Question

Labels have become detached from cylinders of two gases, one of which is known to be propane, ${\mathrm{C}}_3{\mathrm{H}}_8,$ and the other butane, ${\mathrm{C}}_4{\mathrm{H}}_{10}$. The densities of the two gases are compared at the same temperature and pressure. The density of gas A is $1.37\mathrm{g}/\mathrm{L}$, and the density of gas $\mathrm{B}$ is higher. Which gas is A and which is B? What is the density of B?

Short answer

Gas A is propane with 1.37 g/L; Gas B is butane with 1.79 g/L density.

Step-by-step solution

Understand the Problem

We are given two gases, propane ${\mathrm{C}}_3{\mathrm{H}}_8$ and butane ${\mathrm{C}}_4{\mathrm{H}}_{10}$, and their densities at the same condition. Gas A has a density of 1.37 g/L, while gas B has a higher density. We need to determine which gas corresponds to A and B and calculate the density of gas B.

Calculate Molar Masses

Determine the molar mass of each gas. ${\mathrm{C}}_3{\mathrm{H}}_8$ has a molar mass of $3\times 12.01+8\times 1.008=44.1\text{g/mol}$. ${\mathrm{C}}_4{\mathrm{H}}_{10}$ has a molar mass of $4\times 12.01+10\times 1.008=58.14\text{g/mol}$.

Relationship Between Density and Molar Mass

The density of a gas ($d$) at constant temperature and pressure is directly proportional to its molar mass ($M$), following the formula $d=\frac{PM}{RT}$. Thus, a higher molar mass results in a higher density.

Match Densities with Gases

Since butane has a larger molar mass than propane, butane should have a higher density than propane. Therefore, gas A (1.37 g/L), which has a lower density, is propane, ${\mathrm{C}}_3{\mathrm{H}}_8$. Gas B, which has a higher density, is butane, ${\mathrm{C}}_4{\mathrm{H}}_{10}$.

Calculate Density of Gas B

The density ratio between butane and propane can be estimated using the ratio of their molar masses because $\frac{d_B}{d_A}=\frac{M_{\text{butane}}}{M_{\text{propane}}}$. Substituting the values, we get $\frac{d_B}{1.37}=\frac{58.14}{44.1}$. Solving for $d_B$, we find $d_B\approx 1.79\text{g/L}$. Thus, the density of gas B (butane) is approximately 1.79 g/L.

Key concepts

Propane Density

Propane is a commonly used hydrocarbon gas with the chemical formula ${\text{C}}_3{\text{H}}_8$. When considering the density of propane, it's essential to understand the environmental conditions such as temperature and pressure because density can vary with these factors.
In the problem above, gas A is identified as propane due to its given density of 1.37 g/L, which is lower than that of butane. This aligns with the expectation since propane, having a lower molar mass than butane, should naturally exhibit a lower density when both gases are compared under the same conditions.

• Propane density: 1.37 g/L
• Temperature and Pressure: Constant across comparison

Butane Density

The density of butane, ${\text{C}}_4{\text{H}}_{10}$, is higher than that of propane under similar conditions due to its greater molar mass. Butane, as identified in the problem, corresponds to gas B which possesses a higher density than gas A (propane).
Using the results from the exercise, we find the density of butane to be approximately 1.79 g/L. This illustrates the concept that butane is indeed denser than propane at a given temperature and pressure.

• Butane density: Higher than propane
• Calculated as: ~1.79 g/L

Molar Mass Calculation

Molar mass is crucial when comparing the density of gases. It represents the mass of one mole of a substance and is calculated based on the atomic masses of the elements comprising the compound. In our context:

- Propane (${\text{C}}_3{\text{H}}_8$) has a molar mass calculated as follows: $$\text{Molar Mass of Propane}=3\times 12.01+8\times 1.008=44.1\text{g/mol}$$ - Butane (${\text{C}}_4{\text{H}}_{10}$) molar mass calculation is: $$\text{Molar Mass of Butane}=4\times 12.01+10\times 1.008=58.14\text{g/mol}$$
These calculations are critical as they directly influence the relative densities of the gases when compared at equal conditions.

Ideal Gas Law

The ideal gas law formula $PV=nRT$ is a cornerstone for understanding the behavior of gases. In density discussions, it is often rearranged to highlight the relationship between density and molar mass:

$$d=\frac{PM}{RT}$$
In this rearranged form, $d$ is the density, $P$ is the pressure, $M$ is the molar mass, $R$ is the gas constant, and $T$ is the temperature.
This equation shows how density is proportional to molar mass when pressure and temperature are constant. Therefore, gases with greater molar masses—such as butane compared to propane—will naturally exhibit greater densities under similar conditions.

Density and Molar Mass Relationship

Understanding the direct relationship between density and molar mass is essential. In the context of gases, this relationship can be expressed through the equation derived from the ideal gas law:
$$d=\frac{PM}{RT}$$
This equation highlights that as the molar mass $M$ increases, the gas's density $d$ also increases when temperature and pressure are constant. This explains why in our exercise, butane has a higher density compared to propane due to its larger molar mass.
Predictably, this means:

• Higher molar mass results in higher density
• Propane, with a lower molar mass, is less dense than butane

Understanding this relationship helps us accurately predict and explain differences in gases' densities under given conditions.