Problem 14 - Chemistry The Molecular Nature of Matter

Question

Use the ideal gas law to show that the partial pressure of a gas is directly proportional to its molar concentration. What is the proportionality constant?

Step-by-Step Solution

Answer

The partial pressure of a gas is directly proportional to its molar concentration with the proportionality constant being RT, where R is the universal gas constant and T is the temperature.

Step 1: Write down the ideal gas law

The ideal gas law 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 universal gas constant, and T is the temperature.

Step 2: Express the molar concentration (C) in terms of n and V

The molar concentration C is defined as the number of moles of gas per unit volume, C = n/V.

Step 3: Rearrange the ideal gas law to solve for P

By rearranging the ideal gas law equation, we can write P = (nRT)/V.

Step 4: Substitute C into the rearranged ideal gas law

Substituting the expression for C from Step 2 into the equation from Step 3, we get P = CRT, showing that the pressure P is directly proportional to the molar concentration C.

Step 5: Identify the proportionality constant

The proportionality constant between the partial pressure P and molar concentration C is RT, where R is the universal gas constant and T is the temperature.

Key Concepts

Understanding Partial Pressure

Partial pressure is a fundamental concept in the study of gases, particularly when dealing with mixtures. It refers to the pressure that a single gas component in a mixture would exert if it occupied the entire volume by itself while maintaining the same temperature. Imagine a container with several types of gases; each one exerts its own partial pressure, and the total pressure is the sum of these partial pressures.

In the context of the ideal gas law, we use partial pressure to describe the behavior of one specific gas among others. For a gas in a mixture, the ideal gas law still applies, but the pressure P in the equation PV=nRT is considered to be the partial pressure of that specific gas. Recognizing the direct relation between partial pressure and molar concentration is essential in applications such as calculating the concentration of oxygen in our atmosphere or understanding the composition of various gas mixtures in chemical reactions.

Molar Concentration and Its Significance

Molar concentration, often denoted as C, is a measurement of the amount of a substance within a specified volume. To put it plainly, it tells us how 'crowded' the molecules or atoms are in a given space. The higher the molar concentration, the more particles you have in a unit volume. For gases, the ideal gas law provides a bridge to understanding the relationship between molar concentration and pressure.

When we look at the steps to solve the problem using the ideal gas law, it's revealed that molar concentration is directly related to the number of moles (n) and the volume (V). In laboratory and industrial settings, knowing the molar concentration can be crucial for predicting how gases will react with each other, as reactions often depend on the concentration of the reactants.

Role of the Universal Gas Constant (R)

The universal gas constant, denoted as R, serves a crucial role in the ideal gas law. It's essentially the bridge that relates pressure, volume, and temperature to the number of moles. Think of R as a translator, converting the language of physical quantities into something that can be universally understood in the context of gases.

This constant has a specific value which is the same for all ideal gases, at approximately 8.314 J/(mol·K) (Joules per mole per Kelvin). Whether you're examining the atmosphere of planets or manipulating conditions in a laboratory chamber, R is used to calculate various properties of gases under different conditions. It’s also a fundamental part in determining the proportionality constant in the relationship between partial pressure and molar concentration.

A proportionality constant is a number that connects two directly proportional variables. In the case of the partial pressure and molar concentration, the proportionality constant is RT, where R is the universal gas constant and T is the absolute temperature. This means that if you know the molar concentration of a gas and the temperature, you can easily find the pressure that it would exert if it were alone in a container — its partial pressure.

This concept is extremely useful in chemical engineering, environmental science, and many other fields that work with gas laws. For various calculations, knowing that the proportionality constant is RT means that changes in temperature will affect the pressure exerted by the gas. Hence, if the temperature increases, so does the pressure, and vice versa, provided the molar concentration stays constant.