Question

A 93-L sample of dry air cools from \(145^{\circ} \mathrm{C}\) to \(-22^{\circ} \mathrm{C}\) while the pressure is maintained at 2.85 atm. What is the final volume?

Short answer

The final volume is approximately 55.8 L.

Step-by-step solution

- Write Down the Initial Parameters

Given:Initial Volume, \( V_1 = 93 \text{ L} \)Initial Temperature, \( T_1 = 145^{\circ}\text{C} \)Converted to Kelvin: \( T_1 (K) = 145 + 273.15 = 418.15 \text{ K} \)Final Temperature, \( T_2 = -22^{\circ}\text{C} \)Converted to Kelvin: \( T_2 (K) = -22 + 273.15 = 251.15 \text{ K} \)Pressure, \( P = 2.85 \text{ atm} \).

- Use the Combined Gas Law

The Combined Gas Law relates pressure, volume, and temperature: \[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \].Since the pressure remains constant, we can simplify it to Charles's Law which states: \[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \].

- Rearrange for the Final Volume

Rearrange Charles's Law to solve for \( V_2 \):\[ V_2 = V_1 \times \frac{T_2}{T_1} \].

- Substitute Known Values

Substitute the known values into the equation:\[ V_2 = 93 \text{ L} \times \frac{251.15 \text{ K}}{418.15 \text{ K}} \].

- Calculate the Final Volume

Perform the calculation:\[ V_2 \approx 55.8 \text{ L} \].

Key concepts

Gas Laws

Gas laws are a set of scientific principles that describe the behavior of gases. These laws help us understand how variables such as pressure, volume, and temperature interact with each other in a gas.
The various laws you might encounter include Boyle's Law, Charles's Law, and Avogadro's Law. Each law isolates one or two variables while holding the others constant.
When all gas laws are combined, they form the Ideal Gas Law represented as \textbf{PV = nRT}, where P stands for pressure, V for volume, T for temperature, n is the number of moles of gas, and R is the gas constant. For this exercise, we'll focus on Combined Gas Law derived from these fundamental principles. It combines Boyle’s, Charles’s, and Gay-Lussac’s laws into one formula:

• \[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \]

This equation is useful to solve problems involving changes in more than one factor – in this case, both temperature and volume.

Charles's Law

Charles's Law is a specific gas law that highlights the direct relationship between the volume and temperature of a gas held at constant pressure. Mathematically, it's represented as:

• \[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]

This relationship implies that when the temperature of a gas increases, its volume increases proportionally, and vice versa, provided the gas pressure is unchanged.
In the given exercise, since the pressure remains constant, we use Charles's Law to find the final volume of the gas after a change in temperature. This simplification helps us transition from the more general Combined Gas Law to a more straightforward problem-solving approach.

Temperature Conversion

Temperature conversion is crucial when dealing with gas laws because temperatures must be in absolute terms, i.e., Kelvin (K). The Kelvin scale is used since it starts at absolute zero, which is the theoretical point where particle motion stops.
To convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature:

• \[ T(K) = T(^{\text{C}}) + 273.15 \]

In this exercise, the initial temperature was \(145^{\text{C}}\), which converts to \[418.15 \text{ K} \] and the final temperature was \(-22^{\text{C}}\), converting to \[251.15 \text{ K} \].
Converting correctly ensures that we apply gas laws accurately without discrepancies due to incompatible units.

Pressure-Volume Relationship

The pressure-volume relationship in gases is generally concerned with how the volume of a gas changes with pressure. According to Boyle's Law, for a fixed amount of gas at constant temperature, the volume is inversely proportional to pressure:

• \[ P_1 V_1 = P_2 V_2 \]

However, in our exercise, the pressure remains constant at 2.85 atm throughout the process, so we do not need to compute changes in pressure.
Simplifying the Combined Gas Law under constant pressure conditions allows us to use Charles's Law directly. Changing either volume or temperature necessitates consistent units and accurate calculations for practical outcomes.
Understanding these interrelationships gives us better insight into gas behavior in various practical and theoretical applications.