Question

The density of a gas of unknown molar mass was measured as a function of pressure at \(0^{\circ} \mathrm{C},\) as in the table that follows. (a) Determine a precise molar mass for the gas. [Hint: Graph \(d / P\) versus \(P . ](\mathbf{b})\) Why is \(d / P\) not a constant as a function of pressure? $$\begin{array}{lllll}{\text { Pressure (atm) }} & {1.00} & {0.666} & {0.500} & {0.333} & {0.250} \\ \hline \text { Density (g/L) } & {2.3074} & {1.5263} & {1.1401} & {0.7571} & {0.5660}\end{array}$$

Short answer

The molar mass M of the gas can be determined by plotting \(d/P\) against \(P\) and estimating the slope of the graph. The molar mass can be calculated using the equation \(M = \frac{R \cdot T \cdot \text{slope}}{1}\), with the ideal gas constant \(R = 0.0821 \frac{\text{L} \cdot \text{atm}}{\text{mol} \cdot \text{K}}\) and the temperature \(T = 273\, \text{K}\). The \(d/P\) ratio is not constant as a function of pressure due to the presence of intermolecular forces that could influence the density under different pressure conditions, causing deviations from the Ideal Gas Law's assumptions.

Step-by-step solution

Write down the Ideal Gas Equation

Write the Ideal Gas Law as follows: \(PV = nRT\)

Express the amount of moles in terms of density, mass, and molar mass

We can write the amount of moles \(n\) as the ratio of mass to molar mass, \(\frac{m}{M}\). From the given data, density can be related to mass and volume as follows: \(d = \frac{m}{V} \Rightarrow m = dV\), so we have: \(n = \frac{dV}{M}\)

Substitute the expression for the amount of moles into the Ideal Gas Law

Using the expression for \(n\) from Step 2, substitute into the Ideal Gas Law: \(PV = \frac{dVRT}{M}\)

Isolate the molar mass M

Isolate molar mass M by reordering the equation in the previous step: \(M = \frac{dVRT}{PV}\)

Calculate \(d/P\) for each data point

Calculate the ratio of density to pressure for each data point in the given table.

Graph \(d/P\) against \(P\)

Plot the calculated \(d/P\) values from the previous step on the y-axis against their corresponding pressure values on the x-axis.

Determine the slope of the graph

Estimate the value of the slope of the graph obtained from Step 6.

Calculate the molar mass of the gas

Use the slope value from Step 7 and the temperature given as \(0^{\circ} \mathrm{C}\) (convert to Kelvin by adding 273, i.e., 273 K) to find the molar mass of the gas using the equation: \(M = \frac{dVRT}{PV}\). Since the slope represents the \(d/P\) ratio, we can write this as: \(M = \frac{R \cdot T \cdot \text{slope}}{1}\), and inserting \(R = 0.0821 \frac{\text{L} \cdot \text{atm}}{\text{mol} \cdot \text{K}}\) (the ideal gas constant).

Explain why \(d/P\) is not constant as a function of pressure

The reason behind \(d/P\) not being constant as a function of pressure is related to the Ideal Gas Law's assumptions. Since the given data does not strictly adhere to the Ideal Gas Law (due to the unknown compound not being a perfect ideal gas), there might be intermolecular forces present that could influence the density under different pressure conditions. Therefore, the relationship between density and pressure is not perfectly linear, and thus \(d/P\) is not constant as a function of pressure.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics that provides a clear relationship between the pressure (P), volume (V), temperature (T), and the amount of moles (n) of a gas. Expressed as the equation
\(PV = nRT\), it describes the behavior of an 'ideal' gas, or a hypothetical gas that perfectly follows the described relationship without any deviations due to molecular interactions or individual gas properties.

Understanding the Ideal Gas Law is crucial when dealing with problems in which you need to calculate or estimate properties of a gaseous substance under various conditions. For example, in determining the molar mass of an unknown gas, the law allows us to connect the measurable quantities of pressure, volume, and temperature to the number of moles and, subsequently, to the molar mass through rearrangement of the equation. This paves the way for experimental calculations such as those presented in the given exercise, where density and pressure readings are instrumental.

Density of Gas

The density of a gas is defined as its mass per unit volume, commonly expressed in grams per liter (g/L). It's a particularly important property in the field of gas analysis and can be determined experimentally.

In the context of the Ideal Gas Law, the molar mass of an unknown gas can be closely related to its density. As shown in the step-by-step solution, density (\(d\))) plays an integral role when calculating the amount of the gas present, leading to the determination of its molar mass.

By manipulating the equation \(d = \frac{m}{V}\), where \(m\)) is mass and \(V\)) is volume, and integrating it into the Ideal Gas Law, we can isolate and determine the molar mass, \(M\)), through experimental data. This understanding of density is essential for students and provides a practical approach to connecting theoretical concepts to real-world applications, reinforcing the relevancy of chemistry in daily life.

Relationship Between Pressure and Gas Density

Pressure and density are inherently linked when it comes to the behavior of gases. Pressure is a measure of the force exerted by the gas molecules against the walls of its container, while density gives us an idea of how much mass of the gas is contained within a certain volume.

The relationship between pressure and gas density can be complex, as it is not always linear, particularly for real gases. Ideal gases, in theory, have a direct proportionality between pressure and density, holding temperature constant. However, for real gases, the interaction between molecules and the actual volume the gas molecules occupy results in deviations from the ideal behavior, as illustrated in the exercise's hint that the ratio \(d/P\)) is not constant with varying pressure.

This non-constancy can also be seen when working through the ideal gas equation rearranged to relate molar mass with the ratio of density to pressure, leading to insightful discussions on gas behavior and the impact of different factors such as temperature, intermolecular forces, and the real volume of gas particles on the properties of a gas. This exploration aids students in developing a deeper understanding of the physical chemistry governing gas properties.