Molar Mass
Molar mass is a fundamental concept in chemistry that represents the mass of one mole of a substance, usually measured in grams per mole (g/mol). It is the sum of the atomic masses of all the atoms in a molecule. In the context of a gas mixture, like the one in the laser discharge cell, we cannot merely take the molar mass of a single component since it is a composite of different gases. To calculate the molar mass of a gas mixture, we must determine the weighted average. This means taking into account the proportion of each gas in the mixture as expressed by its mole percent.
The formula to compute the molar mass of a mixture (MM) involves multiplying the molar mass of each component by its respective mole percent and summing these products. For example, with the molar masses CO2, N2, and He being 44.01 g/mol, 28.02 g/mol, and 4.00 g/mol respectively, and their mole percents being 11%, 5.3%, and 84%, we calculate the molar mass of the mixture as follows:
MM = (44.01 g/mol \( \times \) 0.11) + (28.02 g/mol \( \times \) 0.053) + (4.00 g/mol \( \times \) 0.84).
Clearly, understanding the concept of molar mass is crucial when dealing with mixtures, as it directly affects the calculations for other properties such as density, as we'll see in the following sections.
Density Calculation
Density is a physical property that can be defined as the mass per unit volume of a substance, often expressed in units of grams per liter (g/L). The calculation of the density of a gas mixture requires knowledge of its molar mass, as well as the conditions of temperature and pressure the gas is under.
To calculate the density of a gas, we can use the Ideal Gas Law, which is a cornerstone in understanding gas behaviors. The law is usually represented by the equation \( PV = nRT \), where P stands for pressure, V for volume, n for the number of moles, R for the gas constant, and T for temperature. For density calculations, the equation is rearranged to the form \( \rho = \frac{{PM}}{{RT}} \), where \( \rho \) represents the density and M is the molar mass of the gas mixture.
We then substitute in the known values for pressure, molar mass, the gas constant, and temperature to find the gas mixture's density. For instance, after converting pressure from mmHg to atmospheres and temperature from Celsius to Kelvin, we use the already calculated molar mass of the mixture to determine the density. Follow these ordered steps, and you can compute the density for any gas under specific conditions, as shown in the detailed solution above.
Ideal Gas Law
The Ideal Gas Law is integral to understanding and predicting the behavior of gases, particularly when dealing with changes in temperature, pressure, and volume. This principle suggests that a gas's behavior can be mathematically modeled with the equation \( PV = nRT \), which connects the pressure (P), volume (V), and temperature (T) of a gas, with the amount of substance in moles (n) and the ideal gas constant (R).
In practical terms, this law means that if we know any three of these variables, we can find the fourth. The constant R has different values depending on the units used for pressure, volume, and temperature, with a common value being 0.0821 L.atm/mol.K in the case of using atmospheres, liters, and Kelvins.
To find the density, the original equation is adjusted, as observed in step 3 of the provided solution, so that the density (\( \rho \)) can be calculated as \( \frac{{PM}}{{RT}} \), where M is the molar mass of the gas. The Ideal Gas Law, when manipulated correctly, works superbly for gases at low pressures and high temperatures, where gas behavior approximates ideality.
