Problem 44 - Chemistry Matter and Change

Question

Calculate the molar mass of a gas that diffuses three times faster than oxygen under similar conditions.

Step-by-Step Solution

Answer

The molar mass of the unknown gas that diffuses three times faster than oxygen under similar conditions is 3.56 g/mol.

Step 1: Write the formula for Graham's Law of Diffusion

The formula for Graham's law of diffusion is given by: \[ \frac{Rate_{Gas1}}{Rate_{Gas2}} = \sqrt{\frac{MolarMass_{Gas2}}{MolarMass_{Gas1}}} \]

Step 2: Apply the known values to the formula

We know that the unknown gas diffuses three times faster than oxygen and the molar mass of oxygen is 32 g/mol. We can plug these values into the formula: \[ \frac{Rate_{UnknownGas}}{Rate_{Oxygen}} = \frac{3}{1} = \sqrt{\frac{32}{MolarMass_{UnknownGas}}} \]

Step 3: Solve for the unknown molar mass

We can square both sides of the equation to get rid of the square root: \[ \left(\frac{3}{1}\right)^2 = \frac{32}{MolarMass_{UnknownGas}} \] Now, we need to multiply both sides by the MolarMass of the unknown gas: \[ 9 \cdot MolarMass_{UnknownGas} = 32 \] Finally, divide both sides by 9: \[ MolarMass_{UnknownGas} = \frac{32}{9} \]

Step 4: Calculate the molar mass of the unknown gas

Now we simply perform the division to find the molar mass of the unknown gas: \[ MolarMass_{UnknownGas} = \frac{32}{9} = 3.56 \; g/mol \] The molar mass of the unknown gas that diffuses three times faster than oxygen under similar conditions is 3.56 g/mol.

Key Concepts

Molar Mass

Molar mass is a fundamental property of substances in chemistry, defined as the mass of one mole of a given substance. It is numerically equal to the substance’s average atomic or molecular weight and is expressed in units of grams per mole (g/mol). When it comes to gases, molar mass becomes a key player in understanding their behavior, as it helps us predict how they will diffuse.

In essence, the molar mass allows us to quantify the amount of matter in a given sample of a gas. It's like having a scale at the molecular level. Understanding molar mass is vital not just for solving problems relating to gas diffusion, but for a plethora of chemical calculations involving stoichiometry, reacting masses, and the use of the ideal gas equation.

Rate of Diffusion

The rate of diffusion refers to how quickly a gas spreads out to uniformly occupy a space. It's an invisible race, where different gases move at different speeds depending on their physical characteristics. Graham's Law of Diffusion gives us insight into this race by providing a simple relationship: lighter gases diffuse faster than heavier ones.

The intriguing part about the rate of diffusion is that it inversely relates to the square root of the molar mass of the gas. This means that even a slight difference in molar mass between two gases can have a significant effect on their respective rates of diffusion. When you think about it, the rate of diffusion is quite a vital concept in our everyday life; it impacts everything from the smell of baking cookies spreading throughout a house to essential industrial processes like the purification of natural gas.

Gas Diffusion

Gas diffusion is a phenomenon where gas molecules spread from an area of higher concentration to an area of lower concentration, until they are evenly distributed. It's one of nature's democratic processes, ensuring that every space is equally shared by gas particles. This process is driven by the kinetic energy of the gas molecules which are constantly in random motion.

Understanding gas diffusion is not just an academic exercise; it has real-world implications. For instance, our lungs are able to absorb oxygen and expel carbon dioxide through diffusion. Another example is the way a hazardous gas leak can spread through the atmosphere. This illustrates why comprehending gas diffusion is crucial for safety in various industries, and more broadly, for understanding the behavior of gases in both natural and controlled environments.