Question

For each of the following sets of volume/temperature data, calculate the missing quantity. Assume that the pressure and the mass of gas remain constant. a. \(V=2.01 \times 10^{2} \mathrm{~L}\) at 1150 " \(\mathrm{C} ; V=5.00 \mathrm{~L}\) at? \(\mathrm{C}\) b. \(V=44.2 \mathrm{~mL}\) at \(298 \mathrm{~K} ; V=?\) at \(0 \mathrm{~K}\) c. \(V=44.2 \mathrm{~mL}\) at \(298 \mathrm{~K} ; V=?\) at 0 'C

Short answer

The missing quantities for the given sets of volume/temperature data are as follows: a. The missing temperature is 81.6 °C. b. It is not possible to determine the missing volume at 0 K under ideal gas conditions. c. The missing volume at 0 °C is 40.9 mL.

Step-by-step solution

Convert temperature to Kelvin

First, convert the given temperature 1150°C to Kelvin: \(T_1 = 1150 + 273 = 1423 K\)

Apply Charles' Law equation

Plug in the given values into the equation: \(\frac{2.01 \times 10^{2} \mathrm{~L}}{1423 \mathrm{~K}} = \frac{5.00 \mathrm{~L}}{T_2}\)

Solve for the missing temperature

Cross-multiply and solve for \(T_2\): \(T_2 = \frac{5.00 \mathrm{~L} \times 1423 \mathrm{~K}}{2.01 \times 10^{2} \mathrm{~L}} = 354.6 K\)

Convert temperature to Celsius

Finally, convert the new temperature in Kelvin back to Celsius: \(T_2 = 354.6 - 273 = 81.6 \mathrm{~C}\) b. Find the missing volume at 0 K

Set up the equation for Charles' Law

Since the volume is asked at \(0 K\), it is impossible because for ideal gas it can't have volume at Absolute zero (0 K). Therefore, this situation is not possible under ideal gas conditions. c. Find the missing volume at 0°C

Convert temperature to Kelvin

First, convert the given temperatures to Kelvin: \(T_1 = 298 K\) (already in Kelvin), \(T_2 = 0 + 273 = 273 K\)

Apply Charles' Law equation

Plug in the given values into the equation: \(\frac{44.2 \mathrm{~mL}}{298 \mathrm{~K}} = \frac{V_2}{273 \mathrm{~K}}\)

Solve for the missing volume

Cross-multiply and solve for \(V_2\): \(V_2 = \frac{44.2 \mathrm{~mL} \times 273 \mathrm{~K}}{298 \mathrm{~K}} = 40.9 \mathrm{~mL}\) So the missing quantities are: a. 81.6 °C b. Not possible c. 40.9 mL

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental concept in chemistry and physics that relates the pressure, volume, and temperature of an ideal gas. It is expressed by the formula \( PV = nRT \), where \( P \) stands for pressure, \( V \) for volume, \( n \) for the number of moles of gas, \( R \) for the ideal gas constant, and \( T \) for absolute temperature measured in Kelvin. This equation helps scientists understand how gases behave under different conditions of temperature and pressure.

Though commonly used, the Ideal Gas Law is an approximation, best applied to gases at high temperatures and low pressures where the interactions between gas molecules are minimal. It is important to remember that real gases deviate from this behavior, especially under extreme conditions such as very high pressures or low temperatures. Despite these limitations, the Ideal Gas Law remains a valuable tool for calculating changes in a gas's state when comparing initial and final conditions, as seen in the exercise where Charles' Law component is applied.

Volume and Temperature Relationship

Charles' Law is key to understanding the volume and temperature relationship in gases. It states that at constant pressure, the volume of an ideal gas is directly proportional to its temperature in Kelvin. Mathematically, this relationship is expressed as \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \). This means that if the temperature of a gas increases, so does its volume, provided that the pressure is constant.

For example, in the exercise, we applied Charles' Law multiple times to determine unknown volumes or temperatures. By converting temperatures to Kelvin, and maintaining a constant pressure, we could cross-multiply and solve for the unknowns. This relationship makes it easier to predict how a gas will respond to temperature changes in practical situations, like inflating a balloon or the behavior of automotive tires under different weather conditions.

Absolute Zero

Absolute zero is a theoretical temperature where a gas would have zero volume and no internal energy. It is defined as 0 Kelvin, equivalent to -273.15°C. At this point, the motion of atoms theoretically stops entirely. Understanding absolute zero is crucial because it represents a fundamental limit reached by cooling gases within the framework of classical thermodynamics.

In the context of the given exercise, determining a volume at 0 K highlights the practical limitations of applying Charles' Law to real gases. As volume cannot truly reach zero, the attempt to calculate gas volume at absolute zero demonstrates the boundary where classical equations are rendered invalid. This thought experiment reminds us that, although the laws of gases provide valuable guidance for understanding their properties, they are not without their constraints when pushed to theoretical extremes.