In a certain chemical reaction 1.00 gram of product is produced in 15 minutes when the reaction is carried out at 20oC. Assuming that there is sufficient starting material available, how much product will be produced in the same length of time if the reaction is carried out in a boiling water bath at 100oC?
The reaction rate approximately doubles for every 10 degree rise in T. (100-20)/10 = 8 and 2^8 = 256. Therefore, APPROXIMATELY 4 x 256 g should be produced in 15 minutes.
To determine how much product will be produced in a boiling water bath at 100°C, we need to use the concept of reaction rate and the Arrhenius equation.
The Arrhenius equation is as follows:
k = Ae^(-Ea/RT)
Where:
k = rate constant
A = pre-exponential factor
Ea = activation energy
R = gas constant (8.314 J/mol·K)
T = temperature in Kelvin
First, let's convert the given temperature values from Celsius to Kelvin:
20°C + 273.15 = 293.15 K
100°C + 273.15 = 373.15 K
Now, let's assume that the reaction rate at 20°C is given by k1, and the reaction rate at 100°C is given by k2. Since the reaction times (15 minutes) and the starting material are the same, we can compare the two rates to find the ratio of the product formed.
The ratio of the rate constants is given by:
k2 / k1 = e^[(Ea/R) * (1/T1 - 1/T2)]
Substituting the known values:
k2 / k1 = e^[(Ea/8.314) * (1/293.15 - 1/373.15)]
Now, we need to find the ratio of the product formed at each temperature. Since the reaction times are the same, the ratio of the product formed is the same as the ratio of the rate constants:
Product at 100°C / Product at 20°C = k2 / k1
To find the product at 100°C, we need to know the product formed at 20°C. According to the question, 1.00 gram of product is produced in 15 minutes at 20°C.
Now, let's calculate the ratio of the product formed:
Product at 100°C / 1.00 g = k2 / k1
To find the product at 100°C, we multiply the ratio by the amount of product formed at 20°C:
Product at 100°C = (Product at 100°C / 1.00 g) * 1.00 g
Simplifying the equation will give us the final answer.
To determine how much product will be produced at a higher temperature, we can make use of the Arrhenius equation. The Arrhenius equation describes the relationship between the rate constant of a reaction and the temperature at which the reaction occurs.
The Arrhenius equation is given by:
k = A * e^(-Ea/RT)
where:
- k is the rate constant
- A is the pre-exponential factor
- Ea is the activation energy of the reaction
- R is the ideal gas constant
- T is the absolute temperature in Kelvin
We are assuming that the reaction has sufficient starting material available, so it is safe to assume that the reaction is not limited by the availability of reactants.
Given that the reaction at 20oC produces 1.00 gram of product in 15 minutes, we can calculate the rate constant at this temperature. Let's take the natural logarithm of the Arrhenius equation:
ln(k) = ln(A) + (-Ea/RT)
Now we can use the given information to solve for the rate constant at 20oC. The temperature must be converted to Kelvin:
T = 20 + 273 = 293 K
We have:
ln(k1) = ln(A) + (-Ea/(R * 293))
Similarly, at 100oC:
T = 100 + 273 = 373 K
ln(k2) = ln(A) + (-Ea/(R * 373))
Since we want to find the ratio of the product produced at 100oC to the product produced at 20oC for the same time interval, let's divide the two equations:
ln(k2) / ln(k1) = (-Ea/(R * 373)) / (-Ea/(R * 293))
The ln(A) terms cancel out.
Simplifying, we get:
ln(k2/k1) = (Ea/R) * (1/293 - 1/373)
Now we can solve for k2/k1 using the fact that ln(k2/k1) = ln(x), where x is the ratio of the product produced at 100oC to the product produced at 20oC. Let's find this ratio:
x = e^[(Ea/R) * (1/293 - 1/373)]
Now we are ready to find the amount of product produced at 100oC for the same time interval. Since the reaction rate is directly proportional to the rate constant, we can express the amount of product produced at 100oC as a ratio to the amount produced at 20oC:
product_100°C = x * product_20°C
Given that 1.00 gram of product is produced at 20oC, we can calculate the amount of product produced at 100oC:
product_100°C = x grams * 1.00 gram
Substitute the value of x from our previous calculation to get the answer.