A DNA molecule is 2.24 $\mu$m long. The molecule becomes singly ionized, with one end negative and the other positive. This causes the molecule to compress by 1.34%. We need to find the effective spring constant of the molecule.

1. Problem Description

A DNA molecule is 2.24 $\mu$m long. The molecule becomes singly ionized, with one end negative and the other positive. This causes the molecule to compress by 1.34%. We need to find the effective spring constant of the molecule.

2. Solution Steps

First, we need to find the amount of compression, $\Delta x$, given the original length $x$ and the percentage of compression. The original length is $x = 2.24 \ \mu\text{m} = 2.24 \times 10^{-6} \ \text{m}$. The compression percentage is 1.34%, so the compression is given by:

$\Delta x = 0.0134 \times x = 0.0134 \times 2.24 \times 10^{-6} \ \text{m} = 3.0016 \times 10^{-8} \ \text{m}$.

Next, we need to calculate the force, $F$, acting on the molecule. Since the molecule has a single positive and negative charge at each end, we can calculate the electrostatic force between them. The charge on each end is the elementary charge, $e = 1.602 \times 10^{-19} \ \text{C}$. The force can be calculated using Coulomb's Law:

$F = k \frac{q_1 q_2}{r^2}$,

where $k = 8.9875 \times 10^9 \ \text{N m}^2 \text{/ C}^2$ is Coulomb's constant, $q_1$ and $q_2$ are the charges, and $r$ is the distance between the charges. In this case, $q_1 = q_2 = e = 1.602 \times 10^{-19} \ \text{C}$, and $r = x = 2.24 \times 10^{-6} \ \text{m}$. Therefore:

$F = (8.9875 \times 10^9) \frac{(1.602 \times 10^{-19})^2}{(2.24 \times 10^{-6})^2} = \frac{8.9875 \times 10^9 \times 2.566404 \times 10^{-38}}{5.0176 \times 10^{-12}} = \frac{2.3064 \times 10^{-28}}{5.0176 \times 10^{-12}} \approx 4.6 \times 10^{-17} \ N$.

Finally, we can find the spring constant, $k_s$, using Hooke's Law:

$F = k_s \Delta x$,

$k_s = \frac{F}{\Delta x} = \frac{4.6 \times 10^{-17}}{3.0016 \times 10^{-8}} \approx 1.53 \times 10^{-9} \ \text{N/m}$.

3. Final Answer

The effective spring constant of the molecule is approximately $1.53 \times 10^{-9} \ \text{N/m}$.