Problem 44 - Chemistry

Question

At 741 torr and \(44^{\circ} \mathrm{C}, 7.10 \mathrm{~g}\) of a gas occupy a volume of \(5.40 \mathrm{~L}\). What is the molar mass of the gas?

Step-by-Step Solution

Answer

The molar mass of the gas is approximately 31.8 g/mol.

Step 1: Convert all quantities to SI units

First, convert the given pressure from torr to atm by using the conversion factor 1 atm = 760 torr. This gives \(P = 741 \, \text{torr} \times \frac{1 \, \text{atm}}{760 \, \text{torr}} = 0.975 \, \text{atm}\). Next, convert the temperature from Celsius to Kelvin by adding 273.15. This gives \(T = 44^{\circ} \mathrm{C} + 273.15 = 317.15 \, \mathrm{K}\). The given volume is already in liters, so no conversion is necessary.

Step 2: Plug the values into the ideal gas law, solve for n

Replace P, V, and T in the ideal gas law equation \(PV = nRT\) with the values obtained in Step 1. Use the value of R is 0.0821 L atm/mol K. This gives \(0.975 \, \text{atm} \times 5.40 \, \text{L} = n \times 0.0821 \, \text{L atm/mol K} \times 317.15 \, \text{K}\). Solve for n to obtain \(n = \frac{0.975 \, \text{atm} \times 5.40 \, \text{L}}{0.0821 \, \text{L atm/mol K} \times 317.15 \, \text{K}}\).

Step 3: Calculate the molar mass

The molar mass of the gas is the mass of the gas divided by the number of moles. So, \(\text{molar mass} = \frac{7.10 \, \text{g}}{n \, \text{mol}}\).

Key Concepts

Pressure Conversion

When working with gases in chemistry, pressure is a crucial variable. It measures the force that the gas exerts on the walls of its container. For consistency, we often convert the pressure into atmospheres (atm), which are standard units in gas calculations. In the exercise, the pressure is initially given in torr but needs to be converted into atm using the conversion factor: 1 atm = 760 torr. This allows for easy insertion into equations like the ideal gas law. For conversions, multiply the given pressure by the conversion factor:\[P = 741 \, \text{torr} \times \frac{1 \, \text{atm}}{760 \, \text{torr}} = 0.975 \, \text{atm}\] Doing this ensures accuracy when using formulas or comparing different pressure measurements. It simplifies calculations and helps align with the universal standard for scientific studies.

Temperature Conversion

Temperature is a measure of how hot or cold something is and plays a key role in gas calculations. The ideal gas law requires the temperature to be in Kelvin. While science typically references Celsius, Kelvin is preferred in chemistry because it starts at absolute zero, making it a more natural scale for equations. To convert Celsius to Kelvin, add 273.15 to the Celsius temperature: \[T = 44^{\circ} \mathrm{C} + 273.15 = 317.15 \, \mathrm{K}\] Using Kelvin ensures that the temperature scale aligns with the conditions gases are studied under, providing consistent and reliable results when performing any calculations. Always remember: no Kelvin, no accurate gas laws!

Molar Mass Calculation

In chemistry, molar mass is the mass of one mole of a substance, usually expressed in grams per mole (g/mol). It's calculated using the mole concept, which links mass with moles for a given sample. The ideal gas law - a core equation that relates pressure, volume, temperature, and moles of a gas - helps us determine the amount of substance present in terms of moles. The example uses this formula:\[\text{PV} = nRT\]Rearrange to solve for moles \((n)\):\[n = \frac{P \times V}{R \times T}\]By substituting the known values, we find how much gas (in moles) occupies a specific volume under certain conditions.Lastly, use the mass given (like 7.10 g) divided by the mole quantity to find molar mass:\[\text{Molar mass} = \frac{\text{Mass}}{n}\]This is crucial for fully identifying a gas or understanding its properties, significant in contexts like chemical reactions or industrial applications.

SI Units

The International System of Units (SI) is the globally accepted system for measurement. It keeps scientific calculations clear and precise, as it standardizes how measurements are taken and expressed. For chemistry and physics, especially when working with gas laws, it's important that pressure, volume, and temperature be expressed in their respective SI units.

• Pressure should be in atmospheres (atm)
• Volume in liters (L)
• Temperature in Kelvin (K)

The ideal gas constant \(R\) (0.0821 L atm/mol K) is derived based on these units, ensuring all parts of the gas laws work in harmony. Consistency in using SI units across various calculations fosters universal understanding and application, allowing scientists from different parts of the world to replicate experiments and verify results. By mastering these units, you'll be better prepared to tackle complex problems and real-world applications.