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 concept of the first-order reaction and the half-life to determine the rate constant (k). Once you have the rate constant, you can use it to calculate the concentration of the gas at 2.18 hours and then convert it to pressure. Here are the steps to follow:
Step 1: Determine the rate constant (k)
We know that the half-life of the reaction is 27 minutes. The relationship between the rate constant (k) and the half-life (t1/2) of a first-order reaction is given by the formula:
k = 0.693 / t1/2
Plugging in the value, we get:
k = 0.693 / 27 min
Step 2: Calculate the concentration of (CH3)2O at 2.18 hours
Since we know the initial concentration of (CH3)2O is unknown, we can assume it as "A" (moles/L). At any time t, the concentration of (CH3)2O remaining can be expressed as A × e^(-kt), where e is the mathematical constant approximately equal to 2.718.
To calculate the concentration at 2.18 hours, we need to convert it to minutes: 2.18 hours * 60 minutes/hour = 130.8 minutes.
The equation becomes: (CH3)2O = A × e^(-k * 130.8)
Step 3: Calculate the partial pressure of (CH3)2O at 2.18 hours
Now that we have the concentration of (CH3)2O at 2.18 hours, we can use the ideal gas law to calculate the partial pressure. The ideal gas law equation is:
PV = nRT,
where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvins.
Since the volume and number of moles are fixed, we can express the equation as:
P = nRT / V
Since we are given pressure in terms of atmospheres, we can use R = 0.0821 L.atm/mol.K as the ideal gas constant.
Step 4: Plug in values and calculate the total gas pressure
Now, we can calculate the partial pressure of (CH3)2O at 2.18 hours using the concentration (from step 2) in the ideal gas law equation.
The total gas pressure will be the sum of the partial pressures of all the gases in the reaction vessel, which are (CH3)2O, CH4, H2, and CO.
Add together the partial pressures of CH4, H2, and CO (which will be given in the problem) and the partial pressure of (CH3)2O (calculated in step 4) to get the total gas pressure.
By following these steps, you should be able to find the total gas pressure at 2.18 hours.