The reaction X + Y --> products was studied using the method of initial rates. The initial rate of consumption of X was measured in three different experiments. What is the value of the rate constant, k?
*concentrations are in mol/L and initial rates are in molL-1s-1
Exp 1:
[X]initial=0.6
[Y]initial=0.6
Initial Rate=3.306
Exp 2:
[X]initial=1.2
[Y]initial=0.6
Initial Rate=1.653
Exp 3:
[X]initial=0.6
[Y]initial=1.8
Initial Rate=29.75
My first step was to determine the rate law. However, I got r=k[X]^-1[Y]^2
I check this over numerous times and don't know what I'm doing wrong because you can't have a negative exponent in a rate law.. can you?!
In general, the orders (exponents) in a rate law can turn out to be positive or negative rational numbers, so -1 is a possibility.
Ah, the wonders of chemistry! It seems like you might have made a little mistake while determining the rate law. Fear not, I shall help you figure it out with a touch of humor!
Now, let's juggle some equations. The general form of the rate law equation is:
rate = k * [X]^a * [Y]^b
Where 'a' and 'b' are the unknown exponents we need to determine.
In your case, you found that the rate law equation is r = k * [X]^-1 * [Y]^2. However, you're right, negative exponents are no fun in rate laws!
But don't worry, we just need to play with the algebra of exponents for a bit. If we invert [X]^-1, it becomes [X]^1, which is simply equal to [X]. So now we have:
r = k * [X] * [Y]^2
And that's more like it, right? No pesky negative exponents in the equation!
Now, we can compare the initial rates from the three experiments and their respective concentrations to determine the values of 'a' and 'b'. Let's plug in the values:
Exp 1: 3.306 = k * (0.6)^a * (0.6)^b
Exp 2: 1.653 = k * (1.2)^a * (0.6)^b
Exp 3: 29.75 = k * (0.6)^a * (1.8)^b
Since these equations involve exponential functions, we need to find a way to eliminate either 'a' or 'b' so we can solve for the remaining variable. We can do this by taking ratios!
Let's divide Exp 3 by Exp 1 to eliminate 'a':
(29.75/3.306) = [(0.6)^a * (1.8)^b] / [(0.6)^a * (0.6)^b]
Now, we can cancel out (0.6)^a, leaving us with:
8.999 = (1.8)^b / (0.6)^b
Simplifying further:
8.999 = 3^b
Now, I can see you're in a spot of reasoning here. You know that 3^2 is 9, but 3^1 is simply 3. So, b must equal 2. Aha!
Now that we have 'b', we can substitute it back into any of the earlier equations to solve for 'a'. Let's use Exp 1:
3.306 = k * (0.6)^a * (0.6)^2
3.306 = k * (0.6)^a * 0.36
Dividing by 0.36:
9.183 = k * (0.6)^a
Well, well, well! Now we have an equation that's screaming to solve for 'a'. And guess what? It's your turn to solve this abstractive puzzle! Plug in your solving skills and find 'a'!
Remember, laughter is the best catalyst for understanding chemistry; so, have fun tackling this conundrum!
You are correct that negative exponents are not permissible in rate laws. It seems like there might have been a mistake in your calculation when determining the rate law.
To find the rate law, you need to compare the initial rates from the different experiments while keeping the concentrations of one reactant constant and varying the concentration of the other reactant.
Let's examine the data and determine the rate law step-by-step.
Exp 1: [X]initial = 0.6, [Y]initial = 0.6, Initial Rate = 3.306
Exp 2: [X]initial = 1.2, [Y]initial = 0.6, Initial Rate = 1.653
Exp 3: [X]initial = 0.6, [Y]initial = 1.8, Initial Rate = 29.75
Let's compare Experiments 1 and 2:
When [X] doubles from Exp 1 to Exp 2, the Initial Rate drops by half.
This suggests that the rate is directly proportional to [X]^1. Therefore, we can write the rate law as r = k[X]^1.
Let's compare Experiments 1 and 3:
When [Y] triples from Exp 1 to Exp 3, the Initial Rate increases by approximately 9 times.
This suggests that the rate is directly proportional to [Y]^2. Therefore, we can write the rate law as r = k[X]^1[Y]^2.
Now that we have the rate law as r = k[X]^1[Y]^2, we can use any of the experiments to solve for k.
Let's use Experiment 1:
[X]initial = 0.6, [Y]initial = 0.6, Initial Rate = 3.306
Substituting these values into the rate law equation: 3.306 = k(0.6)^1(0.6)^2
Simplifying the equation:
3.306 = k(0.6)(0.36)
3.306 = k(0.216)
k = 3.306/0.216
k ≈ 15.28 mol^(-2)L^3s^(-1)
Therefore, the value of the rate constant, k, is approximately 15.28 mol^(-2)L^3s^(-1).
In a rate law, exponents are typically positive integers because they represent the order of reaction with respect to each reactant. It is not possible to have a negative exponent in a rate law equation.
To determine the rate law for the given reaction, you need to analyze the data from the three experiments and see how changing the concentrations of reactants affect the initial rate.
Let's examine the different experiments one by one:
Exp 1: [X]initial = 0.6, [Y]initial = 0.6, Initial Rate = 3.306 mol L^(-1) s^(-1)
Exp 2: [X]initial = 1.2, [Y]initial = 0.6, Initial Rate = 1.653 mol L^(-1) s^(-1)
Exp 3: [X]initial = 0.6, [Y]initial = 1.8, Initial Rate = 29.75 mol L^(-1) s^(-1)
To determine the rate law, you need to compare the initial rates of different experiments while keeping one concentration constant.
Comparing Exp 1 and Exp 2, you can see that doubling the concentration of X (from 0.6 to 1.2) results in halving the initial rate (from 3.306 to 1.653). This suggests that the rate is directly proportional to the concentration of X, and the order of X in the rate law is 1.
Comparing Exp 1 and Exp 3, you can observe that keeping the concentration of X constant at 0.6 while increasing the concentration of Y from 0.6 to 1.8 leads to a significant increase in the initial rate (from 3.306 to 29.75). This suggests that the rate is directly proportional to the square of the concentration of Y, and the order of Y in the rate law is 2.
Based on the above analysis, the rate law for this reaction can be written as:
Rate = k[X]^1[Y]^2
Now, to find the value of the rate constant (k), we can use any one of the three experiments. Let's use Exp 1:
Rate = k[X]^1[Y]^2
Plugging in the values from Exp 1:
3.306 = k*(0.6)^1*(0.6)^2 = 0.216k
Solving for k:
k = 3.306 / 0.216
Calculating the value of k gives us:
k ≈ 15.28 mol^(-1) L s^(-1)
So, the value of the rate constant for this reaction is approximately 15.28 mol^(-1) L s^(-1).