MathMaster logo

MathMaster

  1. Home
  2. general
  3. a given polymer whose mean square radius of gyration is 7 54 10 3 a 2 has at its ends two reactive groups capable of emitt
Question

A given polymer, whose mean square radius of gyration is 7.54·10^3 Å^2, has at its ends two reactive groups capable of emitting light when they are less than 5 Å away (regardless of which one it is in particular). this distance). The emission is proportional to the number of molecules that at each moment meet the aforementioned condition and we consider ideal dilute solutions, or “theta” in which the probability of the two terminal groups of different molecules approaching is small. Obtain the ratio between the amount emitted by a solution of a practically monodisperse sample of said polymer and another similar one, and of the same concentration by weight, but in which the mean square radius of gyration of the molecules is 1.32·10^ 4 Å^2.

126

likes
629 views

Answer to a math question A given polymer, whose mean square radius of gyration is 7.54·10^3 Å^2, has at its ends two reactive groups capable of emitting light when they are less than 5 Å away (regardless of which one it is in particular). this distance). The emission is proportional to the number of molecules that at each moment meet the aforementioned condition and we consider ideal dilute solutions, or “theta” in which the probability of the two terminal groups of different molecules approaching is small. Obtain the ratio between the amount emitted by a solution of a practically monodisperse sample of said polymer and another similar one, and of the same concentration by weight, but in which the mean square radius of gyration of the molecules is 1.32·10^ 4 Å^2.

Expert avatar
Cristian
4.7
119 Answers
To calculate the ratio R of intensities between the two polymers with different radii of gyration, we need to evaluate the integrals involving the probabilities P_1(r) and P_2(r) over the range from 0 to 5 Å.

Given:
R_{g1}^2 = 7.54 \times 10^3 \, Å^2
R_{g2}^2 = 1.32 \times 10^4 \, Å^2

We know that R_e^2 = 6R_g^2 . Therefore, the mean square end-to-end distances for the two polymers are:
R_{e1}^2 = 6 \times 7.54 \times 10^3 = 4.524 \times 10^4 \, Å^2
R_{e2}^2 = 6 \times 1.32 \times 10^4 = 7.92 \times 10^4 \, Å^2

The probability density functions P_1(r) and P_2(r) are given by:
P_1(r) = \left(\frac{3}{2\pi \times 4.524 \times 10^4}\right)^{3/2} \exp\left(-\frac{3r^2}{2 \times 4.524 \times 10^4}\right)
P_2(r) = \left(\frac{3}{2\pi \times 7.92 \times 10^4}\right)^{3/2} \exp\left(-\frac{3r^2}{2 \times 7.92 \times 10^4}\right)

Now, we need to calculate:
R = \frac{I_1}{I_2} = \frac{\int_{0}^{5} P_1(r) \, 4\pi r^2 dr}{\int_{0}^{5} P_2(r) \, 4\pi r^2 dr}

Integrating, we get:
R = \frac{\int_{0}^{5} P_1(r) \, 4\pi r^2 dr}{\int_{0}^{5} P_2(r) \, 4\pi r^2 dr} = 2.32

\boxed{R = 2.32}

Frequently asked questions (FAQs)
Math question: Identify the basic shape of the graph of the equation y = 2x^3 - 5x^2 + 3x - 4.
+
What is the probability of getting at least one head when flipping a fair coin three times?
+
Question: "Factorize 12x^2 + 10xy - 6xz - 15y + 9z using the distributive property.
+
New questions in Mathematics
calculate the derivative by the limit definition: f(x) = 6x^3 + 2
I want to divide R$ 2200.00 between Antônio, Beto and Cássia, so that Beto receives half from Antônio and Cássia receives a third of Beto. Under these conditions, how much more will Beto receive than Cássia?
12-6x=4x+2
A book is between 400 and 450 pages. If we count them 2 at a time there is none left over, if we count them 5 at a time there is none left over and if we count them 7 at a time there are none left over, how many pages does the book have?
two particles start at the origin and move along the x axis. for 0 <= t <= 10, their respective position functions are given by x1 = cos(t) and x2 = (e^-3t) + 1. for how many values of t do the particles have the same velocity?
STUDENTS IN A CLASS LEARN ONLY ONE FOREIGN LANGUAGE. two-sevenths of the students learn German, half of the students learn Spanish, and the remaining six students learn Italian. what is the number of students in this class? detail your reasoning carefully.
a) A tap can supply eight gallons of gasoline daily to each of its 250 customers for 60 days. By how many gallons should each customer&#39;s daily supply be reduced so that it can supply 50 more customers for twenty more days?
The mean temperature for july in H-town 73 degrees fahrenheit. Assuming that the distribution of temperature is normal what would the standart deviation have to be if 5% of the days in july have a temperature of at least 87 degrees?
1 plus 1
Analyze the following situation Juan is starting a new business, he indicates that the price of his product corresponds to p=6000−4x , where x represent the number of tons produced and sold and p It is given in dollars. According to the previous information, what is the maximum income that Juan can obtain with his new product?
3(2•1+3)4
To celebrate the five-year anniversary of a consultancy specializing in information technology, the administrator decided to draw 3 different qualification courses among its 10 employees. Considering that the same employee cannot be drawn more than once, the total number of different ways of drawing among employees is:
∫ √9x + 1 dx
TEST 123123+123123
X³-27
You want to study incomes in a large city. You take a simple random sample of 5012 households and find that the distribution of household incomes is skewed right. If you calculate the mean of the 5012 household incomes will the distribution of mean scores be skewed right as well? Hint: this involves the Central Limit Theorem.
Find the equation of a straight line that has slope 3 and passes through the point of (1, 7) . Write the equation of the line in general forms
Kayla started a book club at her school. The number of girls in the book club was one more than twice the number of boys. If there are 15 girls in the book club, how many boys are in the club?
It costs a manufacturer $2,500 to purchase the tools to manufacture a certain homemade item. If the cost for materials and labor is 60¢ per item produced, and if the manufacturer can sell each item for 90¢, find how many items must he produce and sell to make a profit of $2000?
Find the number of liters of water needed to reduce 9 liters of lotion. shave containing 50% alcohol to a lotion containing 30% alcohol.

You might be interested in