How would I go about solving this problem? Can someone show me an equation that can be used?
The decomposition of dimethyl ether at 504°C is a first-order reaction with a half-life of 27 min.
(CH3)2O(g) CH4(g) + H2(g) + CO(g)
What will be the total gas pressure after 2.18 h? [Hint: (CH3)2O and its decomposition products are the only gases present in the reaction vessel.
I found the pressure at 2.18 h to be 23.54. I do not know where to go from there
To solve this problem, you can use the first-order reaction equation and the concept of half-life to determine the amount of (CH3)2O remaining after 2.18 hours.
The half-life of the reaction is given as 27 minutes, which means that after every 27 minutes, the concentration of (CH3)2O will be reduced by half. To convert this to hours, we divide the half-life by 60: 27 min / 60 min = 0.45 hours.
Now, we need to calculate the number of half-lives that have passed in 2.18 hours: 2.18 hours / 0.45 hours = 4.8444.
Since we can't have a fraction of a half-life, let's round down to the nearest whole number of half-lives, which is 4. This means that after 4 half-lives, the concentration of (CH3)2O will have reduced to 1/2^4 = 1/16 of its initial concentration.
Now, to find the total pressure of the gases after 2.18 hours, we need to consider the stoichiometry of the reaction. According to the balanced equation, for every one mole of (CH3)2O that decomposes, one mole of CH4, one mole of H2, and one mole of CO are produced.
Since (CH3)2O is the only gas present initially, its concentration will be equal to the total gas pressure.
Let's say the initial pressure of (CH3)2O is P0. After 4 half-lives, the concentration of (CH3)2O will be P0/16, and the concentrations of the other gases will be P0/16 each.
Now, according to Dalton's law of partial pressures, the total pressure of the gases is the sum of the partial pressures:
Total pressure = Pressure of (CH3)2O + Pressure of CH4 + Pressure of H2 + Pressure of CO
Total pressure = P0/16 + P0/16 + P0/16 + P0/16
Total pressure = 4*(P0/16)
Total pressure = P0/4
Therefore, the total gas pressure after 2.18 hours is P0/4.
Since the initial pressure of (CH3)2O is not provided, we cannot calculate the exact numerical value of the total gas pressure. However, we know that the pressure at 2.18 hours is 23.54 (as you mentioned), so the total gas pressure will be 23.54/4 = 5.885 atm (rounded to three decimal places).
So, the total gas pressure after 2.18 hours is 5.885 atm (approximately).