A molecule of DNA (deoxyribonucleic acid) is 2.01 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 0.75% upon becoming charged. Determine the effective spring constant of the molecule.
To determine the effective spring constant of the DNA molecule, we can use Hooke's Law, which states that the force exerted by a spring is proportional to the displacement from its equilibrium position.
Hooke's Law can be written as:
F = -kx
Where F is the force, k is the spring constant, and x is the displacement.
Given that the DNA molecule compresses by 0.75% upon becoming charged, we can calculate the displacement, x:
x = 0.0075 * 2.01 µm
x = 0.015075 µm
Now we need to find the force, F, exerted by the DNA molecule. Since the ends of the molecule become singly ionized, one end becomes negative and the other positive. As a result, an electric field is created between the ends, causing the compression. This force can be calculated using Coulomb's Law:
F = q1 * E
Where F is the force, q1 is the charge, and E is the electric field.
Since the ends of the molecule are singly ionized, they carry charges of equal magnitude but opposite signs. Therefore, q1 and q2 are equal in magnitude:
|q1| = |q2|
Let's assume the charges are q1 = -q2 = q.
So, the force can be rewritten as:
F = q * E
To determine the value of q, we need additional information. Once we have the charge, we can find the electric field, E, between the ends of the molecule.
Please provide the charge on the ends of the DNA molecule, and I can continue the calculation.