Question

A gas is in a container whose volume is variable. The container is in an ice bath at \(0.00^{\circ} \mathrm{C}\), and there are \(2.0\) moles of gas in it. What must the volume in liters be if the gas has a pressure of \(2.5 \mathrm{~atm}\) ?

Short answer

The volume of the gas container at \(0.00^{\circ} \mathrm{C}\), 2.0 moles of gas, and a pressure of \(2.5 \mathrm{~atm}\) is approximately 17.98 liters.

Step-by-step solution

Identify the given variables

We are given the following information: - Number of moles (n) = 2.0 moles - Pressure (P) = 2.5 atm - Temperature (T) = 0.00°C (which is equal to 273.15 K in Kelvin) We can also use the Ideal Gas Constant for the appropriate units: - R (in liter-atm units) = 0.0821 L atm / (mol K)

Use the ideal gas law to find the volume

Using the ideal gas law formula, PV = nRT, we can solve for the volume (V). V = (nRT) / P Substitute the given values and the Ideal Gas Constant into the equation: V = (2.0 moles * 0.0821 L atm / (mol K) * 273.15 K) / (2.5 atm)

Evaluate the expression for the volume

Now, carry out the calculations to find the volume: V = (2.0 * 0.0821 * 273.15) / 2.5 V ≈ (44.95663) / 2.5 V ≈ 17.98265 L

Report the answer

The volume of the gas container, under the given conditions, is approximately 17.98 liters.

Key concepts

Gas Laws

Understanding the behavior of gases is a fundamental part of physical chemistry and the 'gas laws' are equations that relate the variables of a gas sample: pressure, volume, temperature, and number of moles. These laws describe how gases respond to changes in the environment and help us predict their behavior under different conditions.

The Ideal Gas Law, which is derived from combining several simpler gas laws, is the equation that is particularly relevant when you're dealing with problems like the one in the exercise. It expresses the direct proportionality of volume (V) to the number of moles (n) and temperature (T), and the inverse proportionality to pressure (P), summarized by the equation PV = nRT where R is the universal gas constant.

Physical Chemistry

Physical chemistry involves the study of how matter behaves on a molecular and atomic level and how chemical reactions occur. Within this field, understanding the properties of gases is essential. An important application of physical chemistry is the use of the ideal gas law to predict the behavior of gases under varying conditions.

To apply the ideal gas law correctly, you have to be precise with units, converting temperatures to Kelvin and using consistent units for pressure, volume, and the ideal gas constant (R). It's this depth of understanding that allows us to solve complex problems and carry out precise calculations, just as in the exercise provided.

Moles and Gas Volume

The mole is a unit of measurement used in chemistry to express amounts of a chemical substance. One mole corresponds to the number of atoms found in 12 grams of carbon-12, which is approximately 6.022 x 10^23 atoms (Avogadro's number).

In the context of gases, understanding the relationship between moles and gas volume is crucial. According to Avogadro's law, one mole of any gas at standard temperature and pressure (STP) occupies 22.4 liters. While conditions in real-world problems are often different from STP, this concept helps illustrate how more moles will generally result in a greater volume of gas, assuming pressure and temperature remain constant.

Temperature and Pressure Relationships

Temperature and pressure have a straightforward but profound relationship in gas behavior. From a microscopic viewpoint, increasing the temperature of a gas means the gas particles move more rapidly and collide with the walls of their container with greater force, which can lead to an increase in pressure if the volume is constant – this is known as Gay-Lussac's law.

Conversely, Boyle's law tells us that at a constant temperature, the volume of a gas is inversely proportional to its pressure. When it comes to solving problems, it's important to consider these relationships and how altering one variable will affect the others, especially when your container's volume is variable, as in our exercise scenario.