Question

A cylinder contains 28.5 L of oxygen gas at a pressure of 1.8 atm and a temperature of 298 \(\mathrm{K}\) . How much gas (in moles) is in the cylinder?

Short answer

There are approximately 2.2 moles of oxygen gas in the cylinder.

Step-by-step solution

Identify Given Information

List the known values from the problem. Volume (\( V \)) is 28.5 L, pressure (\( P \)) is 1.8 atm, and temperature (\( T \)) is 298 K.

State the Ideal Gas Law

Write down the Ideal Gas Law equation, which is \( PV = nRT \) where \( n \) is the number of moles, \( R \) is the gas constant with a value of 0.08206 L·atm/K·mol, and \( T \) is the temperature in Kelvin.

Solve for the Number of Moles (\( n \) )

Rearrange the Ideal Gas Law equation to solve for \( n \) which gives \( n = \frac{PV}{RT} \) and plug in the known values.

Calculate the Number of Moles of Gas

Insert the known values into the rearranged formula: \ \( n = \frac{(1.8 \ text{ atm})(28.5 \ text{ L})}{(0.08206 \text{ L·atm/K·mol})(298 \text{ K})} \) and perform the calculations.

Key concepts

Molar Volume of Gas

The molar volume of a gas refers to the volume occupied by one mole of a gas at a specific temperature and pressure. For gases, it's often assumed that they behave ideally, which is the case under standard temperature and pressure (STP), defined as 0°C (273.15 K) and 1 atm.

Under these conditions, one mole of any ideal gas occupies approximately 22.4 liters, known as the molar volume of an ideal gas at STP. It's crucial for students to understand that this value can change with varying temperature and pressure, which is why calculations must take into account the actual conditions, as in our exercise. In this case, the volume of oxygen gas provided is 28.5 L - which is not at STP - hence why we apply the Ideal Gas Law for accurate calculations.

Gas Constant (R)

The gas constant, symbolized by the letter R, is a fundamental component in various gas laws, including the Ideal Gas Law. This constant provides the link between the physical quantities of pressure, volume, temperature, and moles within the equation.

R has a value of 0.08206 L·atm/K·mol when pressure is in atmospheres, volume is in liters, temperature in kelvins, and the amount of gas in moles. It's important to match units across the entire calculation to ensure that they align with the units of the gas constant. Since different conditions can require different units, R also has other values that correspond to different unit combinations. However, in our exercise, we use the value that matches the given units, reinforcing to students that careful attention to unit consistency is key in gas law calculations.

Gas Law Calculations

Gas law calculations are mathematical methods used to relate the physical properties of gases—pressure (P), volume (V), temperature (T), and the amount in moles (n). The Ideal Gas Law, given by the equation PV = nRT, is a central equation for these calculations and applies to ideal gases.

To solve for any unknown in the equation, rearrange the Ideal Gas Law to isolate the variable of interest. For instance, to find the number of moles as in our exercise, the equation is rearranged to = \(\frac{PV}{RT}\). Then plug in the known values for pressure, volume, and temperature, along with the gas constant, to find the desired quantity. It's a good practice to check for unit consistency and remember that temperature must always be in kelvins for these calculations. Through systematic application of these steps, students can confidently address a variety of problems involving gas properties.