Question

State whether the pressure of a gas in a sealed container increases or decreases with the following changes: (a) The volume changes from \(2.50 \mathrm{~L}\) to \(5.00 \mathrm{~L}\). (b) The temperature changes from \(20^{\circ} \mathrm{C}\) to \(100{ }^{\circ} \mathrm{C}\). (c) The moles of gas change from \(0.500 \mathrm{~mol}\) to \(0.250 \mathrm{~mol}\).

Short answer

(a) Pressure decreases; (b) Pressure increases; (c) Pressure decreases.

Step-by-step solution

Analyze Volume Change

According to Boyle's Law, which states that pressure is inversely proportional to volume (\( P \propto \frac{1}{V} \) for constant temperature and moles), if the volume of a gas increases from \(2.50 \, \mathrm{L}\) to \(5.00 \, \mathrm{L}\), the pressure will decrease. This is because the gas particles will have more space to move, leading to fewer collisions with the walls of the container.

Analyze Temperature Change

According to Charles's Law, which states that pressure is directly proportional to temperature (in Kelvin) when volume and moles are constant (\( P \propto T \)), increasing the temperature from \(20^{\circ} \mathrm{C}\) (or \(293 \, \mathrm{K}\)) to \(100^{\circ} \mathrm{C}\) (or \(373 \, \mathrm{K}\)) will increase the pressure of the gas. Higher temperature means increased kinetic energy, thus more collisions and higher pressure.

Analyze Moles of Gas Change

According to Avogadro's Law, pressure is directly proportional to the number of moles of gas when volume and temperature are constant (\( P \propto n \)). Decreasing the moles from \(0.500 \, \mathrm{mol}\) to \(0.250 \, \mathrm{mol}\) will result in a decrease in pressure since there are fewer particles to exert force on the container walls.

Key concepts

Boyle's Law

Boyle's Law reveals the fascinating relationship between the pressure and volume of a gas. It simply states that when the temperature and the number of moles of gas remain constant, the pressure of a gas is inversely proportional to its volume. This can be expressed as:\[ P \propto \frac{1}{V} \]In other words, if you increase the volume of a gas, the pressure decreases, and vice versa. Imagine a sealed balloon: as you inflate it, you increase its volume, and if you measure the pressure inside, it will lower. The gas particles have more space, so they collide less with the container walls.When applying Boyle's Law to scenarios like the exercise, always ensure that the temperature and the amount of gas stay constant. This helps understand how gases behave under different volume changes. As seen in the example, doubling the volume reduces the pressure, making this concept crucial for calculations where gas behavior is affected by volume changes.

Charles's Law

Charles's Law is our go-to when understanding how the temperature and pressure of a gas relate. When you keep the volume and number of moles constant, pressure is directly proportional to the temperature, which means:\[ P \propto T \]But remember, temperature must always be in Kelvin to accurately reflect this relationship. Kelvin starts at absolute zero, ensuring you avoid negative temperatures in calculations.So, if you heat a gas, its pressure will increase. The gas particles move faster at higher temperatures, hitting the container walls more frequently and with greater force, resulting in increased pressure. In the exercise we see, raising the temperature from \(293 \, \mathrm{K}\) to \(373 \, \mathrm{K}\) leads to a pressure increase. This principle is evident in everyday life. Think of a car tire on a hot sunny day. As the air inside warms up, the tire pressure rises. By understanding Charles's Law, you appreciate these everyday phenomena and become adept at predicting how gases respond to temperature changes.

Avogadro's Law

Avogadro's Law opens up a world of understanding about the relationship between the amount of gas and its pressure. According to this law, when the volume and temperature remain constant, the pressure of a gas is directly proportional to the number of moles. This can be shown as:\[ P \propto n \]This means that if you add more gas particles into a container, the pressure will go up, given that there's no change in volume or temperature. Similarly, if you remove some gas, the pressure decreases. Consider the exercise where the moles decrease from \(0.500 \, \mathrm{mol}\) to \(0.250 \, \mathrm{mol}\). With fewer molecules to collide against the walls, the pressure naturally decreases. This concept is crucial when dealing with reactions and processes in closed systems, like gas cylinders. It helps predict how alterations in the amount of gas affect system pressure, refining our ability to manage and utilize gases in various applications.