Cyclopropane mixed in the proper ratio with oxygen can be used as an anesthetic. At \(755 \mathrm{~mm} \mathrm{Hg}\) and \(25^{\circ} \mathrm{C}\), it has a density of \(1.71 \mathrm{~g} / \mathrm{L}\). (a) What is the molar mass of cyclopropane? (b) Cyclopropane is made up of \(85.7 \% \mathrm{C}\) and \(14.3 \% \mathrm{H}\). What is the molecular formula of cyclopropane?

The ideal gas law is a critical concept for anyone studying chemistry or physics, as it describes the behavior of an 'ideal' gas under various conditions. It is represented by the equation PV=nRT, where P stands for pressure, V for volume, n for the amount of substance in moles, R for the universal gas constant, and T for temperature in Kelvin.

To use the ideal gas law effectively, it's essential to ensure that the units used are consistent. For pressure, atmospheres (atm) are often preferred, and the volume must be in liters (L) to align with the gas constant R, which has units of L*atm/(mol*K). When dealing with temperature, always use Kelvin, as it is the SI unit for thermodynamic temperature scale.

In the context of calculating the molar mass of cyclopropane, the ideal gas law helps to determine the molar volume at which one mole of gas occupies under given pressure and temperature. This foundational understanding is the stepping stone towards finding the molar mass, as the density is connected to molar volume through the equation \( \rho = \frac{mass}{volume} \).

The distinction between empirical and molecular formulas is fundamental in chemistry. An empirical formula represents the simplest whole number ratio of the elements in a compound, while the molecular formula shows the actual number of atoms of each element in a molecule. To determine these formulas, the percentage composition of each element must be known.

Taking the example of cyclopropane, the empirical formula is derived by converting percentages into moles, based on the assumption of a 100g sample. Then, the moles of each element are divided by the smallest number of moles to achieve the simplest ratio. For many compounds, the empirical formula is a building block for the molecular formula. To bridge the gap from the empirical formula to the molecular formula, the molar mass (obtained from molar volume and density via the ideal gas law) is divided by the mass of the empirical formula unit. The resulting ratio is used to multiply the subscripts in the empirical formula to obtain the molecular formula, giving a true representation of the substance's composition.

The molar mass of a substance is a fundamental physical property defined as the mass of one mole of that substance. It is a key piece of information for many chemical calculations. In the case of gases, molar mass can be determined from the density using the ideal gas law. By manipulating the law to express n/V (moles per volume), it becomes possible to solve for the molar mass when density (mass per volume) is given.

In the solution for the molar mass of cyclopropane, density (\( \rho \)) is utilized along with the calculated molar volume under the specified conditions of temperature and pressure. The formula \( molar~mass = \frac{\rho*molar~volume}{n} \) directly relates density to molar mass, simplifying the process of calculation. The determined molar mass not only provides insight into the molecular weight of the gas but also serves as a stepping stone to finding the molecular formula. By combining the understanding of molar mass, empirical formula, and molecular formula, one gains a comprehensive overview of the compound's identity in terms of its compositional structure.