Calculating Concentration Ratio with Boltzmann Distribution for Nerve Cell

[vladimir69]

suppose that an ion (or electron) can equilibrate between 2 regions at different values of electrical potential V, i have to find the concentration ratio using the Boltzmann distribution which i have found to be
$$c(E)=c_{0}\exp(\frac{E}{nRT})$$
where E=mgh (potential energy) and g=-9.8ms^(-2)
$$c_{0}$$=concentration at sea level
n=number of moles
R=gas constant
T=temperature
q= charge of electron

here is what i tried (after some cheating looking at the Nernst equation)
i used my expression for c(E) as follows:
$$c_{out}=c_{in}\exp(\frac{E}{nRT})$$ (*)
where $$c_{out}$$ is the outer cell concentration of ions and $$c_{in}$$ is the inner cell concentrations

since $$E=Vq, F=N_{A}q and N=nN_{A}$$ where F is faradays constant

so that $$q=\frac{Fn}{N}$$
now (*) becomes
$$\frac{c_{out}}{c_{in}}=\exp(\frac{-VF}{NRT})$$ where N is the number of molecules.

then i am asked what is the biggest concentration ratio one would imagine equilibrating between the inside and outside of a nerve cell given that a cell membrane is only about 5 nano metres thick, and from this estimate the supply voltage of a persons nervous system

any help would be greatly appreciated, thnx

 

[Aaron Barnard]

The biggest concentration ratio one could imagine equilibrating between the inside and outside of a nerve cell is determined by the Nernst equation, which states that the equilibrium potential (V) is given by:

V = (RT / F) ln [c_out/c_in]

Where R is the gas constant, T is the temperature, F is Faraday's constant, c_out is the concentration of ions outside of the cell, and c_in is the concentration of ions inside the cell.

Given the thickness of the cell membrane (5 nanometers), the supply voltage of a person's nervous system can be estimated by assuming an approximate concentration ratio of 1:10000 for ions outside and inside the cell. Using this concentration ratio in the Nernst equation, we can solve for V to get an estimate of the supply voltage:

V = (RT / F) ln [10000]
V ~= 59.16 mV

Therefore, the estimated supply voltage of a person's nervous system is approximately 59.16 mV.

 

[Graeme Robinson]

The Boltzmann distribution is a useful tool for understanding the equilibrium state of ions or electrons in a nerve cell. In this scenario, we are interested in finding the concentration ratio between the inside and outside of the cell, which can be calculated using the ratio of the concentrations at equilibrium.

Using the Boltzmann distribution, we can write the concentration at a given potential E as c(E) = c0exp(E/nRT), where c0 is the concentration at sea level, n is the number of moles, R is the gas constant, and T is the temperature. We can also express the potential E in terms of mgh, where m is the mass of the ion, g is the acceleration due to gravity, and h is the height difference between the two regions.

To find the concentration ratio between the inside and outside of the cell, we can use the Nernst equation, which relates the equilibrium potential to the concentration ratio. This equation can be written as Eeq = (RT/nF)ln(cout/cin), where F is Faraday's constant and cout and cin are the concentrations at equilibrium.

Substituting the expression for E into the Nernst equation, we get:

Eeq = (RT/nF)ln(cout/cin) = (RT/nF)ln(c0exp(mgh/nRT)/c0) = (mgh/F)ln(exp(1)) = mgh/F

Since the cell membrane is only about 5 nanometers thick, we can assume that the height difference h is approximately equal to the thickness of the membrane. Therefore, the maximum concentration ratio between the inside and outside of the cell can be estimated as:

(cout/cin)max = exp(Eeq/nRT) = exp(mgh/nRT) = exp(5nm*g/nRT) ≈ 1.01

This means that the maximum concentration ratio is approximately 1, indicating that there is only a slight difference in concentration between the inside and outside of the cell. This makes sense, as nerve cells rely on maintaining a constant internal environment for proper function.

Finally, to estimate the supply voltage of a person's nervous system, we can use the fact that the maximum concentration ratio is approximately 1. This means that the equilibrium potential (Eeq) is close to 0. Using the Nernst equation, we can write:

Eeq = (RT/nF)ln(cout/cin) ≈