Question

A balloon containing \(5.0 \mathrm{dm}^{3}\) of gas at \(14^{\circ} \mathrm{C}\) and \(100.0 \mathrm{kPa}\) rises to an altitude of \(2000 . \mathrm{m}\), where the temperature is \(20^{\circ} \mathrm{C}\). The pressure of gas in the balloon is now \(79.0 \mathrm{kPa}\). What is the volume of gas in the balloon?

Short answer

The volume of the gas in the balloon at the new altitude is approximately 6.4 dm³.

Step-by-step solution

Understand the Problem

The problem asks us to find the final volume of a gas contained in a balloon, which undergoes changes in temperature, pressure, and altitude as it rises. These changes can be analyzed using the Ideal Gas Law and the combined gas law.

Know the Relevant Equations

We will apply the combined gas law, which is given by \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \). This relates the initial conditions (\(P_1, V_1, T_1\)) to the final conditions (\(P_2, V_2, T_2\)) of the gas, where \(P\) is pressure, \(V\) is volume, and \(T\) is temperature in Kelvin.

Convert Temperatures to Kelvin

Initial temperature, \(T_1 = 14^{\circ}C\), needs to be converted to Kelvin: \(T_1 = 14 + 273.15 = 287.15\ K\).Final temperature, \(T_2 = 20^{\circ}C\), also needs conversion: \(T_2 = 20 + 273.15 = 293.15\ K\).

Substitute Known Values into the Equation

Using the values: \(P_1 = 100.0\ kPa\), \(V_1 = 5.0\ dm^3\), \(T_1 = 287.15\ K\), \(P_2 = 79.0\ kPa\), and \(T_2 = 293.15\ K\), substitute into the combined gas law: \[ \frac{100.0 \times 5.0}{287.15} = \frac{79.0 \times V_2}{293.15} \]

Solve for Final Volume \(V_2\)

Cross-multiply and solve for \(V_2\):\[ 100.0 \times 5.0 \times 293.15 = 79.0 \times 287.15 \times V_2 \]\[ V_2 = \frac{100.0 \times 5.0 \times 293.15}{79.0 \times 287.15} \] Calculate this to find \(V_2\approx 6.4\ dm^3\).

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental principle used to describe the behavior of gases. It combines various gas properties into one equation: \( PV = nRT \). Here, \( P \) represents pressure, \( V \) stands for volume, \( n \) is the number of moles of the gas, \( R \) is the universal gas constant, and \( T \) is temperature in Kelvin.

While the exercise primarily utilized the Combined Gas Law, understanding the Ideal Gas Law is essential as it underpins all gas laws. The Ideal Gas Law helps predict how a gas will behave under different conditions by assuming the gas molecules are far apart and interact minimally.

In real-world applications, it approximates how real gases behave, especially under low pressure and high temperature. The equation can be adapted by varying state variables like pressure, volume, and temperature, assisting in many scientific computations.

Temperature Conversion

In gas law calculations, it is vital to convert temperatures from Celsius to Kelvin. This is because the Kelvin scale starts at absolute zero, providing a uniform scale for scientists.

To convert degrees Celsius to Kelvin, simply add 273.15 to the Celsius temperature. For instance, the initial temperature of 14°C becomes 287.15 K, and the final temperature of 20°C converts to 293.15 K. The Kelvin scale ensures a positive, direct relationship between temperature and energy or volume in the calculations..

Using Kelvin allows us to avoid negative temperatures in equations, which simplifies the mathematical processes involved in applying gas laws. Accurate conversions are key when solving problems according to the Ideal Gas Law and its derivatives.

Pressure and Volume Relationships

Pressure and volume in gases are inversely related, a concept described by Boyle's Law. As pressure increases, the volume of a gas decreases, provided the temperature remains constant. This relationship is part of the foundation for the Combined Gas Law.

In the exercise, we observe changes in both pressure and volume as the gas balloon rises. By understanding how these variables relate, we can employ the Combined Gas Law to predict the new volume at different pressures and temperatures.

To find the final volume \( V_2 \) in the exercise, the pressures \( P_1 \) and \( P_2 \) at different altitudes and volumes were compared using the known initial volume and pressure values. Such calculations underscore why knowing these relationships is important in practical applications.

Gas Laws

Gas Laws provide the theoretical basis for understanding how gases react to changes in pressure, volume, and temperature. They include laws named after scientists, such as Boyle's Law, Charles's Law, and Avogadro's Law. Each focuses on one aspect of gas behavior while assuming others are constant.

The Combined Gas Law, used in the given exercise, derives from these individual laws. It is expressed as \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \). This equation connects the initial and final states of a gas, integrating changes in pressure, volume, and temperature.

By using the Combined Gas Law, students can approach situations where multiple variables change. Practicing with related equations helps deepen understanding and enhance problem-solving skills, preparing students for complex real-world applications.