Problem 11 - Biofluid Mechanics: An Introduction to Fluid Mechanics, Macrocirculation, and Microcirculation

Question

Calculate the Debye length for a \(120 \mathrm{mM}\) sodium ion \(\left(\sum \mathrm{c}_{\mathrm{i}} \chi_{\mathrm{i}}=5 \mu \mathrm{m}^{-3}\right)\) that is being transported through the interstitial space (where \(\varepsilon_{l}\) is 60 ; this is a relative number with no units) at body temperature.

Step-by-Step Solution

Answer

The Debye length is approximately 2.67 nm.

Step 1: Understanding the Debye Length Formula

The Debye length \( \lambda_D \) is calculated using the formula: \[ \lambda_D = \sqrt{\frac{\varepsilon_0 \varepsilon_l k_B T}{\sum_i n_i z_i^2 e^2}} \] where \( \varepsilon_0 \) is the vacuum permittivity, \( \varepsilon_l \) is the relative permittivity of the medium, \( k_B \) is Boltzmann's constant, \( T \) is the temperature in Kelvin, \( n_i \) is the concentration of ith ion (in this case, sodium ion), \( z_i \) is the valence of the ion, and \( e \) is the elementary charge.

Step 2: Convert Concentration to Appropriate Units

We need to ensure the concentration \( n_i \) is in \( m^{-3} \) for calculation. The given concentration is \( 120 \, \text{mM} = 120 \times 10^{-3} \, \text{mol/L} \). Since \( 1 \, \text{mol/L} = 10^3 \, ext{mol/m}^3 \), we have \( n_i = 120 \times 10^{-3} \times 10^3 = 120 \, ext{mol/m}^3 = 120 \times 10^3 \, ext{ions/m}^3 \) since 1 mol contains Avogadro's number of ions.

Step 3: Assign Known Values and Constants

Let's assign all known values and constants: \( \varepsilon_0 = 8.854 \times 10^{-12} \, ext{F/m} \), \( \varepsilon_l = 60 \), \( k_B = 1.38 \times 10^{-23} \, ext{J/K} \), \( e = 1.602 \times 10^{-19} \, ext{C} \), and \( T = 310 \, \text{K} \) for body temperature.

Step 4: Calculate Ion Contribution

We calculate the denominator \( \sum_i n_i z_i^2 e^2 \). For sodium ions \((z_i = 1)\), this simplifies to \( n_i e^2 = 120 \times 10^3 \times (1.602 \times 10^{-19})^2 \), which evaluates to \( 120 \times 10^3 \times 2.566 \times 10^{-38} = 3.0792 \times 10^{-32} \).

Step 5: Calculate Debye Length

Substitute the values into the Debye length formula to find \( \lambda_D \):\[ \lambda_D = \sqrt{\frac{8.854 \times 10^{-12} \times 60 \times 1.38 \times 10^{-23} \times 310}{3.0792 \times 10^{-32}}} \]. Solve this step-by-step.

Step 6: Simplify and Compute

Evaluate the numerator and divide by the denominator: \( 8.854 \times 10^{-12} \times 60 \times 1.38 \times 10^{-23} \times 310 = 2.191 \times 10^{-31} \). Dividing this by \( 3.0792 \times 10^{-32} \) gives approximately 7.119. Taking the square root, \( \lambda_D \approx 2.67 \times 10^{-9} \, ext{m} \) or \( 2.67 \, ext{nm} \).

Key Concepts

Electrolyte Concentration

Electrolyte concentration refers to the amount of ions in a solution. In this exercise, the given concentration of sodium ions is 120 millimolars (mM). To use this value in calculations, it must be converted to the appropriate units, which is ions per cubic meter.

• First, convert millimoles to moles: 120 mM becomes 120 x 10^-3 mol/L.
• Next, change the volume unit from liters to cubic meters. Since 1 liter is equal to 0.001 cubic meters, 120 x 10^-3 mol/L is equivalent to 120 mol/m³.
• Finally, knowing that 1 mole contains Avogadro's number of ions (approximately 6.022 x 10²³), this gives us 120 x 10³ ions/m³.

These conversions ensure that the concentration is correctly formatted for the calculation of Debye length.

Relative Permittivity

Relative permittivity, also known as the dielectric constant, is a measure of how easily a medium can maintain an electric field. This property affects how ions in the solution interact with each other and the surrounding environment.

• In this exercise, the medium has a relative permittivity of 60. This is unitless.
• The relative permittivity signifies how much electric field will decrease in strength compared to a vacuum.
• In equations, it is often denoted as \(\varepsilon_l\), and it is multiplied by the vacuum permittivity \(\varepsilon_0\) to determine the effective permittivity of the medium.

By impacting the Debye length, relative permittivity is a crucial factor in the behavior of ions within the medium.

Ion Valency

Ion valency is the charge of an ion, typically known as its valence. This is an important factor in the determination of the ion's effect on the solution's electrical properties.

• Sodium, the ion in this example, has a valency of +1, meaning it is a positively charged ion.
• Valency is denoted as \(z_i\) in formulas. In the Debye length equation, it is squared, which underscores the increased effect of ions with higher charges.

The valency of ions directly influences the summation term in the Debye length equation, impacting the overall electrical interactions within the solution.

Thermodynamic Temperature

Thermodynamic temperature is a measure of absolute temperature in kelvins, influencing the thermal motion of particles. This impacts the behavior of ions in the medium.

• For biological systems, the relevant temperature is often body temperature, around 310 K.
• The Boltzmann constant \(k_B\) represents the link between temperature and energy, appearing in the Debye length formula as it multiplies with the temperature.

Understanding thermodynamic temperature's role helps depict the thermal kinetic effects on ionic interactions via the Debye length. This demonstrates how temperature variations influence the extent of ionic shielding effects.