Question

Diethyl ether, \(\left(\mathrm{C}_{2} \mathrm{H}_{5}\right)_{2} \mathrm{O},\) vaporizes easily at room temperature. If the vapor exerts a pressure of \(233 \mathrm{mm}\) Hg in a flask at \(25^{\circ} \mathrm{C},\) what is the density of the vapor?

Short answer

The vapor density of diethyl ether at 25°C and 233 mm Hg is 2.573 g/L.

Step-by-step solution

Identify Given Information

We are provided with several pieces of information for this problem: the chemical formula of diethyl ether, \((\mathrm{C}_{2} \mathrm{H}_{5})_{2} \mathrm{O}\), the vapor pressure at 25°C, which is 233 mm Hg, and the temperature is 25°C. We are tasked with finding the vapor density.

Convert Pressure Units

We convert the given vapor pressure from mm Hg to atmospheres because it is a standard unit in chemical calculations. 1 atm = 760 mm Hg, so we have:\[ P = \frac{233 \, \text{mm Hg}}{760 \, \text{mm Hg/atm}} = 0.3066 \, \text{atm} \]

Convert Temperature to Kelvin

We convert the given temperature from Celsius to Kelvin because ideal gas law calculations require temperature in Kelvin. Using the formula \[ T(K) = T(°C) + 273.15 \]we have:\[ T = 25 + 273.15 = 298.15 \, K \]

Calculate Molar Mass

To find the molar mass of \((\mathrm{C}_{2} \mathrm{H}_{5})_{2} \mathrm{O}\), we sum the atomic masses of all the atoms in one molecule:- Carbon: 12.01 g/mol- Hydrogen: 1.01 g/mol- Oxygen: 16.00 g/mol\[\text{Molar Mass} = 2(2\times12.01) + 2(5\times1.01) + 16.00 = 74.12 \, \text{g/mol} \]

Use Ideal Gas Law to Find Density

The ideal gas law is \( PV = nRT \), where \( P \) = pressure in atm, \( V \) = volume in liters, \( n \) = moles of gas, \( R \) = ideal gas constant (0.0821 L·atm/mol·K), \( T \) = temperature in Kelvin.Density (\( \rho \)) is mass per volume (\( \frac{m}{V} \)), so first find \( n \) in terms of density:\[ n = \frac{m}{M} \Rightarrow m = nM \Rightarrow m = \rho V \Rightarrow n = \frac{\rho V}{M} \]Substitute \( n \) in the Ideal Gas Law:\[ P = \frac{\rho V}{M} RT \Rightarrow \rho = \frac{PM}{RT} \]Now plug in values:\[\rho = \frac{0.3066 \, \text{atm} \times 74.12 \, \text{g/mol}}{0.0821 \, \text{L·atm/mol·K} \times 298.15 \, K} = 2.573 \, \text{g/L}\]

State Final Density

The density of the diethyl ether vapor at 25°C and 233 mm Hg is approximately 2.573 g/L.

Key concepts

Vapor Pressure

Vapor pressure is a measure of the tendency of a liquid to vaporize. For a given substance, it is the pressure exerted by its vapor when the liquid and vapor phases are in equilibrium. This means that when a liquid is at a certain temperature, some of its molecules will have enough energy to escape into the gas phase and exert a pressure. At room temperature, some liquids, like diethyl ether, can have quite a high vapor pressure, which is why they can evaporate easily.

Vapor pressure is influenced by temperature. As the temperature increases, the kinetic energy of the molecules in the liquid increases, leading to more molecules escaping into the vapor phase and, as a result, a higher vapor pressure. This is why substances with high vapor pressure might be used as solvents or for cooling purposes in various applications. Knowing the vapor pressure of a substance can help us predict how it will behave in various environments.

Molar Mass

Molar mass is a fundamental concept in chemistry that refers to the mass of one mole of a given substance. For molecules, it is calculated by adding up the atomic masses of all the atoms in the molecule. This value is essential when using the ideal gas law to calculate properties of gases, such as density.For instance, to determine the molar mass of diethyl ether, \((C_2H_5)_2O\), we need to identify the atomic masses of constituent elements:

• Carbon (C) has an atomic mass of 12.01 g/mol
• Hydrogen (H) is 1.01 g/mol
• Oxygen (O) weighs 16.00 g/mol

To find its molar mass, you multiply the atomic mass of each element by the number of atoms in the molecule and sum the total: \[ \text{Molar Mass} = 2(2 \times 12.01) + 2(5 \times 1.01) + 16.00 = 74.12 \text{ g/mol}\] With the molar mass known, we can proceed to use the ideal gas law to find other properties of the gas.

Temperature Conversion

In scientific calculations, especially those involving gases, it's crucial to use temperature in Kelvin instead of Celsius. The Kelvin scale is an absolute temperature scale, meaning it starts at absolute zero, the theoretically coldest possible temperature where particles have minimal vibrational motion. Converting Celsius to Kelvin is straightforward: \[ T(K) = T(°C) + 273.15 \] For instance, if the temperature is given as 25°C, it can be converted to Kelvin by adding 273.15, which results in 298.15 K. This conversion is necessary because many scientific equations, including the ideal gas law, assume temperature values in Kelvin to maintain proper proportion and ratio.

Density Calculation

Density is defined as the mass per unit volume of a substance. In the case of gases, we often compute density using the ideal gas law, which relates pressure, volume, and temperature with moles of gas, using a constant for ideal gases.To find density using the ideal gas law (\( PV = nRT \)), we first express the number of moles (\( n \)) in terms of density: - since \( n = \frac{m}{M} \) (mass over molar mass) - and \( m = \rho V \) (density times volume)We get: \( n = \frac{\rho V}{M} \) Substituting back into \( PV = nRT \), we get: \( P = \frac{\rho V}{M} RT \) Solving for density (\( \rho \)) gives: \( \rho = \frac{PM}{RT} \)Given this formula, you can calculate the density of any gas if you know its pressure, molar mass, and temperature (in Kelvin). This method provides a straightforward way to estimate how dense a gas will be under specific conditions.