(a) Calculate the density of sulfur hexafluoride gas at 707 torr and \(21^{\circ} \mathrm{C}\) . (b) Calculate the molar mass of a vapor that has a density of 7.135 \(\mathrm{g} / \mathrm{L}\) at \(12^{\circ} \mathrm{C}\) and 743 torr.

Understanding how to calculate the density of a gas involves applying the ideal gas law, which relates the volume (V), pressure (P), temperature (T), and number of moles (n) of a gas. The formula for the ideal gas law is \( PV = nRT \), where R is the universal gas constant.

To find the density of a gas, one must start by finding the number of moles of the gas present in a certain volume. This number can be expressed by deriving \( n \) from the ideal gas law equation, with the relationship \( n = \frac{PV}{RT} \). Density (\( \rho \)) is mass per unit volume, so the equation becomes \( \rho = \frac{m}{V} \), where mass (m) is the product of the number of moles and molar mass (M). So, the density of a gas can be calculated using the formula \( \rho = \frac{nM}{V} \). By substituting the expression for \( n \) from the ideal gas law, the equation for density becomes \( \rho = \frac{P \times M}{RT} \).

For the step by step solution provided to calculate the density of sulfur hexafluoride gas, the molar mass of the gas is first determined, and then the ideal gas law is used to find the moles. Knowing the conditions under which the gas is present (temperature, volume, and pressure), the calculation of density then simplifies to substituting values into the rearranged ideal gas law.

Determining the molar mass of a substance is essential in many chemical calculations. Molar mass is the mass of one mole of a substance, and it is expressed in grams per mole (g/mol). In gases, the molar mass can be determined from the ideal gas law by rearranging the equation to solve for the molar mass (M). Given the density of a gas, the molar mass can be calculated using the formula \( M = \frac{\rho RT}{P} \).

This rearranged form exposes M as a function of the gas's density (\( \rho \)), the universal gas constant (R), the absolute temperature (T), and the pressure (P). This method is pivotal when analyzing vapors or gases for which the molar mass is not initially known. It is particularly useful in situations like the one demonstrated in the b part of the exercise, where the molar mass of a vapor is calculated from its density under specific temperature and pressure conditions. In this manner, knowing the properties of the gas enables us to back out important informational details about the substance's molar mass, therefore, facilitating further analysis or calculations.

The Kelvin scale is the base unit of temperature in the International System of Units (SI) and is a crucial component in gas calculations, particularly when using the ideal gas law. Unlike Celsius or Fahrenheit, Kelvin is an absolute temperature scale where zero Kelvin (0 K) is absolute zero, the theoretical point where particles have minimal thermal motion.

To convert Celsius to Kelvin, the equation \( T(K) = T(°C) + 273.15 \) is used. This step is vital as all gas law equations require temperature to be expressed in Kelvin to ensure consistency and accuracy. For example, in the solution provided, temperatures given in Celsius are converted to Kelvin before being plugged into the ideal gas law. This conversion ensures that temperature is viewed on an absolute scale, allowing direct proportionality between variables like pressure, volume, and the number of moles in the gas law, which would not be the case with Celsius or Fahrenheit scales.