Question

The average atmospheric pressure on earth is approximated as a function of altitude by the relation \(P_{\mathrm{atm}}=\) \(101.325(1-0.02256 z)^{5.256},\) where \(P_{\text {atm }}\) is the atmospheric pressure in \(\mathrm{kPa}\) and \(z\) is the altitude in \(\mathrm{km}\) with \(z=0\) at sea level. Determine the approximate atmospheric pressures at Atlanta \((z=306 \mathrm{m}),\) Denver \((z=1610 \mathrm{m}),\) Mexico City \((z=2309 \mathrm{m}),\) and the top of Mount Everest \((z=8848 \mathrm{m})\)

Short answer

Question: Calculate the atmospheric pressure at the following locations using the given formula: \(P_{\mathrm{atm}} = 101.325(1-0.02256 z)^{5.256}\), where \(P_{\mathrm{atm}}\) is the atmospheric pressure in kPa and \(z\) is the altitude in km. Altitudes: - Atlanta: 306 meters - Denver: 1610 meters - Mexico City: 2309 meters - Mount Everest: 8848 meters Answer: Using the given formula and converting altitudes to kilometers, we find the atmospheric pressures for each location: - Atlanta: 99.034 kPa - Denver: 83.748 kPa - Mexico City: 76.869 kPa - Mount Everest: 33.575 kPa

Step-by-step solution

Understand the formula

The formula given to calculate the atmospheric pressure is \(P_{\mathrm{atm}} = 101.325(1-0.02256 z)^{5.256}\), where \(P_{\mathrm{atm}}\) is the atmospheric pressure in kPa and \(z\) is the altitude in km. Step 2: Convert altitudes to kilometers

Altitude conversion

The altitudes for the four places are given in meters. Before substituting these values into the formula, it is essential to convert them to kilometers by dividing by 1000: - Atlanta: \(z_1 = 306/1000 = 0.306\) km - Denver: \(z_2 = 1610/1000 = 1.610\) km - Mexico City: \(z_3 = 2309/1000 = 2.309\) km - Mount Everest: \(z_4 = 8848/1000 = 8.848\) km Step 3: Calculate the atmospheric pressure for each location

Plug the altitudes into the formula

Now that we have the altitudes in kilometers, simply substitute these values one by one into the formula to calculate the atmospheric pressure for each location. Atmospheric pressure at Atlanta: \(P_{\mathrm{atm1}} = 101.325(1 - 0.02256 \times 0.306)^{5.256} = 99.034\) kPa Atmospheric pressure at Denver: \(P_{\mathrm{atm2}} = 101.325(1 - 0.02256 \times 1.610)^{5.256} = 83.748\) kPa Atmospheric pressure at Mexico City: \(P_{\mathrm{atm3}} = 101.325(1 - 0.02256 \times 2.309)^{5.256} = 76.869\) kPa Atmospheric pressure at the top of Mount Everest: \(P_{\mathrm{atm4}} = 101.325(1 - 0.02256 \times 8.848)^{5.256} = 33.575\) kPa Step 4: OUTPUT - Atmospheric pressures at the four locations

Present the results

By applying the given formula with the respective altitudes, we found the atmospheric pressures for the following locations: - Atlanta has an atmospheric pressure of 99.034 kPa. - Denver has an atmospheric pressure of 83.748 kPa. - Mexico City has an atmospheric pressure of 76.869 kPa. - The top of Mount Everest has an atmospheric pressure of 33.575 kPa.

Key concepts

Altitude Effect on Atmospheric Pressure

Understanding how the atmosphere's pressure changes with altitude is a fundamental concept in atmospheric science. As we ascend from sea level into the higher layers of the atmosphere, the density of air molecules decreases. This drop in density means there are fewer air molecules above us to exert force, resulting in lower atmospheric pressure.

Why does this happen? The air's weight creates pressure at the Earth's surface, so the higher you go, the less air there is above you to 'weigh down,' hence a decrease in pressure. This phenomenon can affect various activities, from hiking at high altitudes to engineering projects, which must account for the changing atmospheric conditions with height.

• At low altitudes, such as Atlanta (306 m), the pressure remains quite close to the standard sea level pressure.
• As we move to cities like Denver (1610 m) and Mexico City (2309 m), the drop in pressure is notable.
• At extreme altitudes, such as Mount Everest (8848 m), the pressure is significantly lower, which can cause altitude sickness due to decreased oxygen levels in the body.

Pressure-Altitude Relationship

The pressure-altitude relationship is quantitatively described by a mathematical formula, like the one given in our exercise: \(P_{\text{atm}} = 101.325(1 - 0.02256 z)^{5.256}\), where \(P_{\text{atm}}\) represents atmospheric pressure in kPa and \(z\) is the altitude in kilometers. This type of relation is derived from the barometric formula, which models how atmospheric pressure decreases with an increase in altitude.

The relationship shows us an exponential decrease in pressure as the altitude increases, which means the rate of pressure decrease is not constant but rather changes with altitude. This connection is vital for understanding weather patterns, flying aircraft, and even designing buildings. To apply the formula to real-world scenarios, one must:

• Convert all measurements to consistent units (e.g., altitude in kilometers).
• Understand that the coefficients in the formula represent fixed atmospheric conditions and Earth's gravity.
• Recognize that local weather fluctuations can shift the actual pressures from these known standards.

Thermodynamics

In the context of atmospheric pressure and altitude, thermodynamics plays a vital role. Thermodynamics involves the study of energy transformations, and in the atmosphere, these transformations are often seen in the form of heat exchange.

The temperature of the atmosphere typically decreases with altitude in the troposphere, which is the lowest layer of the atmosphere where weather occurs. This cooling effect is due to several factors, including the decrease in air pressure, which allows air molecules to spread out and cool as they rise. The ideal gas law, a fundamental equation in thermodynamics given by \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is the amount of substance, \(R\) is the ideal gas constant, and \(T\) is temperature, helps to explain this phenomenon. As the air pressure \(P\) decreases with altitude, the temperature \(T\) also drops, influencing how pressure varies with altitude.

Atmospheric Science

Atmospheric science is an interdisciplinary field studying the Earth's atmosphere's physical and chemical properties and how they interact with energy and forces. It encompasses various topics, including but not limited to, atmospheric pressure changes with altitude.

Key principles of atmospheric science relevant to our exercise include:

• The understanding of how the Earth's gravitational force causes the accumulation of gases, forming the atmosphere.
• Knowledge of how different atmospheric layers behave and impact environmental conditions.
• Insights into predicting weather patterns, climate change models, and even helping to understand the atmospheres of other planets.

When we apply atmospheric science to our daily lives, it informs us how we can predict weather patterns, prepare for flights, or construct buildings that can withstand different atmospheric pressures at various altitudes.