Concentration at particle surface. Unit: mg/mL (or mg/cm³)
Current concentration in the medium. Unit: mg/mL (or mg/cm³)
Unit: cm (Centimeters)
Calculated Dissolution Rate (dm/dt):0.00 mg/s
This implies mg of substance is dissolving per second.
function calculateDissolutionRate() {
// 1. Get input elements explicitly
var dInput = document.getElementById("diffusionCoeff");
var aInput = document.getElementById("surfaceArea");
var csInput = document.getElementById("satConc");
var cbInput = document.getElementById("bulkConc");
var hInput = document.getElementById("layerThickness");
var resultBox = document.getElementById("resultBox");
var resultValue = document.getElementById("resultValue");
var errorDisplay = document.getElementById("errorDisplay");
var massChangeDesc = document.getElementById("massChangeDesc");
// 2. Parse values
var D = parseFloat(dInput.value);
var A = parseFloat(aInput.value);
var Cs = parseFloat(csInput.value);
var Cb = parseFloat(cbInput.value);
var h = parseFloat(hInput.value);
// 3. Validation Logic
errorDisplay.style.display = "none";
resultBox.style.display = "none";
if (isNaN(D) || isNaN(A) || isNaN(Cs) || isNaN(Cb) || isNaN(h)) {
errorDisplay.innerText = "Please fill in all fields with valid numbers.";
errorDisplay.style.display = "block";
return;
}
if (h <= 0) {
errorDisplay.innerText = "Diffusion layer thickness (h) must be greater than 0.";
errorDisplay.style.display = "block";
return;
}
if (D < 0 || A < 0 || Cs < 0 || Cb Saturation (Supersaturation context or error)
// We will allow calculation but the rate will be negative (precipitation), handled in display logic.
// 4. Calculation: Noyes-Whitney Equation
// Rate = (D * A * (Cs – Cb)) / h
var concentrationGradient = Cs – Cb;
var numerator = D * A * concentrationGradient;
var rate = numerator / h;
// 5. Formatting Result
// Since Inputs: D (cm2/s), A (cm2), C (mg/cm3), h (cm)
// Rate Unit check: (cm2/s * cm2 * mg/cm3) / cm = (cm4 * mg / (s * cm3)) / cm = (cm * mg / s) / cm = mg/s
// Display Logic
resultValue.innerText = rate.toFixed(6) + " mg/s";
massChangeDesc.innerText = rate.toFixed(6);
if (rate < 0) {
resultValue.style.color = "#dc3545"; // Red for precipitation
massChangeDesc.parentElement.innerHTML = "Result is negative, indicating precipitation rather than dissolution.";
} else {
resultValue.style.color = "#007bff";
massChangeDesc.parentElement.innerHTML = "This implies " + rate.toExponential(4) + " mg of substance is dissolving per second.";
}
resultBox.style.display = "block";
}
Understanding Dissolution Rate and the Noyes-Whitney Equation
In pharmaceutical sciences, chemistry, and chemical engineering, the dissolution rate defines the speed at which a substance dissolves into a solvent. This process is critical for determining the bioavailability of drugs, as a drug must dissolve in body fluids before it can be absorbed into the systemic circulation.
This calculator utilizes the classic Noyes-Whitney equation to estimate the rate of dissolution based on physicochemical properties of the solute and solvent.
The Noyes-Whitney Equation
The dissolution rate is mathematically expressed as:
dm/dt = (D × A × (Cs – Cb)) / h
Where:
dm/dt: The dissolution rate (mass per unit time, e.g., mg/s).
D: The diffusion coefficient of the solute (cm²/s). This represents how easily molecules move through the solvent.
A: The surface area of the undissolved solid (cm²). Higher surface area leads to faster dissolution.
Cs: The saturation solubility of the solute in the diffusion layer (mg/cm³ or mg/mL).
Cb: The concentration of the solute in the bulk solvent at time t (mg/cm³ or mg/mL).
h: The thickness of the diffusion layer surrounding the solid particle (cm).
Key Factors Affecting Dissolution
1. Surface Area (A)
There is a direct linear relationship between surface area and dissolution rate. This is why micronization (reducing particle size) is a common technique in pharmaceutical formulation. By grinding a drug into a fine powder, you exponentially increase the total surface area ($A$), thereby increasing the dissolution rate.
2. Diffusion Coefficient (D)
The value of $D$ is inversely proportional to the viscosity of the medium. A highly viscous solvent (like syrup) will result in a lower diffusion coefficient and a slower dissolution rate compared to water. Temperature also affects $D$; generally, higher temperatures increase molecular movement, increasing $D$ and the overall rate.
3. Concentration Gradient (Cs – Cb)
The driving force of dissolution is the difference between the saturation concentration ($C_s$) and the concentration already in the bulk ($C_b$).
Sink Conditions: If $C_b$ is kept very low (typically less than 10-20% of $C_s$), the system is in "sink conditions," and dissolution proceeds at its maximum rate.
As $C_b$ approaches $C_s$, the rate slows down. If $C_b$ exceeds $C_s$, precipitation occurs.
4. Diffusion Layer Thickness (h)
The diffusion layer is a stagnant layer of liquid surrounding the particle. The rate of agitation (stirring or mixing speed) significantly impacts $h$. Faster stirring thins the diffusion layer (reduces $h$), which reduces the distance molecules must travel to enter the bulk solvent, thus speeding up dissolution.
How to Use This Calculator
To perform a calculation, ensure your units are consistent. The standard scientific unit set used here is:
Length/Area: Centimeters (cm) and Square Centimeters (cm²)
Time: Seconds (s)
Mass/Concentration: Milligrams (mg) and mg/mL (which is equivalent to mg/cm³)
Example Calculation:
If you have a particle with a surface area of 2.5 cm², a diffusion coefficient of 5×10⁻⁶ cm²/s, a saturation solubility of 10 mg/mL, and you drop it into a fresh solvent (Bulk Conc = 0) with a diffusion layer thickness of 0.005 cm: