Question

For each case, rearrange the Ideal Gas Law Equation to show that it is consistent with the given law or hypothesis and obtain an expression for the corresponding constant. a) Boyle's Law, \(\mathrm{k}_{\mathrm{B}}\) b) Charles' Law, \(\mathrm{k}_{\mathrm{C}}\) c) Avogadro's Hypothesis, \(\mathrm{k}_{\mathrm{A}}\)

Short answer

The constants for Boyle's Law (\(k_B\)), Charles' Law (\(k_C\)), and Avogadro's Hypothesis (\(k_A\)) derived from the Ideal Gas Law are respectively: \(\mathrm{k}_{\mathrm{B}}=\frac{PV}{nT}\), \(\mathrm{k}_{\mathrm{C}}=\frac{V}{T}\), \(\mathrm{k}_{\mathrm{A}}=\frac{V}{n}\)

Step-by-step solution

Boyle's Law

Boyle's law states that the pressure of a gas is inversely proportional to its volume, given constant temperature and amount of gas. Thus to obtain an equation consistent with Boyle's law, we should rearrange the Ideal Gas Law to have \(\mathrm{k}_{\mathrm{B}} = PV\). From the Ideal Gas Law \(PV=nRT\), we divide both sides by \(nT\) to isolate \(k_B\), giving \(\mathrm{k}_{\mathrm{B}}=\frac{PV}{nT}\)

Charles' Law

Charles' law states that the volume of a gas is directly proportional to its temperature, given constant pressure and amount of gas. Thus, we should rearrange the Ideal Gas Law to have \(\mathrm{k}_{\mathrm{C}} = \frac{V}{T}\). From the Ideal Gas Law \(PV=nRT\), we divide both sides by \(nP\) to isolate \(k_C\), giving \(\mathrm{k}_{\mathrm{C}}=\frac{V}{T}\)

Avogadro's Hypothesis

Avogadro's hypothesis states that equal volumes of all gases, at the same temperature and pressure, contain the same number of molecules. Thus, we should rearrange the Ideal Gas Law to have \(\mathrm{k}_{\mathrm{A}} = \frac{V}{n}\). From the Ideal Gas Law \(PV=nRT\), we divide both sides by \(PR\) in order to isolate \(k_A\), giving \(\mathrm{k}_{\mathrm{A}}=\frac{V}{n}\)

Key concepts

Understanding Boyle's Law

Boyle's Law is a fundamental principle in the study of gases, and it is crucial for students to understand how it correlates with the Ideal Gas Law. Boyle's Law articulates that for a given amount of gas at a constant temperature, the pressure of the gas is inversely proportional to its volume.

When you compress a gas, reducing its volume without changing its temperature, the pressure increases, and vice versa. This relationship is captured in the equation \(P_1V_1 = P_2V_2\), where \(P_1\) and \(V_1\) are the initial pressure and volume, and \(P_2\) and \(V_2\) are the pressure and volume after the change.

To link this to the Ideal Gas Law (\(PV=nRT\)), you can see that if the temperature (\(T\)) and the amount of gas (\(n\)) are held constant, the product of pressure and volume should also be constant (\(k_B = PV\)). The Ideal Gas Law thus backs up Boyle's Law when you rearrange it to isolate this product, making it a powerful tool to predict the behavior of gases under pressure changes.

Exploring Charles' Law

Charles' Law offers another perspective on gas behavior, focusing on the relationship between volume and temperature. This law states that the volume of a given amount of gas is directly proportional to its temperature when pressure and amount of gas remain constant.

To grasp this concept, picture a balloon on a warm day; as the temperature rises, the gas inside the balloon expands, increasing the volume. Conversely, in a colder environment, the volume of the gas decreases. Mathematically, Charles' Law is expressed as \(V_1/T_1 = V_2/T_2\), with \(V_1\) and \(T_1\) being the initial volume and temperature, and \(V_2\) and \(T_2\) being the final volume and temperature after a change.

Using the Ideal Gas Law, this direct relationship is illustrated by the ratio \(V/T\) being constant (\(k_C\)). Notice that when we rearrange the Ideal Gas Law by dividing both sides by the amount of gas (\(n\)) and pressure (\(P\)), we obtain the expression \(k_C = V/T\). Thus, Charles' Law is neatly shown to be a special case of the broader Ideal Gas Law.

Interpreting Avogadro's Hypothesis

Avogadro's Hypothesis is essential for understanding the nature of gases at the molecular level. The hypothesis asserts that equal volumes of different gases, under the same conditions of temperature and pressure, contain an equal number of particles, or in other words, molecules or moles of gas.

This concept introduces the idea that the volume of a gas (\(V\)) is directly proportional to the number of moles of gas (\(n\)) present when the temperature and pressure are held constant. Such an understanding leads us to the simple relationship \(V_1/n_1 = V_2/n_2\) when comparing the same gas under different conditions or even when comparing different gases.

The interpretation of Avogadro's Hypothesis through the Ideal Gas Law is quite confirming. By rearranging the law to isolate the ratio of volume to moles, you obtain the constant \(k_A = V/n\). It's an elegant expression that serves as a cornerstone for gas calculations and theoretical concepts in chemistry, linking the microscopic world of atoms and molecules to the macroscopic properties of gases we can observe and measure.