Spring constant of molecule

In summary, a molecule of DNA with a length of 1.12 μm becomes singly ionized, resulting in a compressed length of 1.11552E-9 μm. The effective spring constant of the molecule is 1.685*109 N/m, taking into account the compressed length of 4.48 nm.
  • #1
bastige
15
0

Homework Statement


A molecule of DNA lies along a straight line. It is 1.12 μm long. The ends of the molecule become singly ionized; negative on one end, positive on the other. The helical molecule acts as a spring and compresses .4% upon becoming charged.
The Coulomb constant is 8.99 X 109 N*m2/C2.
Determine the effective spring constant of the molecule. Take into account the compressed length when calculating the distance between the ends of the molecule. Answer in units N/m.


Homework Equations


Fe= ke * ( q1q2 / d^2 )



The Attempt at a Solution


d = 1.2*10-6 - (1.2*10-6 * .004) = 1.1952 μm
Fe = ke * ( q1q2 / d2 ) = (8.99*109) * (( 1.6*10-19 )2) / (1.1952*10-6)2 = 2014 N
Fs = kx
x = d
Fe = Fs ---> 2014 = (1.1952*10-6) k ---> k = 1.685*109 N/m
 
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  • #2
bastige said:
d = 1.2*10-6 - (1.2*10-6 * .004) = 1.1952 μm
Is the distance 1.2 or 1.12 μm?
Fe = ke * ( q1q2 / d2 ) = (8.99*109) * (( 1.6*10-19 )2) / (1.1952*10-6)2 = 2014 N
Fs = kx
x = d
Fe = Fs ---> 2014 = (1.1952*10-6) k ---> k = 1.685*109 N/m
In Hooke's law (F = kx), x is the amount of compression from the uncompressed/unstretched position. So x does not equal d!
 
  • #3
distance is 1.12μm , so
d=(1.12E-9) - (1.2E-9 * .004) =1.11552E-9

& i totally butchered the rest. I'm not sure what its supposed to look like
Fe=8.99E+9 * ? / 1.11552E-9

Then
Fs=kx
Fs*x =K
 
  • #4
bastige said:
distance is 1.12μm , so
d=(1.12E-9) - (1.2E-9 * .004) =1.11552E-9
Good.

& i totally butchered the rest. I'm not sure what its supposed to look like
Fe=8.99E+9 * ? / 1.11552E-9
You had the right idea before:
[tex]F_e = k_e q^2/d^2[/tex]

Then
Fs=kx
Fs*x =K
F = kx
so, k = F/x.

What's x? (By what distance is the "spring" compressed?)
 
  • #5
Doc Al said:
Is the distance 1.2 or 1.12 μm?

In Hooke's law (F = kx), x is the amount of compression from the uncompressed/unstretched position. So x does not equal d!

This guy is right. x does not equal d. x equals (1.12 μm*.004)
 

What is the spring constant of a molecule and why is it important?

The spring constant of a molecule, also known as the force constant, is a measure of the strength of the bond between atoms in a molecule. It is important because it determines the frequency and energy of molecular vibrations, which in turn affects the physical and chemical properties of the molecule.

How is the spring constant of a molecule determined?

The spring constant of a molecule is determined by measuring the vibrational frequency of the molecule using spectroscopic techniques such as infrared or Raman spectroscopy. The frequency is then used in the equation k = mω^2 to calculate the spring constant, where k is the spring constant, m is the reduced mass of the molecule, and ω is the vibrational frequency.

What factors can affect the spring constant of a molecule?

The spring constant of a molecule can be affected by various factors such as the type of bond between atoms, the length of the bond, and the mass of the atoms involved. Additionally, external factors such as temperature and pressure can also affect the spring constant of a molecule.

Can the spring constant of a molecule change?

Yes, the spring constant of a molecule can change under certain conditions. For example, increasing the temperature of a molecule can lead to an increase in its vibrational frequency and therefore a change in the spring constant. Additionally, chemical reactions and interactions with other molecules can also cause changes in the spring constant of a molecule.

How is the spring constant of a molecule related to its potential energy?

The spring constant of a molecule is directly related to its potential energy. A higher spring constant means a stiffer bond and therefore a higher potential energy, while a lower spring constant indicates a weaker bond and lower potential energy. This relationship is described by the equation E = 1/2kx^2, where E is the potential energy, k is the spring constant, and x is the displacement from equilibrium.

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