1\. A sample of methane has a volume of \(17.5 \mathrm{~L}\) at \(100.0^{\circ} \mathrm{C}\) and \(1.72 \mathrm{~atm}\). How many moles of methane are in the sample? 2\. A \(0.0500 \mathrm{~L}\) sample of a gas has a pressure of \(745 \mathrm{~mm} \mathrm{Hg}\) at \(26.4^{\circ} \mathrm{C}\). The temperatureis now raised to \(404.4 \mathrm{~K}\) and the volume is allowed to expand until a final pressure of \(1.06\) atm is reached. What is the final volume of the gas? 3\. When \(128.9\) grams of cyclopropane \(\left(\mathrm{C}_{3} \mathrm{H}_{6}\right)\) are placed into an \(8.00 \mathrm{~L}\) cylinder at \(298 \mathrm{~K}\), the pressure is observed to be \(1.24\) atm. A piston in the cylinder is now adjusted so that the volume is now \(12.00 \mathrm{~L}\) and the pressure is \(0.88 \mathrm{~atm} .\) What is the final temperature of the gas?

The process of calculating the moles of methane involves the ideal gas law, which is expressed through the formula PV = nRT. In this case, to determine the number of moles (), we have the given volume (V) of methane gas, expressed in liters, the pressure (P) in atmospheres, and the temperature (T) converted into Kelvin by adding 273.15 to the given Celsius temperature. With the values for P, V, and T known, and the ideal gas constant (R) as 0.0821 L\(\cdot\)atm/(mol\(\cdot\)K), the moles of methane can be solved by rearranging the ideal gas equation to n = PV/RT.

By inserting the specific values into the rearranged formula, we find the exact number of moles present in the sample. It's important to ensure that all units match those required by the ideal gas equation, particularly temperature in Kelvin and pressure in atmospheres, to obtain the correct calculation. This conversion consistency is essential when applying the ideal gas law.

Conversions play a key role in applying gas laws effectively. In the case of the combined gas law, pressures may need to be converted between different units, such as from mmHg to atmospheres, because the gas law requires consistency in units. The conversion factor between mmHg and atm is that 1 atm equals 760 mmHg.

To apply the combined gas law correctly, it's also critical to convert temperatures to Kelvin. This is done by adding 273.15 to the Celsius temperature. Additionally, the volume and number of moles may need to be expressed in liters and moles respectively, consistent with the values for R, the ideal gas constant. Ensuring these conversions before applying the gas laws ensures accurate results and avoids common calculation errors.

The combined gas law is a cornerstone when dealing with the three-state variables of a gas, which are pressure, volume, and temperature. It integrates Charles's Law, Boyle's Law, and Gay-Lussac's Law, and is represented as P1V1/T1 = P2V2/T2, where P is pressure, V is volume, T is temperature, and the subscripts 1 and 2 refer to the initial and final states of the gas, respectively.

In practice, the combined gas law allows us to calculate the unknown final state of a gas when the other conditions are known. For instance, if the initial pressure, volume, temperature, and final pressure and temperature are known, this law can be used to compute the final volume. It is essential to ensure all variables are in the correct units before calculation. Being a versatile tool, its application extends to a multitude of problems in thermodynamics and environmental science.

Analyzing temperature changes in gases such as cyclopropane involves the ideal gas equation once more. When the volume and pressure of a gas change, its temperature change can be determined, assuming we know the amount of gas in moles. In the case of cyclopropane, we first have to find the number of moles from its given mass and molar mass. After that, we can use the ideal gas law with the initial conditions to find the initial temperature and then apply it again with the final conditions to discover the new temperature after the change.

The temperature change can give us insights into the effects of pressure and volume changes on a gas within a confined space, such as a cylinder with a piston. This concept is pivotal in fields like anesthesia, where cyclopropane was historically used, demonstrating the practical implementations of the ideal gas law in real-world situations.