A 2.50-L container is filled with \(175 \mathrm{~g}\) argon. a. If the pressure is \(10.0 \mathrm{~atm}\), what is the temperature? b. If the temperature is \(225 \mathrm{~K}\), what is the pressure?

The ideal gas law is a fundamental equation bridging the pressure, volume, amount, and temperature of an ideal gas. The equation is given as:
\( PV = nRT \)
where

• \(P\) represents the pressure of the gas,
• \(V\) stands for the volume it occupies,
• \(n\) is the number of moles,
• \(R\) is the ideal gas constant, and
• \(T\) is the absolute temperature in Kelvin.

The real power of this equation lies in its ability to analyze a sample of gas under various conditions. By rearranging the equation, we can solve for any one of the five variables if the other four are known. Whether you're determining the pressure exerted by a gas in a container at a certain temperature or calculating the needed volume for a gas at a certain pressure, the ideal gas law is the go-to tool.

Calculating gas pressure involves understanding the forces that gas particles exert on the walls of their container due to their random motion. To find the pressure of a confined gas using the ideal gas law, you rearrange the formula to solve for the pressure (\(P\)) as such:
\( P = \frac{nRT}{V} \)
By inputting the number of moles (\(n\)), the temperature in Kelvin (\(T\)), and the volume (\(V\)), and using a suitable value for the ideal gas constant (\(R\)), the resulting calculation gives you the pressure. This calculation assumes the behavior of an ideal gas, which is a simplified model where the gas particles do not interact with each other except through elastic collisions and occupy no volume themselves. Real gases may diverge from this model, especially under high pressures or low temperatures, but the ideal gas law provides a useful first approximation for many conditions.

Converting mass to moles is a routine conversion in chemistry that allows for the use of the ideal gas law, among other equations. To convert from grams to moles, you use the molar mass of the substance, which is the mass of one mole of its particles (usually in grams per mole). The formula is straightforward:
\( n = \frac{mass}{molar\;mass} \)
For example, with argon (\(Ar\)), which has a molar mass of approximately \(39.95\,g/mol\), a sample weighing \(175\,g\) would be calculated as:
\( n = \frac{175\,g}{39.95\,g/mol} \approx 4.38\,mol \).
This conversion is critical for understanding the amount of substance you are dealing with. Since many properties and reactions in chemistry are dependent on the number of particles present, rather than their mass, converting mass to moles allows chemists to use the ideal gas law and other mole-based formulas to predict gas behavior.

Gas temperature calculation in the context of the ideal gas law involves determining the absolute temperature at which a gas must be to occupy a specified volume, exert a certain pressure, and contain a set number of moles. The temperature in the ideal gas law is always expressed in Kelvin, which is a scale that starts at absolute zero (the theoretical point where particles have no thermal energy). To find the temperature (\(T\)), you rearrange the ideal gas law as:
\( T = \frac{PV}{nR} \)
By substituting in the known pressure (\(P\)), the volume (\(V\)), the moles of gas (\(n\)), and the ideal gas constant in the correct units (\(R\)), you can find the temperature. It's crucial for this value to be in Kelvin for accurate calculations, and if you're given a Celsius temperature, you must convert it by adding \(273.15\) to the Celsius degree to get Kelvin. Being precise in this step is crucial as gas properties are highly temperature-sensitive.