Ethane reacts with oxygen to form carbon dioxide and water.
a. Derive a system of equations for the unknowns x1, x2, x3 and x4, which represent the number of molecules of ethane, oxygen, carbon dioxide and water
b. Solve the system of equations
To derive a system of equations for the reaction of ethane with oxygen, we can start by writing the balanced chemical equation:
C2H6 + O2 -> CO2 + H2O
Now, let's assign variables to represent the number of molecules of each substance involved in the reaction. Let:
x1 = number of molecules of ethane (C2H6)
x2 = number of molecules of oxygen (O2)
x3 = number of molecules of carbon dioxide (CO2)
x4 = number of molecules of water (H2O)
Based on the balanced equation, we can establish the following stoichiometric ratios:
C2H6:O2 = 1:6
C2H6:CO2 = 1:2
C2H6:H2O = 1:3
Now, we can express these ratios as equations:
Equation 1: x1 = x2/6
Equation 2: x1 = x3/2
Equation 3: x1 = x4/3
Additionally, we know that the total number of molecules in the reactants equals the total number of molecules in the products, according to the law of conservation of mass. This gives us another equation:
Equation 4: x1 + x2 = x3 + x4
We now have a system of four equations (Equations 1-4) with four unknowns (x1, x2, x3, x4).
To solve this system of equations, we can substitute the value of x1 from Equation 1 into Equations 2, 3, and 4.
Using Equation 1:
x1 = x2/6
Substituting in Equation 2:
x2/6 = x3/2
Simplifying:
x2 = 3x3
Substituting in Equation 3:
x2/6 = x4/3
Simplifying:
x2 = 2x4
Substituting in Equation 4:
x1 + x2 = x3 + x4
x2/6 + x2 = x3 + x4
Simplifying:
x2/6 = x3 + x4 - x2
Substituting the value of x2 from Equation 3 and Equation 4 into Equation 2:
3x3 = x3 + x4 - 2x4
Simplifying:
3x3 = -x4
From the above equation, we can see that x4 = -3x3.
Now we have equations for each variable in terms of the other:
x2 = 3x3
x1 = x2/6
x1 = x3/2
x4 = -3x3
To find the values of x1, x2, x3, and x4, we can solve this system of equations. This can be done using various methods, such as substitution or elimination.