91Ó°ÊÓ

Plot the following data for the densities of liquid and gaseous ethane in equilibrium with each other as a function of temperature, and determine the critical temperature of ethane. $$\begin{array}{cccccc} T / \mathrm{K} & \rho^{1} / \mathrm{mol} \cdot \mathrm{dm}^{-3} & \rho^{8} / \mathrm{mol} \cdot \mathrm{dm}^{-3} & T / \mathrm{K} & \rho^{1} / \mathrm{mol} \cdot \mathrm{dm}^{-3} & \rho^{8} / \mathrm{mol} \cdot \mathrm{dm}^{-3} \\ \hline 100.00 & 21.341 & 1.336 \times 10^{-3} & 283.15 & 12.458 & 2.067 \\ 140.00 & 19.857 & 0.03303 & 293.15 & 11.297 & 2.880 \\ 180.00 & 18.279 & 0.05413 & 298.15 & 10.499 & 3.502 \\ 220.00 & 16.499 & 0.2999 & 302.15 & 9.544 & 4.307 \\ 240.00 & 15.464 & 0.5799 & 304.15 & 8.737 & 5.030 \\ 260.00 & 14.261 & 1.051 & 304.65 & 8.387 & 5.328 \\ 270.00 & 13.549 & 1.401 & 305.15 & 7.830 & 5.866 \end{array}$$

Step by step solution

01

Understand the Problem

The task involves plotting given data for densities of liquid (\( \rho^1 \)) and gaseous ethane (\( \rho^8 \)) at specified temperatures \( T \). The goal is to visualize these densities against temperature and determine the critical temperature where the properties of both phases converge.

02

Organize Data

Separate the data into two sets: one for the density of liquid ethane and another for gaseous ethane at given temperatures. This will facilitate the plotting process.

03

Plot the Data

Using a graphing tool or software (such as Python with Matplotlib or Excel), plot two curves on the same graph: Temperature (in Kelvin) on the x-axis against Density (in mol/dm³) on the y-axis. One curve should correspond to the liquid densities, and the other to the gas densities.

04

Analyze the Graph

Inspect the plot for the point where the liquid and gas density curves approach or meet. The temperature at this point represents the critical temperature of ethane, where the properties of the liquid and vapor phase are indistinguishable.

05

Determine Critical Temperature

From the plot, the curves come closest to meeting or are indistinguishable around a temperature of 305.15 K. This temperature is identified as the critical temperature of ethane.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Density Plotting

When handling data about substances like ethane, understanding how to plot the densities of liquid and gaseous phases is crucial. Density plotting means representing the density of a substance, such as ethane, against another variable, typically temperature, on a graph. This type of plot can reveal important phase change information.

• **Prepare Your Data**: Ensure that your data is organized. In this case, separate the data into columns for temperature, and densities for liquid (\(\rho^1\)) and gas (\(\rho^8\)).
• **Choose Your Tools**: You can use graphing software like Excel or Matplotlib in Python.
• **Plot the Curves**: On the x-axis, set temperature in Kelvin, and on the y-axis, set the density in mol/dm³.

This visual representation helps identify trends and intersections between phases, essential for understanding ethane's behavior under varying conditions.

Phase Equilibrium

Phase equilibrium occurs when a substance like ethane exists in both liquid and gaseous forms but shows no net change in the quantity of each phase. At equilibrium, important physical properties become balanced across phases.

• **Balance of Forces**: The forces in each phase balance each other out, meaning conditions like temperature and pressure won't cause more liquid to form from gas, or vice versa.
• **Critical Temperature**: This is the temperature at which the gas and liquid phases of a substance become indistinguishable. For ethane, this point is identified when the density of both phases equalizes, seen at around 305.15 K.

Recognizing these equilibrium points is key to predicting how a substance behaves under changing environmental conditions.

Ethane Properties

Ethane, a simple hydrocarbon with the formula \(C_2H_6\), has distinct properties that vary between its liquid and gaseous states. Understanding these properties helps guide decisions in industrial applications and scientific experiments.

• **Molar Mass**: Ethane has a molar mass of about 30.07 g/mol, influencing its density.
• **Phase Transitions**: At the critical temperature of 305.15 K, unique properties emerge, such as the density of the gas and liquid phases becoming similar.
• **Applications**: Due to its phase-shifting properties and simple molecular structure, ethane is widely used in the chemical industry for synthesis processes and as a fuel component.

Knowledge of ethane's properties is indispensable for chemists and engineers dealing with these materials in both practical and theoretical contexts.