Problem 179 - Chemistry The Science in Context

Question

The severity of a tropical storm is related to the depressed atmospheric pressure at its center. In August \(1985,\) Typhoon Odessa reached maximum winds of about \(90 \mathrm{mi} / \mathrm{h}\) and the pressure was 40 mbar lower at the center than normal atmospheric pressure. In contrast, the central pressure of Hurricane Andrew (Figure \(\mathrm{P} 6.179\) ) was 90 mbar lower than its surroundings when it hit southern Florida with winds as high as \(165 \mathrm{mi} / \mathrm{h}\). If a small weather balloon with a volume of \(50.0 \mathrm{L}\) at a pressure of 1.0 atmosphere was deployed above the center of Andrew, what was the volume of the balloon when it reached the surface of the ocean?

Step-by-Step Solution

Answer

Answer: 55.0 liters

Step 1: Convert the given atmospheric pressure into Pascal

First, we need to convert the atmospheric pressure into Pascal, which is the standard unit of pressure.\[1\ \text{atm} = 101325\ \text{Pa}\]The initial pressure, \(P_1\), is given as 1 atm, so we have:\[P_1 = 1\ \text{atm} = 101325\ \text{Pa}\]

Step 2: Calculate the decrease in pressure due to the hurricane

We are given that the central pressure of Hurricane Andrew is 90 mbar lower than its surroundings. To find the pressure at the center of the hurricane, we first convert mbar to Pascal:\[1\ \text{mbar} = 100\ \text{Pa}\]The change in pressure is 90 mbar, so we have: \[\Delta P = 90\ \text{mbar} = 9000\ \text{Pa}\]

Step 3: Determine the final pressure at the surface of the ocean

The pressure at the surface of the ocean, \(P_2\), is the initial pressure minus the change in pressure due to the hurricane:\[P_2 = P_1 - \Delta P = 101325\ \text{Pa} - 9000\ \text{Pa} = 92325\text{Pa}\]

Step 4: Convert volume from L to m³

The initial volume of the balloon is given in liters (L), so we need to convert this to cubic meters (m³) to use SI units. \[1\ \text{L} = 0.001\ \text{m}^3\]The initial volume, \(V_1\), is given as \(50.0\ \text{L}\), so we have:\[V_1 = 50.0\ \text{L} = 0.050\ \text{m}^3\]

Step 5: Calculate the final volume of the balloon using Boyle's Law

Using Boyle's Law, \(P_1V_1 = P_2V_2\), we get our equation to find the final volume \(V_2\):\[V_2 = \frac{P_1V_1}{P_2}\]Plugging in our known values for \(P_1\), \(V_1\), and \(P_2\):\[V_2 = \frac{101325\ \text{Pa}\times 0.050\ \text{m}^3}{92325\ \text{Pa}} = 0.0550\ \text{m}^3\]

Step 6: Convert the final volume back to liters

Finally, we will convert the final volume, \(V_2\), back to liters for a more recognizable value:\[V_2 = 0.0550\ \text{m}^3 = 55.0\ \text{L}\] Thus, the volume of the balloon when it reaches the surface of the ocean in the center of Hurricane Andrew is 55.0 liters.

Key Concepts

Atmospheric Pressure

Atmospheric pressure is the force exerted by the weight of the air above a surface. It is one of the most fundamental concepts when studying the atmosphere. This force is exerted over a certain area, typically measured in units like millibars (mbar) or atmospheres (atm), with 1 atm being the average pressure at sea level under normal conditions.
When you think of atmospheric pressure, imagine the atmosphere as a thick blanket of air pressing down against everything on Earth. It not only affects the weather patterns but is also a crucial factor in understanding how storms form and behave.

• Normal atmospheric pressure at sea level is approximately 1013 mbar or 1 atm.
• This pressure decreases with altitude and can also vary with weather conditions.

Knowing atmospheric pressure helps in analyzing and predicting changes in the weather, including severe weather conditions like hurricanes.

Hurricane Pressure

Hurricanes are among the most powerful and destructive natural events. A significant drop in atmospheric pressure marks the core of a hurricane. In fact, the lower the pressure at a hurricane's center, the more severe the storm usually is.
This pressure drop is essential because it affects both the wind speed and the storm's destructive potential. For example, Hurricane Andrew had a pressure center 90 mbar lower than its surrounding air.

• A lower central pressure means stronger winds and a more intense storm.
• Hurricane pressure can be used to gauge the potential impact of the storm on affected regions.

Understanding this phenomenon helps us better prepare and respond to hurricanes when they approach land.

Volume Conversion

Volume conversion is converting the measure of space that an object occupies from one unit to another. This is an essential process in scientific calculations to ensure that all measurements are standardized and compatible, especially when using SI units.
In our example, the balloon's initial volume is given in liters, a common unit in chemistry and physics.

• Converting liters to cubic meters is necessary for calculations under the metric system because 1 L equals 0.001 m³.
• Doing this conversion ensures that our calculations using Boyle's Law are precise and in the correct units.

Volume conversions like this one are crucial because they accommodate scientific formulas and ensure that measurements are universally understandable.

Pascal Unit Conversion

Converting to Pascal (Pa) is crucial when measuring pressure in scientific disciplines. The Pascal is the SI unit of pressure, allowing scientists and engineers to maintain consistency in their calculations and findings.
For instance, when converting atmospheric pressure from atm to Pa or mbar to Pa, the following equivalents are used:

• 1 atm is equal to 101325 Pa.
• 1 mbar equals 100 Pa.

Performing these conversions is vital when using physical laws like Boyle's Law, which requires pressure in Pascals for computations. Understanding how and why we convert these units helps students and scientists precisely analyze and predict physical behaviors within scientific contexts.