A chemist isolated a gas in a glass bulb with a volume of \(255 \mathrm{~mL}\) at a temperature of \(25.0{ }^{\circ} \mathrm{C}\) and a pressure (in the bulb) of 10.0 torr. The gas weighed \(12.1 \mathrm{mg}\). What is the molar mass of this gas?

The ideal gas law is a crucial equation in the field of chemistry and physics, providing a practical relationship between pressure (P), volume (V), temperature (T), and the amount of substance in moles (n) for an ideal gas. The law is articulated through the equation:
\( PV = nRT \)
where R is the ideal gas constant. In this scenario, we leverage the ideal gas law to determine the number of moles of a gas given its pressure, volume, and temperature.

To apply the ideal gas law correctly, it’s imperative that all units are consistent. Pressure should be in atmospheres, volume in liters, temperature in Kelvin, and R must match these units. In our exercise, after converting each measurement to the requisite units, we rearrange the formula to solve for the number of moles (n):
\( n = \frac{PV}{RT} \)
Utilizing the ideal gas law requires a precise grasp of various units and conversion, making it a perfect example of how diverse concepts in chemistry are intertwined.

In chemistry, it’s essential to work with a standardized set of units to ensure accurate and coherent computations. This exercise involves multiple conversions across different units of measurement, which is a common practice in the subject.

Pressure, initially presented in torr, was converted to atmospheres using the conversion factor \(1 atm = 760 torr\). Similarly, volume was transformed from milliliters to liters, recognizing that \(1 L = 1000 mL\). Lastly, the mass in milligrams was converted into grams with the conversion factor \(1 g = 1000 mg\).

These conversions are pivotal for the following reasons:

• Ensuring consistency with the ideal gas constant’s units.
• Facilitating the application of equations like the ideal gas law.
• Allowing for clear communication of scientific data.

Mastering unit conversions is a cornerstone of chemical problem-solving, and meticulous attention to units can make the difference between the right answer and a significant error.

When working with gases and their behaviors under varying conditions, Kelvin is the temperature scale of choice. It’s the SI (International System of Units) unit for thermodynamic temperature and is preferred because it starts at absolute zero, implying there is no negative temperature in Kelvin, which simplifies calculations.

The conversion from Celsius to Kelvin is straightforward and necessary for all gas law problems:
\( K = ^\circ C + 273.15 \)
For our chemist’s gas bulb at a cozy \(25.0{ }^\circ C\), we find the Kelvin temperature to be: \(25.0 + 273.15\). This step is fundamental because the behavior of gases is directly proportional to the temperature in Kelvin, thereby impacting the gas' pressure and volume.

Temperature in Kelvin is a basic yet vital conversion, transitioning from everyday Celsius scale to the one required for theoretical and experimental gas law analysis. Understanding how to perform this conversion is a key skill in tackling thermodynamic problems in chemistry.