How would you solve this problem?
C4H8(g)--> 2C2H4
Determine the order of the reaction and the rate constant based on the following pressures, which were recorded when the reaction was carried out 430 C in a constant volume vessel.
Time (s) P(C4H8)(mmHg)
0 400
2,000 316
4,000 248
6,000 196
8,000 155
10,000 122
Scroll down about half way to "Determining the rate law ...."
Graph the data to determine which order.
http://www.sparknotes.com/chemistry/kinetics/ratelaws/section2.rhtml
To determine the order of the reaction and the rate constant, we can use the integrated rate law for a first-order reaction:
-ln([A]/[A]0) = kt
Where [A] is the concentration of reactant at a given time t, [A]0 is the initial concentration of reactant, k is the rate constant, and t is time.
In this case, the reactant is C4H8, and we have the values of P(C4H8) at different times. We can assume that the pressure of C4H8 is directly proportional to its concentration.
To calculate the concentration of C4H8 at each time, we need to convert the pressures from mmHg to atm, as the ideal gas law equation uses atmospheres:
1 atm = 760 mmHg
Therefore, we can convert the pressures from mmHg to atm by dividing each value by 760.
Now, we have the concentrations of C4H8 at different times. We can use these values to calculate the natural logarithm of the concentration ratio [A]/[A]0 at each time.
ln([A]/[A]0) = ln(P(C4H8)/P(C4H8)0)
Where P(C4H8) is the pressure of C4H8 at a given time t, and P(C4H8)0 is the initial pressure of C4H8.
Now we have the natural logarithm of the concentration ratio at each time. We can plot the natural logarithm of the concentration ratio against time (in seconds) and fit a linear trendline to the data.
The slope of the linear trendline represents the rate constant (k) of the reaction, and the order of the reaction is given by the power of the concentration term.
For a simple reaction like this, where the reaction is first-order with respect to C4H8, the concentration ratio [A]/[A]0 can be directly related to the pressure ratio P(C4H8)/P(C4H8)0.
So, by fitting a linear trendline to the graph of the natural logarithm of the pressure ratio against time, the slope of the trendline will give the rate constant (k) of the reaction, and the order of the reaction will be 1 (since it is a first-order reaction).
To summarize:
1. Convert the pressures from mmHg to atm by dividing each value by 760.
2. Calculate the natural logarithm of the pressure ratio ln(P(C4H8)/P(C4H8)0) at each time.
3. Plot the natural logarithm of the pressure ratio against time (in seconds) and fit a linear trendline to the data.
4. The slope of the linear trendline represents the rate constant (k) of the reaction, and the order of the reaction is 1 (since it is a first-order reaction).
To determine the order of the reaction and the rate constant, we can use the method of initial rates:
Step 1: Calculate the initial rate for each set of data points by taking the difference in pressure divided by the difference in time.
Initial Rate = (P₂ - P₁) / (t₂ - t₁)
Calculating the initial rates for each set of data points:
Initial Rate₁ = (316 - 400) / (2000 - 0)
Initial Rate₂ = (248 - 316) / (4000 - 2000)
Initial Rate₃ = (196 - 248) / (6000 - 4000)
Initial Rate₄ = (155 - 196) / (8000 - 6000)
Initial Rate₅ = (122 - 155) / (10000 - 8000)
Step 2: Determine the reaction order with respect to C4H8 by comparing the initial rates.
If the reaction order is "x":
Initial Rate₂ / Initial Rate₁ = (C4H8)₂^x / (C4H8)₁^x
Using the calculated initial rates:
(Initial Rate₂ / Initial Rate₁) = [(248 - 316) / (4000 - 2000)] / [(316 - 400) / (2000 - 0)]
Repeat step 2 for Initial Rate₃ / Initial Rate₂, Initial Rate₄ / Initial Rate₃, and Initial Rate₅ / Initial Rate₄ to confirm if the reaction order is constant.
Step 3: Determine the rate constant using the given initial concentration of C4H8 (400 mmHg)
Rate = k * [C4H8]^x
Using the calculated initial rate (Initial Rate₁) and the initial concentration (400 mmHg):
k = Initial Rate₁ / [C4H8]^x
Repeat step 3 for each set of data points to confirm if the rate constant is constant.
By following these steps for all data points, you can determine the order of the reaction (x) and the rate constant (k) based on the provided data.