Debye Length Calculator

Welcome to the Debye length calculator. This tool, designed by Omni Calculator, is here to help you find the Debye length or the Debye radius in a plasma or monovalent electrolyte. Along with this article, we are going to explain to you in detail the following topics:

• What is the Debye length? The Debye length formula.
• The Debye length equation for electrolyte solution.
• How to use the Debye length calculator.
• What is a Debye sphere?
• What is the Debye length of a double layer?
• How do I calculate the Debye length for an electrolyte?
• What does a higher Debye length mean?
• And much more.

So, set your particles to interact and let's deeply understand the concepts behind the Debye length.

The Debye length represents the distance over which electrostatic interactions are screened in a medium, which means that it measures how far the electrostatic effects persist. It is a very important concept in plasma and electrolyte mediums, where multiple charged particles can interact.

In practical terms, the Debye length is the length scale at which the electric potential of a charged particle drops off significantly. This effect is due to other nearby charges of the medium (ions, electrons), which rearrange themselves to neutralize the electric potential. This is why the Debye length is also known as the Debye radius.

The Debye length equation is written in terms of several fundamental constants. We can have two versions of the equation for plasma physics, whose forms are presented below:

where:

• $\epsilon_0 = 8.8542 \times 10^{-12}\,\mathrm{F \, m^{-1}}$ — Vacuum permittivity;
• $k_{B} = 1.3807 \times 10^{-23}\,\mathrm{m^2 \, kg\, s^{-2}\, K^{-1}}$ — Boltzmann constant;
• $n_e$ — Electron number density in $\mathrm{m}^{-3}$;
• $e = 1.602 \times 10^{-19}\,\mathrm{C}$ — Elementary charge of the electron;
• $T$ — Temperature in kelvin; and
• $T(\mathrm{eV})$ — Temperature in electronvolts.

For monovalent electrolyte solutions, the Debye length formula is such that:

where:

• $\epsilon_r$ — Relative permittivity;
• $N_A = 6.02 \times 10^{23}\,\mathrm{mol^{-1}}$ — Avogadro’s number; and
• $I$ — Ionic strength of the electrolyte in $M$.

When we deal with an electrolyte solution, we need to include the contributions of several ions with different concentrations and charge numbers. Therefore, the standard Debye length formula needs to be modified to compute such effects. In this case, the Debye length equation is given by:

where:

• $c_i$ — Molar concentration of the ion $i$; and
• $z_i$ — Charge number of ion $i$.

Yes, we know that it was a lot of theory. Now, let's see how to compute $\lambda_D$ with our amazing Debye length calculator.

Our Debye length calculator is simple and intuitive to use. The best part is that you do not need to remember or search for the values of any fundamental constant. We've set everything so that you can make your calculation quickly and easily. In order to use it, let us consider that you want to compute the Debye length for $I = 1\,\mathrm{M}$ of KCl (potassium chloride), which is in water at room temperature. Therefore, we need the following data:

• $T = 298\,\mathrm{K}$ — Room temperature in kelvin; and
• $\epsilon_r = 72$ — Relative permittivity for KCl.

We can use our calculator to answer this question in the blink of an eye. You just need to choose the electrolyte option for your medium. Then, insert the temperature, the relative permittivity, and last but not least, the ionic strength. Now, you can see that our tool provides you $\lambda_D = 0.29\,\mathrm{nm}$ as the Debye length.

We can take another interesting example by computing the Debye length for the solar wind plasma. In order to calculate that, you need the data below:

• $T = 10^{5}\,\mathrm{K}$ – solar wind plasma temperature; and
• $n_e = 10^{6}\,\mathrm{m}^{-3}$ – electron number density;

By choosing plasma as the medium and including the previous data in our calculator, you will find that $\lambda_D = 21.82 \, \mathrm{m}$. Therefore, in the solar wind, the plasma behaves more like a collection of individual charged particles rather than a "screened fluid" due to the large value of the Debye length.