Question

The volume of a sample of gaseous argon maintained at constant pressure was studied as a function of temperature and the following data were obtained: Temperature, \(\underline{\mathrm{T}}(\stackrel{\circ}{\mathrm{K}})_{-}\) 250 300 350 \begin{tabular}{r} Volume, \(\underline{\mathrm{V}}(\underline{\ell})\) \\ \hline \(0.005\) \\ \(0.006\) \\ \(0.007\) \end{tabular} Calculate and confirm the Charles' Law constant for this system. Determine the temperature corresponding to a valume of \(22.4 \ell\).

Short answer

The Charles' Law constant for the gaseous argon system is \(0.00002 L K^{-1}\), and the temperature corresponding to a volume of \(22.4 L\) is \(1120000 K\).

Step-by-step solution

Calculate Charles' Law constant 'k' for each data point

We have three data points for temperature (T) and volume (V). Let's calculate the values of 'k' for each: \(k = \frac{V}{T}\) Data point 1: T = 250 K, V = 0.005 L \(k_1 = \frac{0.005}{250} = 0.00002\) Data point 2: T = 300 K, V = 0.006 L \(k_2 = \frac{0.006}{300} = 0.00002\) Data point 3: T = 350 K, V = 0.007 L \(k_3 = \frac{0.007}{350} = 0.00002\) As the value of 'k' is the same for all data points, we have successfully confirmed Charles' Law constant to be \(0.00002 L K^{-1}\).

Find the temperature corresponding to a volume of 22.4 L

Now we need to determine the temperature (T) corresponding to a volume (V) of \(22.4 L\). Using Charles' Law equation: \(V = kT\) Let's solve for T: \(T = \frac{V}{k}\) \(T = \frac{22.4 L}{0.00002 L K^{-1}} = 1120000 K\) So, the temperature corresponding to a volume of \(22.4 L\) is \(1120000 K\).

Key concepts

Gas Laws

The gas laws are fundamental principles that describe how gases behave under different conditions of temperature, pressure, and volume. These laws help us understand and predict how gases will react in various environments. One of the key aspects of gas laws is that gases will expand or contract when subjected to changes in temperature or pressure, following specific mathematical relationships.
One of the most well-known gas laws is Charles' Law, which states that the volume of a gas is directly proportional to its temperature, provided the pressure remains constant. This relationship is pivotal in explaining the behavior of gases in everyday situations and in laboratory settings. It implies that when you heat a gas, the volume will increase, and when you cool it down, the volume will decrease, assuming constant pressure.
Understanding gas laws enables scientists and engineers to design systems and processes that rely on the predictable behavior of gases. They form the foundation for fields such as meteorology, respiratory physiology, and chemical engineering.

Argon Gas

Argon is a noble gas, part of Group 18 on the periodic table. It is colorless, odorless, and nonreactive, making it an ideal candidate for various industrial applications. In its gaseous state, argon occupies space like any other gas and adheres to the gas laws, including Charles' Law, with predictable behavior under changing temperatures and pressures.
Argon is used in a variety of applications due to its inert properties. It is commonly employed in welding and in the production of light bulbs, where its lack of reactivity is crucial. Because it doesn't easily form compounds, it's an excellent choice for environments where other gases might react undesirably.

Temperature-Volume Relationship

The temperature-volume relationship, as defined by Charles' Law, states that the volume of a gas is directly proportional to its absolute temperature, assuming constant pressure. This principle can be summarized by the formula: \[ V = kT \] where \( V \) is the volume, \( T \) is the temperature in Kelvin, and \( k \) is a constant specific to the amount of gas and conditions of the experiment.
This relationship is crucial in predicting how a gas will behave when the temperature changes. For example, if you increase the temperature of a gas, its volume will increase proportionally. Conversely, lowering the temperature will result in a decrease in volume. This can be observed in many practical situations, such as how a balloon expands when heated or contracts when cooled.

Gaseous State

In the gaseous state, matter takes the form of gas, characterized by high kinetic energy and minimal intermolecular forces. Gases expand to fill their containers, evenly distributing within the available volume regardless of the amount. This property is one of the reasons gases are referred to as fluids.
The gaseous state is marked by the ability to conform to the shape and volume of its container, responding keenly to temperature and pressure changes. Because gas particles are in constant, rapid motion and far apart compared to solids and liquids, gases can be easily compressed. This makes understanding their behavior through the gas laws, such as Charles' Law, critically important in various applications, such as air conditioning systems, aerodynamics, and general thermodynamics.