Question

State whether the pressure of a gas in a sealed container increases or decreases with the following changes: (a) The volume changes from \(75.0 \mathrm{~mL}\) to \(50.0 \mathrm{~mL}\). (b) The temperature changes from \(0^{\circ} \mathrm{C}\) to \(-195^{\circ} \mathrm{C}\). (c) The moles of gas change from \(1.00 \mathrm{~mol}\) to \(5.00 \mathrm{~mol}\).

Short answer

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

Step-by-step solution

Analyzing volume change

According to Boyle's Law, pressure and volume are inversely related when temperature and moles of gas are constant. The formula is \( P_1 V_1 = P_2 V_2 \). As the volume decreases from 75.0 mL to 50.0 mL, the pressure must increase to compensate for the smaller volume.

Analyzing temperature change

According to Charles's Law, pressure is directly proportional to temperature when volume and moles of gas are constant. The formula, when combined with ideal gas law, can be expressed as \( P_1/T_1 = P_2/T_2 \). The temperature decrease from \(0^{\circ} \mathrm{C}\) to \(-195^{\circ} \mathrm{C}\) (converted to Kelvin: 273 K to 78 K) leads to decreased pressure as temperature decreases.

Analyzing 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. More moles mean more particles exerting pressure. Increasing the moles from 1.00 mol to 5.00 mol causes the pressure to increase as the number of particles increases.

Key concepts

Boyle's Law

Boyle's Law is a fundamental principle in gas behavior, stating that the pressure of a gas is inversely proportional to its volume when the temperature and the number of moles are held constant. This means if you decrease the volume of a gas, its pressure will increase, and vice versa. The law is mathematically expressed as:\[ P_1 V_1 = P_2 V_2 \]Here, \(P_1\) and \(V_1\) refer to the initial pressure and volume, while \(P_2\) and \(V_2\) refer to the final pressure and volume. For instance, if you compress the gas in a sealed container from 75.0 mL to 50.0 mL, the pressure will rise since the gas molecules have less space to move. This smaller volume forces the molecules to hit the container walls more frequently, thus increasing the pressure. Remember, this relationship only holds when the gas temperature and quantity remain unchanged.

Charles's Law

Charles's Law explores the relationship between the temperature and volume of a gas, asserting that they are directly proportional when pressure and the number of moles are constant. Thus, increasing the temperature of a gas will increase its volume, and decreasing the temperature will decrease its volume. Mathematically, it's often seen in this form:\[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]If volume is constant, a similar relationship exists for pressure and temperature. During a temperature change from 0°C to -195°C, converted to Kelvin as 273 K to 78 K, the pressure declines. Why? Lowering the temperature slows down the gas molecules, causing them to hit the container walls less forcefully and frequently. This reduction in kinetic energy explains the pressure drop during cooling. In essence, Charles's Law shows how temperature alterations affect the physical behavior of gases.

Avogadro's Law

Avogadro's Law highlights the relationship between the number of gas molecules (or moles) and the volume (or pressure), maintaining that they are directly proportional if temperature and pressure are constant. This means that as the number of moles increases, so does the pressure or volume of the gas, given other conditions remain stable. The equation can be expressed as:\[ \frac{V_1}{n_1} = \frac{V_2}{n_2} \]Applying this to a situation where the moles increase from 1.00 to 5.00, the pressure also increases significantly. More moles mean more gas particles within the same space, leading to more frequent collisions with the container walls and thus higher pressure. Avogadro's Law essentially shows how the amount of gas can influence its physical properties, reinforcing the intuitive understanding that more gas takes up more space or exerts more pressure.