Problem 7 - University Physics with Modern Physics

Question

One mole of an ideal gas, at a temperature of \(0^{\circ} \mathrm{C}\), is confined to a volume of \(1.0 \mathrm{~L}\). The pressure of this gas is a) \(1.0 \mathrm{~atm}\). b) 22.4 atm. c) \(1 / 22.4 \mathrm{~atm}\) d) \(11.2 \mathrm{~atm}\).

Step-by-Step Solution

Answer

Answer: b) 22.4 atm.

Step 1: Convert temperature to Kelvin

To convert the temperature from Celsius to Kelvin, add 273.15 to the Celsius temperature. In this case, \(0^{\circ}\mathrm{C}\) equals 273.15 K.

Step 2: Use the ideal gas law to solve for pressure

The ideal gas law is given by the formula \(PV = nRT\). We are given n (1 mole), R (\(0.0821\,\mathrm{L\,atm\,K^{-1}\,mol^{-1}}\)), temperature in Kelvin (273.15 K), and volume (1 L). We need to solve for pressure (P). To do this, we can rearrange the formula: \(P = \frac{nRT}{V}\) Now, plug in the given values: \(P = \frac{1\,\mathrm{mol} \times 0.0821\,\mathrm{L\,atm\,K^{-1}\,mol^{-1}} \times 273.15\,\mathrm{K}}{1\,\mathrm{L}}\)

Step 3: Calculate the pressure

After inserting the values into the formula, we can calculate the pressure: \(P = \frac{0.0821\,\mathrm{L\,atm\,K^{-1}\,mol^{-1}} \times 273.15\,\mathrm{K}}{1.0\mathrm{L}} \approx 22.4\,\mathrm{atm}\) Based on our calculation, the pressure of the ideal gas is 22.4 atm. Therefore, the correct answer is: b) 22.4 atm.

Key Concepts

Thermodynamics and the Ideal Gas Law

Understanding thermodynamics, particularly the ideal gas law, is crucial when dealing with problems that involve the conditions of gases. The ideal gas law, a cornerstone in thermodynamics, correlates the pressure, volume, temperature, and number of moles of an ideal gas in a beautifully simple equation: \(PV = nRT\).

This formula embodies the fundamental principles of thermodynamics. It reflects how gas particles interact with their environment, how they respond to changes in temperature, and how they exert force on the walls of their container, which we observe as pressure. In this case, by isolating the pressure variable, you can describe how temperature, volume, and moles influence it. Thermodynamics is essentially the study of energy transformations, and the ideal gas law demonstrates one way these transformations can be quantified and predicted in a gas.

Pressure Calculation in Gases

Even though the term 'pressure' is familiar in everyday language, understanding pressure in a gas context requires grasping its physical implications. Pressure (P) is a measure of force exerted per unit area, and in the case of gases, it quantifies the collective force exerted by gas particles as they collide with container walls. To find the pressure of a gas, one can use the ideal gas law by rearranging the equation to solve for P.

Using the formula \(P = \frac{nRT}{V}\), you can calculate the pressure when you know the number of moles of the gas (n), the universal gas constant (R), the temperature (T in Kelvin), and the volume (V) the gas occupies. Remember, when performing such pressure calculations, it's essential to ensure the temperature is always in Kelvin, as this ensures consistency with the gas constant units and provides an absolute scale for temperature measurement.

The Mole Concept in Chemistry

At the heart of many chemical calculations is the mole concept, serving as a bridge between the microscopic world of atoms and molecules and the macroscopic world of grams and liters. A mole is defined as the amount of substance that contains as many particles (atoms, molecules, ions, etc.) as there are atoms in 12 grams of carbon-12. This number, known as Avogadro's number, is approximately \(6.022 \times 10^{23}\) particles per mole.

In the context of our gas problem, we talk about '1 mole of a gas', which refers to a quantity that contains Avogadro's number of gas particles. This fundamental understanding allows us to use the gas constant (R) appropriately in the ideal gas law and make accurate predictions about the behavior of gases. When dealing with gas laws, remembering the mole concept is vital, for it gives meaning to the R value (the gas constant) and helps in the precise calculation of pressures, volumes, and temperatures.