Question

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

Short answer

The cylinder contains approximately 1.73 moles of oxygen gas.

Step-by-step solution

Identify the known variables

The problem provides the volume of the gas (V) as 28.5 Liters, the pressure (P) as 1.8 atm, and the temperature (T) as 298 K. These are the known variables.

Determine the appropriate gas law to use

Since we are dealing with pressure, volume, temperature, and amount (moles), the ideal gas law is applicable. The ideal gas law is PV = nRT, where n is the number of moles and R is the ideal gas constant.

Identify the ideal gas constant

The ideal gas constant (R) in units of L·atm/(mol·K) is 0.0821.

Rearrange the ideal gas law to solve for n, the number of moles

To find the number of moles (n), rearrange the ideal gas law to n = PV / RT.

Substitute the known values into the rearranged equation

Substitute P = 1.8 atm, V = 28.5 L, R = 0.0821 L·atm/(mol·K), and T = 298 K into the equation to solve for n.

Calculate the number of moles

Using the substituted values, perform the calculation to find n: n = (1.8 atm * 28.5 L) / (0.0821 L·atm/(mol·K) * 298 K).

Perform the arithmetic

Multiply 1.8 atm by 28.5 L, divide that by the product of 0.0821 L·atm/(mol·K) and 298 K to get the number of moles.

Key concepts

Gas Law Calculations

Understanding gas law calculations is essential for solving problems that involve the properties of gases. These calculations are based on relationships between volume (V), pressure (P), temperature (T), and the amount of gas in moles (n).

When dealing with gases, it's important to consider the conditions they are under, as gases can compress and expand with changes in temperature and pressure. The ideal gas law combines several individual gas laws and provides a good approximation for the behavior of real gases under many conditions. By rearranging the ideal gas law, various unknown quantities can be calculated as long as the other variables are known.

For example, if we need to find the amount of gas present in a container, as seen in the exercise provided, we start with the known volume, pressure, and temperature, and then compute the number of moles of the gas using the ideal gas law.

Molar Volume

Molar volume is a very useful concept when working with gases. It refers to the volume that one mole of any gas occupies at a certain temperature and pressure. At standard temperature and pressure (STP), which is 0°C (273.15 K) and 1 atm pressure, one mole of an ideal gas occupies 22.4 liters.

This concept is important because it allows for direct comparison of different gases under standardized conditions, enabling more straightforward stoichiometric calculations. In the context of the exercise, once the number of moles is calculated, you could further use the molar volume to determine the volume the gas would occupy at STP or compare it with other gases.

PV=nRT Formula

The PV=nRT formula is the mathematical representation of the ideal gas law, where P is pressure, V is volume, n is the number of moles of gas, R is the ideal gas constant, and T is temperature. The R value's units vary based on the units used for pressure and volume. Commonly, R = 0.0821 L·atm/(mol·K) for pressure in atmospheres and volume in liters.

To use this formula effectively, one must ensure that all units are consistent. This formula can be manipulated to solve for any of the variables when the others are given. In the given exercise, the formula is rearranged to solve for the number of moles (n), illustrating how versatile and powerful the PV=nRT equation is in understanding gas behavior.

Stoichiometry

Stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. It is based on the law of conservation of mass and the mole concept. Stoichiometry calculations often use the balanced chemical equation to determine the amount of reactants needed or products formed.

In relation to gases, stoichiometry involves using the ideal gas law, molar volumes, and molar ratios to predict the outcome of gas reactions. For example, if a reaction yields a gas as a product, stoichiometry would allow us to calculate the expected volume of that gas at given conditions using the ideal gas law. It is an invaluable tool for chemists, allowing for precise chemical reaction design and analysis.