A molecule of DNA (deoxyribonucleic acid) is 2.03 µm long. The ends of the molecule become singly ionized: negative on one end, positive on the other. The helical molecule acts like a spring and compresses 1.13% upon becoming charged. Determine the effective spring constant of the molecule.
calculate force (proportional to Q1*Q2/distance^2)
force = k (2.03*10^-6)(.0113)
To determine the effective spring constant of the DNA molecule, we need to use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position.
The elongation of the DNA molecule can be defined as the change in length, ΔL. Given that the molecule compresses by 1.13% upon becoming charged, we can calculate the change in length as:
ΔL = (1.13/100) * 2.03 µm
ΔL = 0.01139 µm
The effective spring constant, k, can be calculated using Hooke's Law:
F = k * ΔL
However, we do not have the force directly. Instead, we have information about the charge on the ends of the molecule.
When a molecule becomes charged, it experiences an electrostatic force due to the interaction between the charges. This force can be calculated using Coulomb's law:
F = k_e * (q1 * q2) / r^2
Where:
- k_e is the electrostatic constant (k_e = 8.99 x 10^9 N m^2/C^2)
- q1 and q2 are the charges on the ends of the molecule (one is negative, and the other is positive)
- r is the separation between the charges (equal to the original length of the DNA molecule, L)
Since we are given that the ends of the molecule become singly ionized (negative on one end and positive on the other), the charges can be considered as +q and -q, respectively.
Setting the electrostatic force equal to the spring force:
k_e * (q * (-q)) / L^2 = k * ΔL
We can rearrange the equation to solve for k:
k = (k_e * q^2) / (L^2 * ΔL)
Now, we substitute the given values:
k_e = 8.99 x 10^9 N m^2/C^2
q = electronic charge = 1.6 x 10^-19 C
L = 2.03 µm = 2.03 x 10^-6 m
ΔL = 0.01139 µm = 0.01139 x 10^-6 m
Plugging in these values and calculating:
k = (8.99 x 10^9 N m^2/C^2 * (1.6 x 10^-19 C)^2) / ((2.03 x 10^-6 m)^2 * 0.01139 x 10^-6 m)
k ≈ 3.64 N/m
Therefore, the effective spring constant of the DNA molecule is approximately 3.64 N/m.
To determine the effective spring constant of the molecule, we can use Hooke's law. According to Hooke's law, the force exerted by a spring is directly proportional to the displacement from its equilibrium position.
First, let's calculate the change in length of the DNA molecule when it becomes charged. The molecule compresses by 1.13% of its original length, which can be calculated as follows:
Change in length = (1.13/100) * 2.03 µm
Next, we need to express the change in length in meters since the SI unit of length is meters:
Change in length = (1.13/100) * 2.03 µm * (1 m / 10^6 µm)
Now, we can calculate the effective spring constant. The spring constant is a measure of how stiff or flexible the spring is:
Spring constant (k) = Force / Displacement
The force can be calculated using Coulomb's Law, which states that the force between two charged particles is proportional to the product of their charges and inversely proportional to the square of the distance between them.
In this case, the force is between the singly ionized ends of the DNA molecule, and since one end is positive and the other end is negative, we can assume they have equal and opposite charges.
Using Coulomb's Law, the force can be calculated as:
Force = (k_e * q^2) / d^2
Where k_e is the electrostatic constant (9 * 10^9 N.m^2/C^2), q is the charge of one ion (in Coulombs), and d is the distance between the ions (in meters).
Since the force is proportional to the displacement, we can equate the force to the spring force:
(k_e * q^2) / d^2 = k * change in length
Now, we rearrange the equation to solve for the spring constant (k):
k = (k_e * q^2) / (d^2 * change in length)
Plugging in the known values:
k = (9 * 10^9 N.m^2/C^2) * (charge^2) / (distance^2 * change in length)
Remember, the charge and distance values must be in Coulombs and meters, respectively.
By substituting the values for charge, distance, and change in length, you can calculate the effective spring constant of the DNA molecule.