Question

If a sample of oxygen gas occupies \(25.0 \mathrm{~mL}\) at \(-25^{\circ} \mathrm{C}\) and \(650 \mathrm{~mm} \mathrm{Hg}\), what is the volume at \(25^{\circ} \mathrm{C}\) and \(350 \mathrm{~mm}\) Hg?

Short answer

The volume at the new conditions is 55.6 mL.

Step-by-step solution

Understand the Problem

We are given the initial volume, pressure, and temperature of a gas and need to find its volume at a different temperature and pressure. We'll use the combined gas law to solve this.

Write the Combined Gas Law

The combined gas law is \[\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\]where \(P\) is pressure, \(V\) is volume, and \(T\) is temperature in Kelvin.

Convert Temperatures to Kelvin

First, we need to convert the given temperatures from Celsius to Kelvin. The conversion formula is \[ T(K) = T(°C) + 273.15 \]For \(-25^{\circ} \mathrm{C}\): \[T_1 = -25 + 273.15 = 248.15 \text{ K}\]For \(25^{\circ} \mathrm{C}\): \[T_2 = 25 + 273.15 = 298.15 \text{ K}\]

Plug Values Into the Combined Gas Law

Given: \[P_1 = 650 \mathrm{~mm} \text{ Hg}\]\[V_1 = 25.0 \mathrm{~mL}\]\[T_1 = 248.15 \text{ K}\]\[P_2 = 350 \mathrm{~mm} \text{ Hg}\]\[T_2 = 298.15 \text{ K}\]We need to find \(V_2\). Plug these into the combined gas law:\[\frac{650 \times 25.0}{248.15} = \frac{350 \times V_2}{298.15}\]

Solve for the New Volume \(V_2\)

Rearrange the equation from Step 4 to solve for \(V_2\):\[V_2 = \frac{650 \times 25.0 \times 298.15}{350 \times 248.15}\]Calculate:\[V_2 = \frac{650 \times 25.0 \times 298.15}{350 \times 248.15} = 55.6 \mathrm{~mL}\]

Key concepts

Gas Laws

Gas laws are key principles in chemistry that describe how gases behave under different conditions of pressure, volume, and temperature. These laws are important for understanding how changes in one parameter can affect another. Among the many gas laws, the combined gas law is particularly essential when looking at scenarios where multiple variables are changing. It incorporates the three fundamental gas laws:

• Boyle's Law: This describes how pressure and volume are inversely related at constant temperature: \[ P_1 V_1 = P_2 V_2 \]
• Charles's Law: This shows that volume and temperature are directly proportional at constant pressure: \[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]
• Gay-Lussac's Law: This explains that pressure and temperature are directly proportional at constant volume: \[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \]

When multiple factors change, as in the given exercise, the combined gas law equation \( \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} \) is used. It helps predict the new state of a gas by connecting all three variables, making it extremely versatile and useful in practical scenarios.

Temperature Conversion

Temperature conversion is a necessary step when working with gas laws. Gas law equations require temperature to be in Kelvin rather than Celsius. This is because Kelvin is an absolute scale starting at absolute zero, where theoretical molecular motion stops. Converting from Celsius to Kelvin is simple and essential for accuracy in calculations.To convert Celsius to Kelvin, use the formula:\[ T(K) = T(°C) + 273.15 \]This step is crucial in any gas law calculation because it ensures temperature is on an absolute scale. For example, in the exercise, the initial temperature of \(-25^{\circ} \mathrm{C}\) was converted to Kelvin as \(248.15 \mathrm{~K}\), and \(25^{\circ} \mathrm{C}\) as \(298.15 \mathrm{~K}\). These converted temperatures allow for correct application of the combined gas law.

Pressure Units

Pressure, a critical factor in gas laws, is often given in various units. In many chemistry problems, pressure is expressed in millimeters of mercury (mm Hg) or atmospheres (atm). The units of pressure need to remain consistent throughout the equation. Otherwise, calculations will be incorrect. In the presented problem, pressure is expressed in mm Hg. If switching to another unit is required, standard conversion between units is necessary. Knowing common conversion factors is helpful. For example,

• 1 atm = 760 mm Hg
• 1 atm = 101.325 kPa

Consistency in pressure units ensures that the combined gas law's proportional relationships hold, leading to accurate predictions of the new state of a gas. In the exercise, both initial and final pressures were given in mm Hg, so no conversion was necessary, simplifying the calculations.