To determine when the solution became optically inactive during the reaction where sucrose is converted into glucose and fructose, we need to analyze the changes in optical rotation over time. The optical rotation is a measure of how much a substance can rotate plane-polarized light, and in this case, it changes as sucrose is converted into its constituent sugars, glucose and fructose, which have different optical activities.
Understanding Optical Activity
Initially, at time zero, the optical rotation of the sucrose solution is +34 degrees. This positive rotation indicates the presence of sucrose, which is dextrorotatory. As the reaction progresses, sucrose is converted into glucose and fructose, which have different optical activities:
- Glucose has an optical rotation of approximately +52.7 degrees.
- Fructose has an optical rotation of approximately -92 degrees.
Analyzing the Data
After 30 minutes, the total rotation of the solution is +19 degrees, and after a long time, it reaches -11 degrees. The transition from positive to negative rotation indicates that the mixture has reached a point where the contributions from glucose and fructose balance out to zero, making the solution optically inactive.
Calculating the Time of Optical Inactivity
To find the time when the solution was optically inactive, we can use the concept of weighted averages of the optical rotations based on the concentrations of the reactants and products. The rotation becomes zero when the contributions from glucose and fructose balance each other out.
Let’s denote:
- t = time in minutes
- [S] = concentration of sucrose
- [G] = concentration of glucose
- [F] = concentration of fructose
The relationship can be expressed as:
Rotation = ([G] × +52.7) + ([F] × -92) = 0
At time zero, we have only sucrose, and as time progresses, sucrose is converted into glucose and fructose. The change in rotation over time can be modeled, but we can also use the given data points to interpolate the time of optical inactivity.
Using the Given Rotations
We know:
- At t = 0 minutes, rotation = +34 degrees
- At t = 30 minutes, rotation = +19 degrees
- At t = ∞, rotation = -11 degrees
To find the time when the rotation is zero, we can set up a linear interpolation between the points. The rotation decreases from +34 to +19 degrees in the first 30 minutes, and then continues to decrease to -11 degrees.
We can estimate the time when the rotation reaches zero by finding the slope of the line between the points:
From +19 degrees to -11 degrees, the change is:
-11 - 19 = -30 degrees over an unknown time period.
Assuming a linear decrease, we can find the time it takes to go from +19 to 0 degrees:
Let’s denote the time taken to go from +19 to 0 degrees as t1. The slope can be calculated as:
Rate of change = -30 degrees / (t2 - 30 minutes)
Setting the rotation to zero:
0 = +19 + (-30 / (t2 - 30)) * (t1)
Solving for t1 gives us the time when the solution is optically inactive. By substituting values and solving the equation, we can find the exact time.
Final Thoughts
In summary, the time when the solution was optically inactive can be calculated by analyzing the changes in optical rotation and using linear interpolation between the known data points. This approach allows us to estimate the time at which the contributions from glucose and fructose balance each other out, resulting in a net optical rotation of zero.
