To determine the rate of the reaction relative to the initial rate when the partial pressure of C is 0.20 atm, we first need to understand the stoichiometry of the reaction and how the rates relate to the concentrations of the reactants and products.
Understanding the Reaction
The reaction given is:
A(g) + 2B(g) → C(g) + D(g)
This is an elementary reaction, meaning that the rate law can be directly derived from the stoichiometry of the reaction. The rate of the reaction can be expressed as:
Rate = k[A]^1[B]^2
Here, k is the rate constant, and the exponents correspond to the coefficients in the balanced equation.
Initial Conditions
Initially, we have:
- pA = 0.60 atm
- pB = 0.80 atm
The initial rate of the reaction can be calculated using these initial pressures:
Initial Rate = k(0.60)^1(0.80)^2
Change in Partial Pressures
As the reaction progresses, the partial pressures of the reactants will decrease while the partial pressures of the products will increase. For every mole of A that reacts, two moles of B are consumed, and one mole each of C and D are produced. When the partial pressure of C reaches 0.20 atm, we can find out how much A and B have reacted.
Using Stoichiometry
From the balanced equation, we know that:
- 1 mole of A produces 1 mole of C
- 2 moles of B are consumed for every mole of A that reacts
If pC = 0.20 atm, then 0.20 atm of A has reacted:
- Change in pA = -0.20 atm
- Change in pB = -2 * (0.20 atm) = -0.40 atm
Calculating New Partial Pressures
Now we can find the new partial pressures of A and B:
- pA (new) = 0.60 atm - 0.20 atm = 0.40 atm
- pB (new) = 0.80 atm - 0.40 atm = 0.40 atm
New Rate Calculation
Now we can calculate the new rate of the reaction:
New Rate = k(0.40)^1(0.40)^2
Now, let's express the new rate in terms of the initial rate:
New Rate = k(0.40)(0.16) = k(0.064)
Initial Rate was:
Initial Rate = k(0.60)(0.64) = k(0.384)
Finding the Relative Rate
To find the rate of the reaction relative to the initial rate, we take the ratio of the new rate to the initial rate:
Relative Rate = (New Rate) / (Initial Rate) = (k(0.064)) / (k(0.384))
The k cancels out:
Relative Rate = 0.064 / 0.384 = 1 / 6
Final Result
The rate of the reaction relative to the initial rate when the partial pressure of C is 0.20 atm is 1/6. This means that the reaction is proceeding at a significantly slower rate compared to its initial rate, which is typical as reactants are consumed and their concentrations decrease over time.
