In chemistry and physics, understanding how fast a gas moves through a tiny opening is crucial for various applications, from isotope separation to industrial processing. This movement is known as effusion. The comparative speed at which two different gases effuse is determined by their molar masses, a relationship defined by Graham's Law of Effusion.
Understanding Graham's Law
Graham's Law states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass. Simply put, lighter gases move faster than heavier gases at the same temperature and pressure.
To calculate the ratio of effusion rates between two gases, we use the following formula:
Rate₁ / Rate₂ = √(M₂ / M₁)
Rate₁: Effusion rate of Gas 1
Rate₂: Effusion rate of Gas 2
M₁: Molar Mass of Gas 1 (g/mol)
M₂: Molar Mass of Gas 2 (g/mol)
Step-by-Step Calculation Example
Let's calculate the ratio of effusion rates between Hydrogen (H₂) and Oxygen (O₂).
Identify Molar Masses:
Hydrogen (Gas 1): Approximately 2.02 g/mol
Oxygen (Gas 2): Approximately 32.00 g/mol
Apply the Formula: Ratio = √(Molar Mass Oxygen / Molar Mass Hydrogen)
Ratio = √(32.00 / 2.02)
Solve: Ratio = √(15.84)
Ratio ≈ 3.98
Result: Hydrogen gas effuses approximately 3.98 times faster than Oxygen gas.
Why does this matter?
The kinetic energy formula (KE = ½mv²) explains this phenomenon. At a constant temperature, all gases have the same average kinetic energy. Therefore, if the mass (m) is lower, the velocity (v) must be higher to maintain the same energy level. This is why lighter molecules zip around much faster than heavier ones.
Applications of Effusion Ratios
Calculating the ratio of effusion rates is frequently used in:
Isotope Separation: Enriching uranium by separating U-235 from U-238 based on slight differences in effusion speeds.
Unknown Gas Identification: Determining the molar mass of an unknown gas by comparing its effusion rate to a known standard.
Leak Detection: Understanding how quickly different gases might escape through microscopic pinholes in a vessel.
function calculateRatio() {
// 1. Get input values
var m1Input = document.getElementById('gas1Mass');
var m2Input = document.getElementById('gas2Mass');
var resultArea = document.getElementById('result-area');
var m1 = parseFloat(m1Input.value);
var m2 = parseFloat(m2Input.value);
// 2. Validate inputs
if (isNaN(m1) || isNaN(m2) || m1 <= 0 || m2 1) {
comparisonText = "Gas 1 is faster";
explanationText = "Gas 1 effuses " + ratioFormatted + " times faster than Gas 2 because it has a lower molar mass.";
} else if (ratio < 1) {
comparisonText = "Gas 2 is faster";
// Calculate inverse for readability
var inverseRatio = (1 / ratio).toFixed(4);
explanationText = "Gas 2 effuses " + inverseRatio + " times faster than Gas 1. Gas 1 is heavier and therefore slower.";
} else {
comparisonText = "Same Rate";
explanationText = "Both gases have the same molar mass, so they effuse at the same rate.";
}
// 5. Update DOM
document.getElementById('displayRatio').innerText = ratioFormatted;
document.getElementById('displayComparison').innerText = comparisonText;
document.getElementById('displayExplanation').innerHTML = explanationText;
// Show result area
resultArea.style.display = 'block';
}