Methanol (CH₃OH) is made industrially in two steps from CO and H₂. It is so cheap to make that it is being considered for use as a precursor to hydrocarbon fuels, such as methane (CH₄):

Step 1. CO(g) + 2 H₂(g) S CH₃OH(l) ΔS° = - 332 J/K

Step 2. CH₃OH₁l₂ → CH₄(g) + 1/2 O₂(g) ΔS° = 162 J/K

(e) In what temperature range is step 1 spontaneous?

Video transcript

Hello. In this problem we are told the industrial preparation of acetic acid is a two step process. It is the chief component of vinegar. Where asked to calculate the temperature at which step two of the reaction becomes spontaneous. And so we are going to be focusing then on this second step of the two step process, we're trying to find the temperature. So we call that our relationship between the change in gift for energy under standard conditions and that of the entropy is given by this equation where the change in strategy is equal to the change in entropy minus our temperature times the change in S. P. So if we find that um our thermodynamic values, we can then solve for the temperature. So we will begin then by finding the change in the entropy for the reaction under standard conditions is equal to the sum of the change in entropy of formation of our products minus the sum of the change in the entropy of formation for our reactant. And so we'll make use of the values that are provided. So we have one mole of our product acetic acid times its value of negative 484.5 jules. Come on And we'll subtract off our reactant. So we have one mole of methanol times its value of negative 201 gold jewels per mole. And then we have One mole of carbon monoxide which has a value of -110.5 joules per mole. So our units of moles will cancel will be left with units of killed jules. So that entropy reaction is equal to then negative 173.0 individuals doing similar to find the change in gibbs. Free energy change and good for energy. Then for our reaction will be equal to the sum of the change in gibbs. Free energy of formation for our products minus the sum of the change in gibbs free energy of formation for our reactant. Making use of the values that are provided. We have then one mole of our product, acetic acid values negative 390 joules per mole. And we're gonna subtract off. Then our reactant, we have one mole of methanol times its value of negative 162.3 Children per mole. And then we have one mole With carbon monoxide which has the value of -137.2 killed joules per mole. Our units will cancel. We'll be left with kill jules. And so this works out to negative 90 0.5 kill jules. So now we'll take the equation we wrote above. It relates our thermodynamic terms and we'll solve for temperature temperature then will be equal to the change in gibbs. Free energy minus change in entropy divided by the negative of the change in our entropy. So are changing gift for energy for the reaction was negative, 90.5 killed jules minus that. The entropy which was negative, 173.0 gold jewels Divided by the change in entropy per step one, which we were told was negative two and 77.4 jewels per kelvin. So we need to convert our jewels to kill jules. So our jewels in the bottom will cancel. Jewels on the top are units of killed jules cancels, and we'll be left with units of Kelvin. And so our temperature, then, which reactions become spontaneous, is equal to 297 Kelvin. Thanks for watching. Hope. This helps.