a gaseous mixture contains 1 mol of O2 and 2 mol of CO2, exerting a total pressure of 1.5 atm. what amount of CH4 must be added to the mixture in order to increase the total pressure to 2 atm. (assume no gas escapes and there is no change in total volume)
the answer is 1 mol but i do not know how to get to that answer!
P1/n1 = P2/n2
P1 is the original total pressure (1.5 atm), n1 is the total moles: (1 mol O2 + 2 mol CO2= 3 moles total.)
P2 is the new total pressure (2 atm), n2 is the new total moles, which is 3 moles + x moles CH4 added.
1.5/3 = 2/ (3+x)
solve for x
x= moles CH4 added= 1 mol
To solve this problem, we can use Dalton's law of partial pressures, which states that the total pressure exerted by a mixture of non-reacting gases is equal to the sum of the pressures that each gas would exert if it were present alone.
Here's how we can approach this problem step by step:
1. List the given information:
- Moles of O2 (oxygen): 1 mol
- Moles of CO2 (carbon dioxide): 2 mol
- Total pressure of the mixture: 1.5 atm
- Desired total pressure: 2 atm
2. Calculate the partial pressures of O2 and CO2:
- The partial pressure of each gas is determined by multiplying its mole fraction by the total pressure.
- The mole fraction of each gas is the ratio of its moles to the total moles of the mixture.
Partial pressure of O2 = (moles of O2 / total moles) * total pressure
= (1 mol / (1 mol + 2 mol)) * 1.5 atm
= (1/3) * 1.5 atm
= 0.5 atm
Partial pressure of CO2 = (moles of CO2 / total moles) * total pressure
= (2 mol / (1 mol + 2 mol)) * 1.5 atm
= (2/3) * 1.5 atm
= 1.0 atm
3. Determine the partial pressure of CH4 (methane):
- Since we want to increase the total pressure to 2 atm, we need to find the partial pressure of CH4 required to make up the difference.
- Let's assume x mol of CH4 is added.
Partial pressure of CH4 = (x mol / (1 mol + 2 mol + x mol)) * 2 atm
4. Set up an equation to solve for x:
Since the sum of the partial pressures equals the total pressure:
Partial pressure of O2 + Partial pressure of CO2 + Partial pressure of CH4 = Total pressure
0.5 atm + 1.0 atm + (x mol / (1 mol + 2 mol + x mol)) * 2 atm = 2 atm
5. Solve the equation for x:
0.5 + 1 + (2x / 3 + x) = 2 // Simplify the equation
1.5 + (2x / 3 + x) = 2 // Multiply both sides by 3 to eliminate the fraction
4.5 + 2x + 3x = 6 // Simplify again
5x + 4.5 = 6 // Combine like terms
5x = 1.5 // Subtract 4.5 from both sides
x = 0.3 mol // Divide both sides by 5
Therefore, the amount of CH4 (methane) that must be added to the mixture in order to increase the total pressure to 2 atm is 0.3 mol, or approximately 0.3 moles.