Question

A sample of an unknown liquid weighing \(0.495 \mathrm{~g}\) is collected as vapor in a \(127-\mathrm{mL}\) flask at \(98^{\circ} \mathrm{C}\). The pressure of the vapor in the flask is then measured and found to be 691 torr. What is the molar mass of the liquid?

Short answer

The molar mass of the unknown liquid is approximately \( 123.75 \mathrm{~g/mol} \).

Step-by-step solution

Convert units

Firstly, convert the given values to the appropriate units for using in the ideal gas law. The pressure needs to be converted from torr to atmospheres using the conversion factor \( 1 \mathrm{~atm} = 760 \mathrm{~torr} \), therefore the pressure in atmosphere is \( \frac{691 \mathrm{~torr}}{760} = 0.91 \mathrm{~atm} \). The volume has to be in liters, so \( 127 \mathrm{~mL} = 0.127 \mathrm{~L} \). The temperature needs to be in Kelvin for the ideal gas law, so convert \( 98^{\circ} \mathrm{C} \) into Kelvin by adding 273 to it which gives \( 371 \mathrm{~K} \).

Apply the ideal gas law to find the number of moles

Arrange the ideal gas law equation to solve for \( n \), the number of moles (\( n = \frac{PV}{RT} \)). Using the given values with \( R = 0.0821 \mathrm{~L \cdot atm/(K \cdot mol)} \), the moles of the unknown gas can be calculated: \( n = \frac{(0.91 \mathrm{~atm})(0.127 \mathrm{~L})}{(0.0821 \mathrm{~L \cdot atm/(K \cdot mol)})(371 \mathrm{~K})} = 0.004 \mathrm{~moles} \).

Calculate the molar mass

The molar mass (M) is calculated by dividing the mass of the liquid by the number of moles: \( M = \frac{mass}{moles} = \frac{0.495 \mathrm{~g}}{0.004 \mathrm{~moles}} = 123.75 \mathrm{~g/mol} \).

Key concepts

Ideal Gas Law

The Ideal Gas Law is an essential equation used in chemistry to relate the pressure, volume, temperature, and number of moles of a gas. It is expressed as \( PV = nRT \), where

• \( P \) represents the pressure of the gas,
• \( V \) is the volume,
• \( n \) stands for the number of moles,
• \( R \) is the universal gas constant (0.0821 L·atm/(K·mol)),
• \( T \) denotes temperature in Kelvin.

When solving problems with the Ideal Gas Law, it's crucial to have all measurements in the correct units. The equation allows us to find one variable if the others are known. By rearranging this equation, we can solve for the unknown variable. For example, rearranging to find \( n \), \[ n = \frac{PV}{RT} \] helps to determine the number of moles based on the known pressure, volume, and temperature. This law is particularly useful when dealing with a gaseous state of a substance, even if that substance is typically a liquid or solid at room temperature.

Unit Conversion

Accurate unit conversion is vital when using equations like the Ideal Gas Law, as the units need to match for the equation to work correctly. In this problem, the pressure was initially given in torr and had to be converted to atmospheres. The conversion factor is

• \(1\) atmosphere = \( 760 \) torr.

Therefore, \[ \text{Pressure in atm} = \frac{691 \, \text{torr}}{760} \approx 0.91 \, \text{atm} \]Likewise, volume measurements must be converted from milliliters to liters:

• \(127\) mL = \( 0.127 \) L.

Temperature is also converted from degrees Celsius to Kelvin by adding \(273\):

• \(98^{\circ} \text{C} = 98 + 273 = 371 \text{ K} \).

These conversions ensure that all units in the Ideal Gas Law align with the constant \( R \), enabling accurate calculations. Paying attention to units and consistently converting them is crucial for solving any chemical problem accurately.

Chemical Problem Solving

In chemistry, problem solving often involves combining several concepts to find a solution. In this exercise, we aimed to determine the molar mass of an unknown liquid by capitalizing on its properties as a gas at certain conditions. Here's how the problem is approached:

• First, convert all values to the correct units needed for the Ideal Gas Law calculation. This makes them compatible with the equation.
• Next, apply the Ideal Gas Law to find the number of moles of the vapor using \( n = \frac{PV}{RT} \). This step helps to quantify the substance in terms of moles, which is a foundational measure in chemistry.
• Finally, calculate the molar mass by dividing the mass of the sample by the number of moles, which provides the molar mass: \( M = \frac{mass}{moles} \).

This problem-solving approach emphasizes logical steps: converting units to fit a specific formula, using the formula to find an unknown, and then deducing another property of the substance (in this case, molar mass). Each step requires attention to detail and a systematic approach, combining measurement and calculation. Mastering such approaches aids students in effectively tackling a wide range of chemical problems.