Question

The density of an unknown gas is \(1.107 \mathrm{g} / \mathrm{L}\) at \(300 \mathrm{K}\) and \(740 \mathrm{mmHg}\) a. What is the molar mass of the gas? b. Could this gas be \(\mathrm{CO}\) or \(\mathrm{CO}_{2} ?\)

Short answer

Answer: The molar mass of the unknown gas is approximately 27.98 g/mol. The gas could be CO, but not CO2.

Step-by-step solution

Write down the given information

Density = \(1.107 g/L\) Temperature = \(300 K\) Pressure = \(740 mmHg\) Now, let's convert the pressure from \(mmHg\) to \(atm\). We know that \(1 atm = 760 mmHg\).

Convert pressure to atm

$$ \text{Pressure (P)} = \frac{740 \text{ mmHg}}{760 \text{ mmHg/atm}} = 0.9737 \text{ atm} $$

Use the Ideal Gas Law to find the molar mass

First, rewrite the Ideal Gas Law as follows. $$ \begin{aligned} P V&=n R T \\ \frac{P V}{T} &=n R \\ \frac{n R T}{V} &= P \end{aligned} $$ We know that the density of a substance is defined as its mass per unit volume. Therefore, we can rewrite the equation using density (\(ρ=m/V\)). $$ \rho=\frac{n M}{V} $$ Where \(ρ\) is the density, \(M\) is the molar mass, and \(n\) is the amount of substance in moles. Substituting the density equation into the Ideal Gas Law, we get: $$ \begin{aligned} \frac{n M}{V} &=\frac{P}{R T} \\ M &=\frac{P V}{n R T} \\ \end{aligned} $$ We can now plug in the given values to find the molar mass (\(M\)).

Plug in the given values

$$ M = \frac{ (\text{density})(R)(T)}{P} = \frac{ (1.107 \frac{\text{g}}{\text{L}})(0.0821\frac{\text{L} \cdot \text{atm}}{\text{mol} \cdot \text{K}})(300 \text{K})}{0.9737 \text{atm}} $$

Calculate M

$$ M = (\frac{1.107 \cdot 0.0821 \cdot 300}{0.9737}) \frac{\text{g}}{\text{mol}} = 27.98 \frac{\text{g}}{\text{mol}} $$ So, the molar mass of the unknown gas is around \(27.98 \frac{\text{g}}{\text{mol}}\).

Compare unknown gas molar mass to CO and CO2 molar masses

The molar mass of CO is: $$ M(\mathrm{CO}) = M(\mathrm{C}) + M(\mathrm{O}) = 12.01 + 16.00 = 28.01 \frac{\text{g}}{\text{mol}} $$ The molar mass of CO2 is: $$ M(\mathrm{CO}_2) = M(\mathrm{C}) + 2M(\mathrm{O}) = 12.01 + 2(16.00) = 44.01 \frac{\text{g}}{\text{mol}} $$ Since the molar mass of the unknown gas is very close to the molar mass of \(\mathrm{CO}\) (28.01 \(\text{g/mol}\)), it could be \(\mathrm{CO}\) but not \(\mathrm{CO}_2\). So, to answer the question: a. The molar mass of the unknown gas is approximately 27.98 \(\frac{\text{g}}{\text{mol}}\). b. The gas could be \(\mathrm{CO}\), but not \(\mathrm{CO}_2\).

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in the study of gases. It combines several individual gas laws, encapsulating the relationships between pressure (P), volume (V), temperature (T), and amount of substance (n) in moles. The law is usually expressed as \[ PV = nRT \], where \( R \) stands for the ideal gas constant. This equation helps in predicting the behavior of an ideal gas under different conditions.
To use the Ideal Gas Law effectively, one must ensure temperature is measured in Kelvin and pressure in atmospheres. This way, the calculations remain consistent with the units of the gas constant \( R = 0.0821 \frac{L \, atm}{mol \, K} \).
In the context of our problem, we've used the Ideal Gas Law to derive a formula that helps find the molar mass of the gas when we know its density and other properties. This manipulation is key to using the law in practical situations beyond theoretical calculations.

Molar Mass

Molar mass is a term used to describe the mass of one mole of a given substance and is usually expressed in \( g/mol \). This concept is crucial in chemistry because it helps link the macroscopic world of grams and liters to the microscopic world of atoms and molecules.
To find molar mass using the Ideal Gas Law, we rearrange the formula and incorporate the given density. This is shown by \[ M = \frac{\text{Density} \times R \times T}{P} \]. By substituting the known values—density, temperature, gas constant, and pressure—we can calculate the molar mass of a substance.
In our exercise, the unknown gas's molar mass was calculated to be approximately 27.98 \( g/mol \). This value is very close to the molar mass of carbon monoxide (CO), which is 28.01 \( g/mol \), indicating the unknown gas is likely CO.

Density Calculation

Density is a measure of how much mass a substance has within a certain volume, commonly expressed as \( g/L \) or \( g/cm^3 \). It is a significant property since it influences how substances interact in various chemical reactions and conditions.
From the Ideal Gas Law, we derived a relationship that connects the density of a gas with its molar mass: \[ \rho = \frac{P \times M}{R \times T} \]. This allows us to establish a formula where density can help us calculate the molar mass by rearranging the variables involved.
In our given example, we start by knowing the density and use it to determine what the gas' molar mass might be. This solved equation is pivotal in identifying unknown gases in experimental settings, as done in the problem where we concluded the gas was likely carbon monoxide due to its calculated properties.