Problem 33 - Chemistry The Molecular Science

Question

Calculate the molar mass of a gas that has a density of \(5.75 \mathrm{~g} / \mathrm{L}\) at STP.

Step-by-Step Solution

Answer

The molar mass of the gas is 128.8 g/mol.

Step 1: Understand STP Conditions

First, we need to recall that Standard Temperature and Pressure (STP) conditions are defined as a temperature of 273.15 K (0 °C) and a pressure of 1 atm. At STP, 1 mole of an ideal gas occupies 22.4 L.

Step 2: Identify the Given Data

We are given the density of the gas as \(5.75 \, \text{g/L}\). We need to find the molar mass (\( ext{M} \)) of the gas.

Step 3: Use the Ideal Gas Law

Remember that under STP, the volume of 1 mole of gas is 22.4 L. Density is mass per unit volume. Thus, the mass for 22.4 L of this gas is \(5.75 \, \text{g/L} \times 22.4 \, \text{L}\).

Step 4: Calculate the Mass of 1 Mole of Gas

Compute the mass of 22.4 L of gas by multiplying density by volume: \(5.75 \, \text{g/L} \times 22.4 \, \text{L} = 128.8 \, \text{g}\).

Step 5: State the Molar Mass

The molar mass is the mass of 1 mole of the gas, which we calculated to be 128.8 g/mol.

Key Concepts

Density

Density is an important concept in chemistry that can help us understand the amount of mass present in a given volume. It is usually denoted by the Greek letter \( \rho \) and the formula for density is expressed as \( \rho = \frac{m}{V} \), where:

• \( m \) is the mass of the substance (measured in grams, g)
• \( V \) is the volume occupied by the substance (measured in liters, L)

In the original exercise, the density of a gas was provided as \( 5.75 \text{ g/L} \). This means that every liter of this gas weighs 5.75 grams. Understanding the density allows us to calculate the total mass of the gas if we know its total volume. By multiplying the density by the volume, we can find the mass of the gas in that specified volume. This mass can then be used in combination with other laws, like the Ideal Gas Law, to find further properties of the gas, such as its molar mass.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry used to describe the behavior of ideal gases. The equation is written as \( PV = nRT \), where:

• \( P \) is the pressure of the gas (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, which is approximately \( 0.0821 \text{ L atm/mol K} \)
• \( T \) is the temperature of the gas (in Kelvin, K)

This equation helps us predict how changes in one of these variables affect the others, assuming the gas behaves ideally. In the exercise, we analyzed a gas at Standard Temperature and Pressure (STP) conditions, where 1 mole of an ideal gas occupies 22.4 liters. By knowing the volume per mole at STP, and the density, the molar mass could be directly calculated since the number of moles and the structure of the gas can be assumed from these conditions.

Standard Temperature and Pressure (STP)

Standard Temperature and Pressure, commonly abbreviated as STP, are conditions set to standardize data and allow scientists to compare results easily. Under STP, the temperature is set to 273.15 Kelvin, which corresponds to 0°C, and the pressure is set to 1 atmosphere (atm). In these conditions, 1 mole of an ideal gas will occupy a volume of 22.4 liters. This is a crucial reference point for chemists.
Knowing the behavior of gases at STP allows us to reliably calculate their properties, such as molar mass if the density of the gas is known. In the original problem, using the values at STP helped simplify the calculation of the molar mass—by multiplying the density by the volume of 1 mole at STP (22.4 L), the mass of one mole of the gas was obtained, which directly provided the molar mass.