Question

At STP, \(0.280 \mathrm{~L}\) of a gas weighs \(0.400 \mathrm{~g}\). Calculate the molar mass of the gas.

Short answer

The molar mass of the gas is 32 g/mol.

Step-by-step solution

Identify Given Information

We are given that at STP (Standard Temperature and Pressure - 1 atm and 273.15 K), a gas has a volume of 0.280 L and a mass of 0.400 g. We need to calculate the molar mass of the gas.

Use Ideal Gas Law

At STP, the volume occupied by 1 mole of any ideal gas is 22.414 L. We'll use this reference to find the number of moles in 0.280 L of the gas.

Determine Moles in Given Volume

Calculate the number of moles in 0.280 L using the ratio:\[\text{moles} = \frac{0.280 \text{ L}}{22.414 \text{ L/mol}}\]This gives the number of moles of the gas.

Calculate Moles

Compute the number of moles using the formula:\[\text{moles} = \frac{0.280}{22.414} \approx 0.0125 \text{ mol}\]This is the number of moles present in 0.280 L of the gas.

Calculate Molar Mass

The molar mass is found using the formula:\[\text{Molar Mass} = \frac{\text{mass of gas}}{\text{moles of gas}}\]Substitute in the known values:\[\text{Molar Mass} = \frac{0.400 \text{ g}}{0.0125 \text{ mol}} = 32 \text{ g/mol}\]

Finalize the Solution

The molar mass of the gas is 32 g/mol. This value indicates the mass of one mole of the gas under standard conditions.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a foundational principle in chemistry that describes the behavior of gases under various conditions. It can be expressed with the equation \( PV = nRT \), where:

• \( P \) is the pressure of the gas, measured in atmospheres (atm).
• \( V \) is the volume of the gas, in liters (L).
• \( n \) is the number of moles of the gas.
• \( R \) is the ideal gas constant \( (0.0821 \, \text{L} \, \text{atm} \, \text{mol}^{-1} \, \text{K}^{-1}) \).
• \( T \) is the temperature in Kelvin (K).

The law provides a relationship between these variables, allowing you to calculate any one of them if the others are known.
It is particularly useful for solving problems involving changes in gas conditions. In this exercise, we're using a simplified form where we're assuming conditions of Standard Temperature and Pressure (STP) to calculate the molar mass.

STP (Standard Temperature and Pressure)

Standard Temperature and Pressure (STP) is a set of predefined conditions that are commonly used in gas calculations to simplify the process. At STP, the standard temperature is 273.15 K, equivalent to 0°C, and the standard pressure is 1 atmosphere (atm).
Under these conditions, 1 mole of an ideal gas occupies a volume of 22.414 L. This is a critical value in calculations because it allows us to easily determine the number of moles in any given volume of gas at STP by using a simple proportion. STP is often employed in gas calculations to establish a common baseline, ensuring that results are uniform and comparable.

Gas Volume

The volume of a gas can change dramatically depending on the conditions of temperature and pressure. At STP, as previously mentioned, a gas behaves predictably, occupying a known volume per mole. In the problem presented, we have a gas with a volume of 0.280 L.
We calculated the number of moles by comparing this volume to the standard molar volume of 22.414 L at STP. This involves a straightforward ratio, allowing us to find how many moles are present in the 0.280 L volume by dividing the given volume by the molar volume:\[\text{moles} = \frac{\text{volume of gas}}{\text{molar volume at STP}}\]This formula gives us a clear pathway to determine the mole quantity when provided with a volume.

Mole Concept

The mole concept is crucial in chemistry as it connects the macroscopic world, which we can measure, to the microscopic world of atoms and molecules. A mole is a specific quantity of substance, defined as containing exactly \(6.022 \times 10^{23}\) entities (usually atoms or molecules). This number is known as Avogadro's Number.Using moles, we can seamlessly convert between mass, volume, and the number of molecules or atoms present.
In the exercise, the number of moles we calculated—from the volume of the gas at STP—was used to determine the molar mass.
Molar mass is the mass of one mole of a substance and is typically expressed in grams per mole (g/mol). By dividing the mass of the gas by the number of moles, we obtained its molar mass, which is a key piece of information about the identity and nature of the substance.