Question

Many altimeters determine altitude changes by measuring changes in the air pressure. An altimeter that is designed to be able to detect altitude changes of \(100 \mathrm{~m}\) near sea level should be able to detect pressure changes of a) approximately \(1 \mathrm{~Pa}\). d) approximately \(1 \mathrm{kPa}\). b) approximately 10 Pa. e) approximately \(10 \mathrm{kPa}\). c) approximately \(100 \mathrm{~Pa}\).

Short answer

Answer: d) approximately 1 kPa

Step-by-step solution

Familiarize yourself with the Barometric Formula

The barometric formula is a mathematical model that relates atmospheric pressure to altitude. Near sea level, it is given by: $$ P(h) = P_{0}e^{\frac{-Mgh}{RT_{0}}}, $$ where \(P(h)\) is the atmospheric pressure at altitude \(h\), \(P_{0}\) is the atmospheric pressure at sea level, \(M\) is the molar mass of Earth's air, \(g\) is the acceleration due to gravity, \(R\) is the ideal gas constant, and \(T_{0}\) is the temperature at sea level.

Calculate the pressure at the initial altitude (0 m)

First, we need to find the atmospheric pressure at sea level (0 m). The given parameters are: - P0 = 101325 Pa - M = 0.029 kg/mol - g = 9.81 m/s^2 - R = 8.314 J/(mol*K) - T0 = 273.15 K (standard temperature at sea level) Using these values and the barometric formula, we get the pressure at sea level: $$ P(0) = 101325e^{\frac{-0.029*9.81*0}{8.314*273.15}} = 101325 \mathrm{~Pa}. $$

Calculate the pressure at the final altitude (100 m)

Now, we need to find the atmospheric pressure at an altitude of 100 m. Using the given parameters and the barometric formula, we get the pressure at 100 m: $$ P(100) = 101325e^{\frac{-0.029*9.81*100}{8.314*273.15}} \approx 100246.2 \mathrm{~Pa}. $$

Determine the pressure change

To find the pressure change that the altimeter should be able to detect, subtract the pressure at the initial altitude (0 m) from the final altitude (100 m): $$ \Delta P = P(100) - P(0) \approx 100246.2 - 101325 \approx -1078.8 \mathrm{~Pa}. $$ Since we are only interested in the magnitude of the pressure change, we can take the absolute value: $$ |\Delta P| \approx 1078.8 \mathrm{~Pa}. $$

Compare the calculated pressure change to the given options

From the calculated pressure change, we can see that the closest option to the correct value is d) approximately 1 kPa. Note that 1 kPa = 1000 Pa, which is close to the calculated value of 1078.8 Pa. So, the correct answer is option d).

Key concepts

Understanding Atmospheric Pressure

Atmospheric pressure is the force exerted by the weight of the air above us and is a fundamental concept in understanding weather, climate, and how altimeters work. It's measured in pascals (Pa) and typically decreases as altitude increases due to the thinning of the air. This is precisely why altimeters, which are instruments used to measure altitude, are based on the detection of these pressure changes. Simplifying this, imagine the atmosphere as a stack of air layers; the higher you go, the fewer layers press down on you, resulting in lower pressure.

For practical applications like aviation, knowing the atmospheric pressure is crucial for pilots to determine their elevation above sea level. Altimeters aid in this process by responding to the pressure changes, allowing pilots to navigate safely by maintaining the correct altitude. It's a fine balance, as the difference between safe flying and potential danger could just be a matter of a few hundred pascals.

The Impact of Altitude Changes

Altitude changes are directly related to changes in atmospheric pressure, and understanding this relationship is essential for various activities ranging from aviation to meteorology. As mentioned earlier, air pressure decreases with an increase in altitude. But why does this matter? For one, hikers and climbers often experience the effects of altitude in the form of 'thin air,' which can make breathing more laborious.

Moreover, the barometric formula shows us that these pressure changes are exponential, not linear. This means that the rate at which pressure decreases is not the same at all altitudes; it's faster at lower altitudes and slower at higher ones. As a result, an altimeter designed to detect changes at sea level needs to be sensitive to relatively small changes in air pressure, as the exercise illustrates.

Interpreting the Ideal Gas Law in Context

The ideal gas law, represented by the equation PV=nRT (where P is pressure, V is volume, n is the number of moles, R is the universal gas constant, and T is temperature), provides the theoretical foundation to understand how gases behave under different conditions of temperature, pressure, and volume. In the context of atmospheric pressure and altitude, the barometric formula can be seen as an application of the ideal gas law tailored to the Earth’s atmosphere.

Diving deeper into the exercise, we see how the ideal gas law is implicitly used to derive the barometric formula for atmospheric pressure at different altitudes. The values for the molar mass of air (M), the gravity (g), and the ideal gas constant (R) work together with the standard temperature at sea level (T0) to predict how pressure decreases with altitude. The barometric formula is a reminder that the air above us behaves according to universal gas laws, and by measuring its pressure, we unlock information about altitude and much more.