Chapter 11: Problem 64
A weather balloon at Earth’s surface has a volume of 4.00 L at 304 K and 755 mm Hg. If the balloon is released and the volume reaches 4.08 L at 728 mm Hg, what is the temperature?
Short Answer
Expert verified
The final temperature is approximately 292.66 K.
Step by step solution
01
Understand the Problem
We are given the initial volume (V_1) = 4.00 L, initial temperature (T_1) = 304 K, and initial pressure (P_1) = 755 mm Hg. We need to find the final temperature (T_2) when the volume (V_2) = 4.08 L and the pressure (P_2) = 728 mm Hg.
02
Use the Combined Gas Law
The combined gas law formula is given as: \(\frac{P_1 \, V_1}{T_1} = \frac{P_2 \, V_2 }{T_2}\). We will use this to relate the initial and final states of the weather balloon.
03
Rearrange for Final Temperature
Rearrange the combined gas law equation to solve for T_2:\[T_2 = \frac{P_2 \, V_2 \, T_1}{P_1 \, V_1}\]
04
Plug in the Known Values
Substitute the known values into the equation:\[T_2 = \frac{728 \, \text{mm Hg} \, \cdot \, 4.08 \, \text{L} \, \cdot \, 304 \, \text{K}}{755 \, \text{mm Hg} \, \cdot \, 4.00 \, \text{L}}\]
05
Calculate the Final Temperature
Calculate the final temperature:\[T_2 = 292.66 \, \text{K}\]}],
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Gas Laws
Gas laws are fundamental principles that describe the behavior of gases. They relate the temperature, pressure, and volume of a gas. The most common gas laws include Boyle's Law, Charles's Law, and Gay-Lussac's Law. These laws were eventually combined into a single equation known as the **Combined Gas Law**. This law is particularly useful when dealing with scenarios where conditions of a gas change, like in weather balloons. By understanding gas laws, you can predict how a gas will respond to changes in its environment. This is crucial in many practical applications, such as meteorology, engineering, and even medicine.
Temperature Calculation
Temperature calculation in gas laws is essential to determine the new temperature of the gas when its pressure or volume changes. With the combined gas law, you can find the temperature by using this formula: \( \frac{P_1 \, V_1}{T_1} = \frac{P_2 \, V_2}{T_2} \). Given the initial conditions and changes, this equation helps to calculate the final temperature. In our weather balloon problem, we started with: \(P_1 = 755 \, \text{mmHg}, V_1 = 4.00 \, \text{L}, T_1 = 304 \, \text{K} \). After the change: \( P_2 = 728 \, \text{mmHg}, V_2 = 4.08 \, \text{L} \). Plugging these values into the equation gives us \(T_2 = 292.66 \, \text{K} \). This calculation helps us understand how the temperature of the balloon changes as it rises.
Pressure-Volume Relationship
The pressure-volume relationship, known as Boyle's Law, states that the pressure of a gas is inversely proportional to its volume when temperature is kept constant. This means if the volume of a gas increases, its pressure decreases, and vice versa. In our example, the weather balloon's volume increased from 4.00 L to 4.08 L while the pressure decreased from 755 mmHg to 728 mmHg. This inverse relationship helps to understand how gases expand and contract under different pressures. By using the combined gas law, we can also incorporate temperature changes into this relationship.
Weather Balloon
A weather balloon is an essential tool in meteorology for measuring atmospheric conditions. It ascends into the atmosphere, expanding as it rises due to the decreasing pressure. This affects its volume and temperature. Using the combined gas law, meteorologists can predict these changes and gather valuable data. For instance, in our problem, the balloon's volume changed and we calculated the final temperature at a higher altitude. This information is crucial for weather forecasting, studying climate patterns, and understanding atmospheric behavior. Accurate predictions about how a weather balloon behaves help scientists with their observations and analyses.