Question

(a) Suppose that \(4.00 \mathrm{~L}\) of methane at a pressure of 800 . Torr is transferred to a vessel of volume \(2.40 \mathrm{~L}\). What is the final pressure of methane if the change occurs at constant temperature? (b) A fluorinated organic gas in a cylinder is compressed from an initial volume of \(936 \mathrm{~mL}\) at \(158 \mathrm{~Pa}\) to \(468 \mathrm{~mL}\) at the same temperature. What is the final pressure?

Short answer

(a) The final pressure of methane is 1333.33 Torr. (b) The final pressure of the fluorinated organic gas is 316 Pa.

Step-by-step solution

Understand Boyle's Law

Review Boyle's Law, which states that for a given mass of gas at constant temperature, the pressure of the gas is inversely proportional to its volume. In formula terms, it appears as: \( P_1V_1 = P_2V_2 \), where \( P_1 \) and \( P_2 \) are the initial and final pressures, and \( V_1 \) and \( V_2 \) are the initial and final volumes.

Solve for Final Pressure of Methane (Part a)

Using Boyle's Law for the methane gas, substitute the known values into the equation: \( (800 \, \text{Torr})(4.00 \, \text{L}) = P_2(2.40 \, \text{L}) \). Solve for \( P_2 \) by dividing the product of the initial pressure and volume by the final volume: \( P_2 = \frac{(800 \, \text{Torr})(4.00 \, \text{L})}{2.40 \, \text{L}} \).

Calculate Final Pressure of Methane (Part a)

Carry out the calculation to find the final pressure of methane: \( P_2 = \frac{(800 \, \text{Torr})(4.00 \, \text{L})}{2.40 \, \text{L}} = \frac{3200 \, \text{Torr}}{2.40 \, \text{L}} = 1333.33 \, \text{Torr} \).

Solve for Final Pressure of Organic Gas (Part b)

Using Boyle's Law for the fluorinated organic gas, substitute the known values into the equation: \( (158 \, \text{Pa})(936 \, \text{mL}) = P_2(468 \, \text{mL}) \). Solve for \( P_2 \) by dividing the product of the initial pressure and volume by the final volume: \( P_2 = \frac{(158 \, \text{Pa})(936 \, \text{mL})}{468 \, \text{mL}} \).

Calculate Final Pressure of Organic Gas (Part b)

Carry out the calculation to find the final pressure of the organic gas: \( P_2 = \frac{(158 \, \text{Pa})(936 \, \text{mL})}{468 \, \text{mL}} = \frac{147888 \, \text{Pa}}{468 \, \text{mL}} = 316 \, \text{Pa} \).

Key concepts

Gas Laws

Gas laws are foundational principles in chemistry that describe the behavior of gases under varying conditions. The most commonly referred gas laws are Boyle's Law, Charles's Law, Avogadro's Law, and the Ideal Gas Law. Each of these laws correlates to how changes in pressure, volume, and temperature affect a gas.

Boyle's Law, in particular, emphasizes the pressure-volume relationship for a gas at constant temperature, asserting that the volume of gas increases as pressure decreases, and vice versa, considering the temperature remains stable. Understanding these relationships is crucial in many scientific and industrial processes, such as filling an air balloon or designing a syringe.

Pressure-Volume Relationship

Delving into the pressure-volume relationship of gases, Boyle's Law states that the pressure of a gas is inversely proportional to its volume when temperature is kept constant. This implies that if the volume of a gas doubles, the pressure will halve, granted that the amount of gas and the temperature do not change.

This relationship is crucial in performing calculations related to gas compressions and expansions. It also informs us about the compressibility of gases as compared to liquids or solids and is used widely in applications involving pneumatic systems and the storage of gases under high pressure.

Chemical Principles

Chemical principles such as Boyle's Law are vital to understanding the behavior of substances and how they interact. They serve as the basis for reactions, stoichiometry, thermodynamics, kinetics, and equilibrium in chemistry. Boyle's Law, for instance, helps chemists predict how gases will behave when they are enclosed in a container and the volume of that container changes.

These principles are not just academic concepts but also have practical implications in the pharmaceutical industry, environmental monitoring, engineering, and beyond. A strong grasping of chemical principles fosters innovation and is essential for advancements in technologies that use gases, like fuel cells and internal combustion engines.

Ideal Gas Behavior

Ideal gas behavior is a theoretical concept where a gas's molecules do not attract or repel each other and occupy no volume themselves. Under this model, the ideal gas law combines Boyle's, Charles's, and Avogadro's laws into one equation: PV=nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature.

While no gas is truly 'ideal', many gases can be approximated as ideal under standard conditions. This makes the calculation and prediction of gas behavior more straightforward in many cases. However, deviations from ideal behavior can occur at high pressures or low temperatures, where intermolecular forces and the volume of the gas molecules become significant.