A compressed gas cylinder contains \(1.00 \times 10^{3} \mathrm{g}\) argon gas. The pressure inside the cylinder is \(2050 .\) psi (pounds per square inch) at a temperature of \(18^{\circ} \mathrm{C}\). How much gas remains in the cylinder if the pressure is decreased to \(650 .\) psi at a temperature of \(26^{\circ} \mathrm{C} ?\)

The Ideal Gas Law is a fundamental equation in chemistry that relates the pressure, volume, number of moles, and temperature of an ideal gas. It is expressed as the formula: \[PV = nRT\]Where:

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

To solve problems involving the Ideal Gas Law, one must ensure all values are in the correct units: pressure in atmospheres, volume in liters, temperature in Kelvin, and the gas constant in the proper units typically \(0.0821 \frac{L\cdot atm}{mol\cdot K}\) for this context. The law is extremely useful for predicting the behavior of gases under various conditions, as seen in the exercise problem where the law is used to deduce the amount of argon gas remaining in a cylinder.

The molar mass is the weight of one mole of a substance, usually expressed in grams per mole (g/mol). It serves as a bridge between the mass of a substance and the number of moles, allowing chemists to convert between these two units. To find out how many moles of a substance you have, you'd divide the mass of the substance by its molar mass.For example, the molar mass of argon is about 39.95 g/mol. If you have \(1.00 \times 10^{3}\) grams of argon, you calculate the number of moles by:\[n = \frac{1.00 \times 10^{3} \text{g}}{39.95 \text{g/mol}}\]Understanding molar mass is crucial for calculations in chemistry, such as when using the Ideal Gas Law or when preparing chemical solutions.

The pressure-temperature relationship is an aspect of the Ideal Gas Law that describes how pressure and temperature are directly proportional when the volume and the number of moles remain constant. This means when the temperature of an enclosed gas increases, the pressure also increases if the gas's volume is kept unchanged, and vice versa. This is captured by the fraction of pressures and temperatures in the Ideal Gas Law manipulations, specifically when you compare two states of the gas:\[\frac{P_1}{P_2} = \frac{n_1T_2}{n_2T_1}\]This relationship is valuable in understanding how gases will behave under different thermal conditions, such as in refrigeration systems, car engines, and even when analyzing the problem at hand involving argon gas in a cylinder.

Temperature conversion between Celsius and Kelvin is a basic, yet critical task in many chemical calculations. In particular, all temperature values must be in Kelvin when applying the Ideal Gas Law. The conversion is straightforward: to convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature:\[T(K) = T(°C) + 273.15\]Applying this in the context of our problem, to convert the initial temperature of \(18°C\) to Kelvin:\[T1 = 18 + 273.15 = 291.15 K\]And for the final temperature of \(26°C\):\[T2 = 26 + 273.15 = 299.15 K\]This step is often where many students make errors, but remembering that 0°C corresponds to 273.15 K can help avoid mistakes in calculations.

Pressure units often need to be converted to match the standard unit required for equations in chemistry, like atmospheres (atm) for the Ideal Gas Law. Pounds per square inch (psi) is commonly used in the United States but must be converted to atm for these calculations. The conversion factor is:\[1 \text{atm} = 14.696 \text{psi}\]To convert psi to atm, divide the pressure in psi by the conversion factor. In our problem, for the initial pressure:\[P1 = \frac{2050 \text{psi}}{14.696 \text{psi/atm}} = 139.57 \text{atm}\]And for the final pressure:\[P2 = \frac{650 \text{psi}}{14.696 \text{psi/atm}} = 44.23 \text{atm}\]Correct conversion is essential as using incorrect units can lead to erroneous results in chemical calculations.