Question

Please explain and show all steps so I can study from it.

You measure the elastic modulus of a stack of pancakes and determine it to be 20000 Pa.

If you assume the interaction energy between the glutenin molecules is U_int=100k_BT=4×10^-19J, what is the characteristic bond length?

Please report your answer in nanometers (nm). Recall there are 1,000,000,000 nm in one meter.

to solve this problem, you need to use the Equation of the Week:

E= Uint
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l3

Answer 1

Hi,

Hope you are doing well.

Well, Here, I'm opting a different method to solve this problem. Fortunately this method will also work out for finding characteristic bond length.

Here, we are assuming the glutenin molecules and the bonds between them as spring system where, bond lengths act as springs.

bond-spring Molecule Molecul<e

Let,

$k$ be the spring constant of the bonds between the glutenin molecules (N/m).

$x$ be the bond length between the molecules (m).

We know that energy of a spring is given by,

$E=\frac{1}{2}k{x}^{2}\Rightarrow (1)$

Where,

$k$ is the spring constant (N/m).

$x$ is the bond length/spring length (m).

Also,We have, for a spring system, young's modulus is given by,

$Y=\frac{k}{x}$

Where,

$k$ is the spring constant (N/m).

$x$ is the bond length/spring length (m).

Given that,

Interaction energy between the molecules,

$U_{int}=E=4\times10^{-19}J$

Young's modulus,

$Y=2\times10^{5}Pa$

$\therefore$From equation (1), we have

$E=\frac{1}{2}k{x}^{2}$

Substituting, $k=Yx$,

$E=\frac{1}{2}Y{x}^{3}$

Also, We have,

$U_{int}=E=4\times10^{-19}J$

$\therefore E=\frac{1}{2}Y{x}^{3}=4\times10^{-19}J$

$\therefore$ Characteristic bond length is given by,

$x=\sqrt[3]{\frac{2E}{Y}}=\sqrt[3]{\frac{2\times4\times10^{-19}J}{2\times10^{4}Pa}}=\sqrt[3]{4\times10^{-23}m^{3}}$

$x=3.42\times10^{-8}\mathbf{m}=34.2\times10^{-9}\mathbf{m}$

$x=34.2\mathbf{nm}$

Hope this helped for your studies. Keep learning. Have a good day.

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