we're going to have minus Okay, And our change is going to be minus 0.8213 for oxygen gas, we have five moles of oxygen gas. So it's going to be negative five times our change which is 0.8 For our carbon dioxide gas, we have three moles of it and we're going to multiply this by the change which is 0.08213. And for our water as a gas we have four moles and the sinus plus. So let's remember our sign. So it's going to be plus four moles of our change, which is 0.08213. So for our equilibrium role, we're going to combine our first two rows are iro and R C ro. So for propane, Our equilibrium is zero for oxygen gas are equilibrium row, we're going to take 0.4928 minus five times 0.8213. And we get 0.82154, our carbon dioxide. When we combine the two rows, we get 0.24639 and four are Water as a gas. Our equilibrium row is going to be 0.32852. So now we know what our equilibrium concentrations are. So what this means then is the total moles of gas is going to equal the summation of our equilibrium row of the ice table. So we're going to get we're going to add everything in our equilibrium in the row. So we have zero plus 0.8215 plus 0.24639 plus 0.32852. So our total modes of gas is equal to 0. moles. So now we know our total moles of gas. Okay, so let's go ahead and convert our heat of reaction hear from killer jewels, two jewels. So we're gonna go from killer jewels to jules. And we're going to do that by multiplying by 1000. So our heat of reaction in jewels is negative 182, 328. Jewels. So now we can go ahead and calculate the heat of the vessel. Okay, so the heat of our vessel is going to equal the negative heat of our reaction. So we're gonna take our heat of our reaction in jewels. So it's going to be positive now because we're switching the signs. So we have 182, 328 0. jewels. And this is going to equal The moles, the total moles. So 0. moles. Right that clear 706 moles. And we're gonna multiply this by We're multiplying this by the heat capacity of the mixture. So we're going to multiply this 14. jules. Her mole degree Celsius Times the final temperature -350°C plus. Okay, so this is gonna be plus. We're told in the problem the mass of the steel container is 15.700 g kilograms. Excuse me. So we want to go from kilograms to grams. So we're going to multiply this by 1000 because there's 1000 g and one kg. And we're going to multiply it by the heat capacity of the steel, which is 0. jewels per gram degree Celsius times the final temperature minus 350°C. Okay, so let's go ahead and simplify this a bit. So if we simplify it, We're going to get for the left side 182,328. jules is going to equal. We're gonna simplify this first part right here that I just underlined. So that first part ends up becoming 9. jewels per degree Celsius times the final temperature minus 350 degrees Celsius plus. Now we're going to simplify the second half. So this is going to be plus 7,049. Jewels per degree C Times the final temperature -350°C. Okay, so remember our goal here, is to calculate the final temperature. Okay, so this is what we're trying to solve for here. Alright, so let's go ahead and simplify this some more. All right, so the left side remains the same. 182,328. Jules is equal to We're going to get 9. 7 3 jewels degree C times the final temperature -3334. jewels plus 7049. jewels degrees Celsius times the final temperature minus 246. Weight 7225 jewels. Okay, so we're going to get we're going to pull out the final temperature or isolate final temperature. T final when we isolate it, we get T final is going to equal the following. Okay, so here we'll see that our units of jewels cancel and we're left with degrees Celsius. So once we do this calculation, our final temperature Equals 375.8299. And we'll go ahead and round this. So our final temperature equals 376°C. So this is going to be the final temperature inside the container after the combustion. That's it for this problem. I hope this was helpful.

Verified Solution

Key Concepts

Stoichiometry of Combustion Reactions

Heat Transfer and Specific Heat Capacity

Thermodynamics and Enthalpy Change