Question

If \(10.0 \mathrm{~L}\) of neon gas exerts a pressure of 125 psi at \(373 \mathrm{~K}\), what is the number of moles of gas?

Short answer

The number of moles of neon gas is approximately 2.78 moles.

Step-by-step solution

Identify the Ideal Gas Law

To find the number of moles of neon gas, we will use the Ideal Gas Law, which is given by the equation \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.

Arrange the Ideal Gas Law to Solve for Moles

Re-arrange the equation \( PV = nRT \) to solve for \( n \), the number of moles: \( n = \frac{PV}{RT} \).

Convert Units

Ensure that all units are compatible with the gas constant \( R \). Pressure should be in atm and volume in liters. Given \( P = 125 \text{ psi} \), convert it to atm: \( P = \frac{125}{14.696} \approx 8.512 \text{ atm} \). Volume is already in liters: \( V = 10.0 \text{ L} \). Temperature is given as \( T = 373 \text{ K} \).

Use the Ideal Gas Constant

Choose \( R \), the ideal gas constant, with appropriate units. For \( atm \), \( L \), and \( K \), \( R = 0.0821 \text{ L atm K}^{-1} ext{mol}^{-1} \).

Calculate the Number of Moles

Substitute the values into the equation \( n = \frac{PV}{RT} \):\[n = \frac{8.512 \times 10.0}{0.0821 \times 373} \approx \frac{85.12}{30.6233} \approx 2.78 \]Thus, the number of moles of neon gas is approximately 2.78 moles.

Key concepts

Mole Calculations

In solving problems involving gases using the Ideal Gas Law, one needs to understand how to calculate the number of moles. The mole is a fundamental concept in chemistry used to express amounts of a chemical substance. It is essential when you're dealing with chemical reactions and quantitative analysis.
To calculate moles using the Ideal Gas Law, the formula is rearranged to solve for \( n \), representing the number of moles: \[ n = \frac{PV}{RT} \]Where:

• \( P \) is the pressure of the gas.
• \( V \) stands for volume.
• \( R \) is the Ideal Gas Constant.
• \( T \) is the temperature in Kelvin.

Each variable must be correctly calculated and converted to ensure they align with the units of \( R \). Specifically, ensure your volume is in liters, pressure in atmospheres, and temperature in Kelvin for this particular constant. In this way, one can determine the number of moles accurately, just like in the exercise where we found 2.78 moles of neon gas.

Gas Conversion Units

To solve problems involving gases and the Ideal Gas Law, converting units correctly is crucial. Every gas problem requires the alignment of the units used with the gas constant, \( R \).
For the exercise given, the pressure was initially in psi, but it needed to be converted to atm because that is the standard unit in the Ideal Gas Law when using the common gas constant \( R = 0.0821 \ \text{L atm K}^{-1} \, \text{mol}^{-1} \). Conversion can be done using the conversion factor: 1 atm = 14.696 psi.
Thus, for 125 psi, conversion to atm is:

• \( P = \frac{125}{14.696} \approx 8.512 \, \text{atm} \)

Volume is already measured in liters, so no conversion is necessary there. However, always ensure temperatures are in Kelvin by adding 273.15 to convert from Celsius.
With the correct units, you can seamlessly apply the Ideal Gas Law to solve for the desired variable!

Ideal Gas Constant

The Ideal Gas Constant, \( R \), is central to applications involving the Ideal Gas Law. It interrelates the physical properties of gases in the equation \( PV = nRT \).
This constant allows us to use a consistent set of units across different calculations, such as liters, atmospheres, Kelvin, and moles. For common calculations involving these units, \( R \) is used as 0.0821 \( \text{L atm K}^{-1} \, \text{mol}^{-1} \).
However, \( R \) can take different values depending on the units involved:

• For \( \ ext{Joule} \) units: \( R = 8.314 \ \text{J K}^{-1} \, \text{mol}^{-1} \)
• For \( \ ext{cm}^3 \) and \( \ ext{torr}: \ R = 62.364 \ \text{cm}^3 \ \, \text{torr} \ \, \text{K}^{-1} \, \text{mol}^{-1} \)

Choosing the correct \( R \) is vital based on the given units in your gas calculations, enabling the correct determination of other gas properties such as pressure or volume.