The two sides of the dna double helix are connected by pairs...

The two sides of the DNA double helix are connected by pairs of bases (adenine, thymine, cytosine, and guanine). Because of the geometric shape of these molecules, adenine bonds with thymine and cytosine bonds with guanine. $\textbf{Figure E21.21}$ shows the bonding of thymine and adenine. Each charge shown is $\pm e$, and the H$-$N distance is 0.110 nm. (a) Calculate the $net$ force that thymine exerts on adenine. Is it attractive or repulsive? To keep the calculations fairly simple, yet reasonable, consider only the forces due to the O$-$H$-$N and the N$-$H$-$N combinations, assuming that these two combinations are parallel to each other. Remember, however, that in the O$-$H$-$N set, the O$^-$ exerts a force on both the H$^+$ and the N$^-$, and likewise along the N$-$H$-$N set. (b) Calculate the force on the electron in the hydrogen atom, which is 0.0529 nm from the proton. Then compare the strength of the bonding force of the electron in hydrogen with the bonding force of the adenine-thymine molecules.

Transcript

00:01 Here we're going to look at a more complicated example of adding the forces of electricity due to point charges on each other.

00:11 So the setup looks fairly complicated, but what's basically going on is a force of attraction between the nh dipole and another negative point charge.

00:29 So the dipole appears twice, and even though the other charge is different, they are both negative charges of amount minus e.

00:52 So we can kind of forget about the complicated double -strand structure and just focus on the force of attraction between each of the dipoles and the negative charge.

01:06 And that attraction will have the same formula, but with slightly different parameters.

01:14 So let's just get started and look at one of the dipoles, and it doesn't matter which one.

01:21 I will focus on the bottom one.

01:26 So we'll develop an expression for that dipole attraction.

01:30 We'll apply it to both the top and the bottom attraction and find the net result.

01:36 But a reminder that kulom's law gives the force of electrical attraction or repulsion between two charges as electrical constant k, q1, q2 over r squared.

01:56 So that's all we need for this.

01:59 But let's focus on the bottom thymine dipole attracting the aden monopole.

02:09 And we'll just call, but we won't get into the molecule.

02:12 We'll just call one of the points negative, the other one, positive e, and they are separated by distance a, which we know is given as part of the structure of the complex molecules.

02:33 Okay, so that's a dipole, and over to the other side is a negative e, and we'll keep everything along the x -axis that will make life.

02:47 Easy.

02:48 So that negative e is going to be attracted to the positive and repelled by the negative.

02:57 And that repulsion is going to be a little bit smaller because the negative charge is a little bit further away.

03:05 But the net attraction of that electron to the dipole is simply f plus minus f minus.

03:28 Okay, and we will separate them by distance d, whereas the bonding distance is distance a.

03:40 So the attraction is going to be k e squared over d minus a.

03:51 The repulsion will be k e squared over d squared.

03:58 So the reason why the attraction wins out is that positive charge is closer to the point.

04:04 Than the negative charge of the dipole.

04:10 Okay, so we're going to apply it for the upper bond and the lower bond, and the only thing that's different is d.

04:19 So the upper thiamine aden bond, thiamine bond.

04:28 We have the force of attraction is equal to a k -e -squared, one over, let's see, distances point.

04:46 280 nanometers.

04:50 The bond length is 0 .110 nanometers.

04:58 And we'll just put that in nanometers, but we'll be careful to square the one over nanometers up to the side.

05:11 And working out the number out in front, that k -e -squared times 10 to the 18th turns out to be 2 .30 times 10 to the minus -tenth.

05:23 And doing everything in si units, we should get newtons as our force...

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