At \(22.0^{\circ} \mathrm{C}\) and a pressure of 755 torr, a gas was found to have a density of \(1.13 \mathrm{~g} \mathrm{~L}^{-1}\). Calculate its molar mass.

The ideal gas law is a fundamental equation in chemistry and physics that relates the pressure, volume, temperature, and amount of an ideal gas. It is expressed as \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.

In practical problems like the one we're examining, the ideal gas law can be reorganized to \( P = (n/V)RT \) or \( d = PM/RT \) when we talk about density, with \( d \) being density and \( M \) the molar mass. By knowing three of the variables, you can always solve for the fourth, which offers a significant insight into the behavior of gases under different conditions.

Gas density refers to the mass of the gas per unit volume, typically expressed in grams per liter (g/L). It can vary greatly with changes in temperature and pressure. For an ideal gas, density can be a useful property when linked with the ideal gas law, as seen in the exercise.

Understanding the relationship between molar mass, pressure, and temperature is crucial when calculating the density of a gas, as one can determine another if the rest are known. In the solution above, by rearranging the ideal gas law to solve for the molar mass, and given the density, we arrive at the formula \( M = dRT / P \) where R is again a constant value. The ability to calculate gas density is essential in many scientific and industrial processes.

Temperature needs to be in Kelvin when using the ideal gas law because the gas constant \( R \) is expressed in these units. Kelvin is the SI base unit for temperature and is used in scientific calculations to avoid negative temperature values. The conversion is straightforward:

Converting Celsius to Kelvin

\( T(K) = T(^\text{C}) + 273.15 \)

It's important to add 273.15 to the Celsius temperature to make the conversion. In the given exercise, a Celsius temperature of 22.0 degrees was converted to Kelvin by adding 273.15, resulting in 295.15 K.

Pressure units often need to be converted to atmospheres (atm) when using the ideal gas law because the gas constant \( R \) is given in terms of atmospheres. To convert from torr (or millimeters of mercury, mmHg) to atmospheres:

Converting Torr to Atmospheres

\( P(atm) = P(torr) / 760 \)

There are exactly 760 torr in one atmosphere. By dividing the pressure in torr by 760, we get the pressure in atmospheres. For example, in the solution provided, the pressure of the gas in torr is divided by 760 to give a pressure in atmospheres, crucial for the calculation of molar mass using the ideal gas law.