Question

What pressure is required to compress \(925 \mathrm{~L}\) of \(\mathrm{N}_{2}\) at \(1.25 \mathrm{~atm}\) into a container whose volume is \(6.35 \mathrm{~L}\) ?

Short answer

The required pressure to compress the nitrogen gas into the smaller container is approximately 182.04 atm.

Step-by-step solution

Identify Known Quantities

We know the initial pressure \(P_{1} = 1.25 \mathrm{~atm}\), the initial volume \(V_{1} = 925 \mathrm{~L}\), and the final volume \(V_{2} = 6.35 \mathrm{~L}\). We have to find the final pressure \(P_{2}\).

Use the Reformulated Ideal Gas Law

Let's use the formula \(P_{1}V_{1} = P_{2}V_{2}\). Plugging in the known values, we get \(1.25 \mathrm{~atm} \cdot 925 \mathrm{~L} = P_{2} \cdot 6.35 \mathrm{~L}\). This can be rearranged to find \(P_{2}\): \(P_{2} = \frac{1.25 \mathrm{~atm} \cdot 925 \mathrm{~L}}{6.35 \mathrm{~L}}\).

Calculate the Final Pressure

We perform the division to find the final pressure. Remember to keep units consistent, so the final pressure will be in atmospheres.

Key concepts

Gas Compression

Understanding gas compression is crucial when analyzing how gases behave under different conditions. Gas compression refers to the process of reducing the volume of a gas, which inherently results in an increase in the gas pressure. This phenomenon occurs because gases are composed of particles that move freely and occupy space. As the space (volume) is reduced, the same number of particles is now constrained into a smaller area. This leads to more frequent particle collisions with the walls of the container, thus increasing the pressure. It's akin to having more people in a room where everybody moves around; the smaller the room, the more likely you are to bump into someone.

In practical applications, gas compression is used in various industries, such as in refrigeration systems where gases are compressed to cool the surroundings, or in internal combustion engines where air is compressed before ignition. The relationship between gas volume and pressure during compression is precisely described by Boyle's Law, which we will delve into further in this article.

Gas Pressure

Gas pressure is a critical concept in understanding gas behaviors. It is defined as the force exerted by the gas molecules against the surface of a container, measured per unit area. This force is the result of gas particles constantly moving and colliding with the container's walls. These microscopic collisions translate into what we can measure as pressure. The SI unit for pressure is the pascal (Pa), but it is commonly measured in atmospheres (atm) or millimeters of mercury (mmHg) in many scientific contexts.

Factors affecting gas pressure include temperature, volume, and the number of gas molecules present. In the given exercise, we are focusing on the effect of volume change on gas pressure, keeping temperature constant. It should be noted that gas pressure can be influenced by outside elements such as altitude and temperature, so experiments and computations often aim to control for these variables to accurately apply the Ideal Gas Law.

Boyle's Law

Boyle's Law is a fundamental principle in chemistry that describes how pressure and volume are inversely related for a given mass of gas at a constant temperature. In mathematical terms, Boyle's Law is stated as: \( P_1V_1 = P_2V_2 \) where \( P_1 \) and \( V_1 \) are the initial pressure and volume, and \( P_2 \) and \( V_2 \) are the final pressure and volume of the gas, respectively. This equation shows that if the volume of the gas decreases (compression), the pressure will increase proportionally, and vice versa.

When applying Boyle's Law to solve problems, one must ensure that the temperature and mass of the gas remain constant, and that the gas conforms to the behavior of an ideal gas. In our textbook exercise, we see Boyle's Law applied to find the new pressure required to compress nitrogen gas. As the volume of the gas is considerably reduced from \(925 \) L to \(6.35 \) L, Boyle's Law predicts a significant increase in pressure which can be calculated using the formula. The concept reinforces the inverse relationship between pressure and volume and is beautifully demonstrated in practical scenarios like syringes and pistons in engines.