Question

At \(46^{\circ} \mathrm{C}\) a sample of ammonia gas exerts a pressure of \(5.3 \mathrm{~atm} .\) What is the pressure when the volume of the gas is reduced to one-fourth of the original value at the same temperature?

Short answer

The new pressure is 21.2 atm.

Step-by-step solution

Understanding the problem

We are given a sample of ammonia gas at a temperature of \(46^{\circ} \mathrm{C}\) with an initial pressure of \(5.3 \mathrm{~atm}\). We are asked to find the new pressure when the volume of the gas is reduced to one-fourth of its original volume while keeping the temperature constant.

Using Boyle's Law

Since the temperature is constant, we can use Boyle's Law, which states that for a fixed amount of gas at a constant temperature, the product of pressure and volume is constant. The law is given as \( P_1V_1 = P_2V_2 \). Here, \(P_1 = 5.3 \mathrm{~atm}\), and the volume is reduced to one-fourth, so \(V_2 = \frac{1}{4}V_1\).

Setting up the equation

According to Boyle's Law, \( P_1V_1 = P_2V_2 \). We substitute the known values into this equation: \(5.3 \mathrm{~atm} \cdot V_1 = P_2 \cdot \frac{1}{4}V_1\).

Solving for the new pressure

The \(V_1\) terms cancel out, simplifying the equation to \(5.3 \mathrm{~atm} = P_2 \cdot \frac{1}{4}\). Solving for \(P_2\), we multiply both sides by 4: \(P_2 = 5.3 \mathrm{~atm} \times 4 \), which yields \(P_2 = 21.2 \mathrm{~atm}\).

Key concepts

Gas Laws

Gas laws are fundamental principles in chemistry that describe the behavior of gases. They help us understand how variables like pressure, volume, and temperature interact in gaseous substances. Gas laws are crucial for predicting how gases will respond to changes in these variables. There are several key gas laws, such as Boyle's Law, Charles's Law, and Avogadro's Law, each focusing on different aspects of gas behavior.

Boyle's Law, for example, deals with the pressure-volume relationship at constant temperature. It shows that pressure increases when volume decreases, and vice versa, as long as the temperature remains unchanged. Meanwhile, Charles's Law describes how the volume of a gas changes with temperature at constant pressure. Understanding these laws is essential for working with gases in both laboratory settings and real-world applications.

Pressure and Volume Relationship

The pressure and volume relationship in gases is a fundamental concept described by Boyle's Law. This law asserts that, for a given quantity of gas at a constant temperature, the pressure of the gas is inversely proportional to its volume. In simpler terms, if you compress a gas by reducing its volume, its pressure increases; likewise, expanding its volume decreases its pressure.

The mathematical expression for Boyle's Law is given as \( P_1V_1 = P_2V_2 \). Here, \(P_1\) and \(V_1\) are the initial pressure and volume, while \(P_2\) and \(V_2\) are the new pressure and volume after a change. This equation highlights that a decrease in volume (as in the case of the exercise with ammonia gas) leads to an increase in pressure. By manipulating this equation, we can determine how changing one parameter affects the other.

• This principle is vital in various applications, such as understanding how gas blanks work in airbags and predicting weather patterns in meteorology.
• It also applies to natural processes and industrial systems where gases are compressed and expanded.

Ammonia Gas

Ammonia (NH₃) is a colorless gas with a distinct, pungent odor commonly used in industrial applications and as a fertilizer in agriculture. It is composed of one nitrogen atom bonded to three hydrogen atoms. Ammonia is a notable compound because of its polar nature, which influences its physical properties and interactions with other compounds.

In the context of the exercise, ammonia gas is used to illustrate the application of Boyle's Law. The behavior of ammonia gas under changing conditions of pressure and volume helps highlight the practical use of gas laws. The properties of ammonia make it an excellent subject for such studies because it is relatively easy to control and observe in laboratory settings. Understanding how ammonia behaves under different conditions can help in industries like refrigeration and chemical manufacturing.

Constant Temperature

A constant temperature setting is essential when applying Boyle's Law to gas calculations. The condition of constant temperature ensures that energy contributions to the gas are stable, allowing changes in pressure and volume to be observed accurately without interference from temperature variations.

When the temperature is held constant, it simplifies the relationship between pressure and volume, as expressed in Boyle's Law. It allows us to focus solely on how these two variables interact, without considering the added complexity that arises when temperature fluctuates.

• This is particularly important in laboratory experiments, where controlling environmental conditions leads to more reliable results.
• In practical terms, maintaining a constant temperature can mean conducting experiments in climate-controlled environments or using equipment that monitors and adjusts temperature levels automatically.

Keeping the temperature constant allows us to draft clear and comprehensible models of gas behavior, essential for both educational purposes and practical applications.