Question

Calculate the volume in liters for each of the following gases at STP: (a) \(0.250 \mathrm{~g}\) of helium, He (b) \(5.05 \mathrm{~g}\) of nitrogen, \(\mathrm{N}_{2}\)

Short answer

(a) 1.40 liters of helium; (b) 4.03 liters of nitrogen at STP.

Step-by-step solution

Define STP and Molar Volume

At Standard Temperature and Pressure (STP), the conditions are defined as a temperature of 273.15 K (0°C) and a pressure of 1 atm. Under these conditions, 1 mole of an ideal gas occupies 22.4 liters. We'll use this information to solve the problem.

Calculate Moles of Helium

First, determine the number of moles of helium using the formula: \( n = \frac{\text{mass}}{\text{molar mass}} \). The molar mass of helium is approximately 4.00 g/mol. Hence, \( n_{\text{He}} = \frac{0.250 \text{ g}}{4.00 \text{ g/mol}} = 0.0625 \text{ mol} \).

Calculate Volume of Helium

Using the molar volume at STP, calculate the volume: \( V = n \times 22.4 \text{ L/mol} \). For helium, \( V_{\text{He}} = 0.0625 \times 22.4 = 1.40 \text{ liters} \).

Calculate Moles of Nitrogen

Similarly, for nitrogen, calculate the moles. The molar mass of nitrogen gas (\( N_2 \)) is approximately 28.02 g/mol. Thus, \( n_{N_2} = \frac{5.05 \text{ g}}{28.02 \text{ g/mol}} \approx 0.180 \text{ mol} \).

Calculate Volume of Nitrogen

Again, using the molar volume at STP, calculate the volume: \( V = n \times 22.4 \text{ L/mol} \). For nitrogen, \( V_{\text{N}_2} = 0.180 \times 22.4 \approx 4.03 \text{ liters} \).

Key concepts

Molar Volume

Molar volume is a key concept in understanding gases. At Standard Temperature and Pressure (STP), one mole of any ideal gas will occupy 22.4 liters. This is a simple and convenient way of relating the amount of gas to its volume under these specific conditions.

STP conditions are defined as 273.15 Kelvin (0 degrees Celsius) and 1 atmosphere of pressure.
Under these conditions, changes in temperature or pressure won't affect the molar volume provided the gas behaves ideally.

• The idea of molar volume is particularly useful in calculations that involve converting amounts of substances (in moles) to volumes of gases, and vice versa.
• This can act as a "shortcut," allowing us to quickly assess how much space a mole of a gas takes up at STP.

Molar volume makes the idea of the mole more tangible through a visual or spatial frame of reference. This can be especially helpful to visualize reactions or to determine how much space a gas will occupy.

Mole Calculations

Mole calculations allow us to convert between the mass of a gas and its volume or number of molecules. Converting mass to moles involves dividing the given mass by the molar mass of the substance.

This means that knowing the molar mass of the substance you are working with is crucial.

For instance, the molar mass of helium is 4 g/mol, which means if you have 1 mole of helium, it will weigh 4 grams. The molar mass of nitrogen gas (which is made up of two nitrogen atoms) is 28.02 g/mol.

• The basic equation for moles is \( n = \frac{\text{mass}}{\text{molar mass}} \).
• Once you obtain the moles, the next step usually involves using the molar volume for gases at STP to find the volume as \( V = n \times 22.4 \text{ L/mol} \).

For calculations involving chemical reactions, knowing how to convert between mass and moles, and then to volume, is a foundational skill. This aids in balancing equations and conducting stoichiometry to predict the amounts of products and reactants involved.

Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and number of moles for an ideal gas. The equation is expressed as \( PV = nRT \), where:

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

This law allows for calculations under various conditions – not just at STP. Whether you're solving for an unknown pressure or determining how temperature changes impact a gas, the ideal gas law provides a consistent method of calculation.

It supports understanding beyond fixed conditions like STP and extends to real-world applications, broadening the way we conceptualize gases. As a student, regularly applying the ideal gas law helps to solidify your grasp of these concepts and gives you a useful tool for many chemistry and physics problems.