A molecule of DNA (deoxyribonucleic acid) is 2.22 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 0.95% 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. The formula for Hooke's Law is:
F = k * Δx
Where:
F is the force applied to the spring,
k is the spring constant, and
Δx is the displacement from the equilibrium position.
First, let's find the displacement of the DNA molecule upon becoming charged. We are given that the DNA molecule compresses by 0.95% of its original length. We can calculate the displacement using the formula:
Δx = (0.95/100) * L
Where L is the original length of the DNA molecule. Plugging in the values, we have:
Δx = (0.95/100) * 2.22 µm
Δx = 0.02109 µm
Now we need to determine the force applied to the DNA molecule upon becoming charged. We know that the molecule becomes singly ionized, with one end negative and the other end positive. This creates an electric field between the ends, which exerts a force on the molecule. The force is given by:
F = q * E
Where:
F is the force,
q is the charge, and
E is the electric field strength.
Since we have a negative charge on one end and a positive charge on the other, the force will be attractive and directed towards the positive end. Therefore, the force can be written as:
F = q * E * (-1)
We can represent the electric field strength as the potential difference (V) divided by the length (L):
E = V / L
The potential difference (V) is proportional to the displacement between the ends of the DNA molecule:
V = k_V * Δx
Where k_V is a proportionality constant. Plugging this into the equation for the force, we get:
F = q * (k_V * Δx) * (-1)
Now, equating this force to Hooke's Law, we have:
k * Δx = q * (k_V * Δx) * (-1)
Simplifying, we get:
k = - (q * k_V)
Therefore, the effective spring constant (k) of the DNA molecule is equal to the negative product of the charge (q) and the proportionality constant (k_V).
To determine the numerical value of the effective spring constant, we need to know the charge (q). However, the question does not provide the charge, so we are unable to calculate the value of k without that information.
force=kqq/d^2 =kx
q= charge on one elctron, d is given k coulombs constant, x=.0095*d
solve for k