Question

Assuming ideal gas behavior, what is the molecular weight of a gas with a density of \(2.50 \mathrm{~g} /\) liter at \(98^{\circ} \mathrm{C}\) and \(.974 \mathrm{~atm}\) ?

Short answer

The molecular weight (M) of the gas is found using the ideal gas law and density information. First, convert the temperature to Kelvin and pressure to Pascal: \(T(K) = 371.15\) and \(P(Pa) = 98654.75\). Calculate the number of moles using \(n = \frac{PV}{RT}\) with the given values. Next, find the mass of the gas by multiplying the density by the volume and converting grams to kilograms. Finally, determine the molecular weight (M) by dividing the mass by the number of moles and convert it to grams per mole (g/mol) for the final result.

Step-by-step solution

Convert the values under SI units

Temperature is given in degree Celsius, and pressure is given in atm. For the ideal gas law, we will have to convert them to Kelvin and Pascal. \[ T(K) = T(^\circ C) + 273.15 \] \[ P(Pa) = P(atm) \times 101325 \]

Calculate the number of moles

Let's rearrange the ideal gas law to find the number of moles (n): \[ n = \frac{PV}{RT} \] Now, plug in the values of pressure (P), volume (V), gas constant (R), and temperature (T).

Calculate the mass of the gas

We are given the density of the gas as 2.50 g/L. To find the mass of the gas, we will multiply the density by the volume (V). Remember that since we are using SI units for other variables, we need to convert the grams to kilograms.

Calculate the molecular weight (M) of the gas

Now, we can find the molecular weight of the gas by dividing the mass (m) by the number of moles (n): \[ M = \frac{m}{n} \] Plug in the values for the mass and number of moles to determine the molecular weight of the gas in SI units, i.e., kilograms per mole (kg/mol). Finally, convert the molecular weight back to grams per mole (g/mol) for the final result.

Key concepts

The Ideal Gas Law

Understanding the ideal gas law is crucial when studying the properties of gases under various conditions. It is a fundamental equation that links pressure (P), volume (V), temperature (T in Kelvin), and number of moles (n) of a gas through the gas constant (R). This relationship is neatly captured in the formula:
\[ PV = nRT \]
The ideal gas law assumes that the gas particles are point masses with no volume and no interactions between them, which is a close approximation for real gases under standard conditions. When applying the ideal gas law, converting all measurements to SI units is essential for consistency, as these are the base units used in the formula. By rearranging this equation, we can solve for various unknowns, such as the number of moles in the given problem.

SI Units Conversion

For accurate scientific calculations, converting measurements to the International System of Units (SI) is a standard practice. In our exercise, we had to convert temperature from degrees Celsius to Kelvin and pressure from atmospheres to Pascals. The conversions are given by:
\[ T(K) = T(^\circ C) + 273.15 \]
and
\[ P(Pa) = P(atm) \times 101325 \]
These transformations allow students to perform calculations within a unified system to maintain precision and avoid errors. It's also vital to remember that the mass should be in kilograms (kg) when applying these to the ideal gas law, even though the final molecular weight may be reported in grams per mole (g/mol).

Gas Density

Gas density is defined as the mass of the gas per unit volume and is a critical factor in determining the molecular weight of a gas. In the context of the ideal gas law, you might be tasked with calculating a gas's density under specified conditions. With the density given, as in the exercise (2.50 \text{g/L}), you can determine the mass of the gas by using the equation:
\[ \text{mass } (m) = \text{density } (\rho) \times \text{volume } (V) \]
After the mass and the number of moles are found, the molecular weight (or molar mass) can be calculated. Remember to be consistent with units throughout the process. This is especially true when dealing with gas density, as the mass of a gas is typically reported in grams while ideal gas law calculations require kilograms.

The Mole Concept

The mole is a fundamental unit in chemistry that measures the amount of substance. One mole is defined as Avogadro's number (6.022 \times 10^{23}) of particles (atoms, ions, or molecules, depending on the substance). It connects the microscale world of atoms and molecules to a macroscale quantity that can be measured and used in calculations. In the context of our exercise, the mole concept allows us to calculate the molecular weight—the weight of one mole of gas particles. Once you know the number of moles (n) and the mass (m) of a gas, you determine the molecular weight with the equation:
\[ M = \frac{m}{n} \]
It's this molar mass that allows chemists to quantitatively compare different substances and convert between mass and moles in reactions or when studying the properties of gases, as with the ideal gas law.