Calculating Atmospheric Pressure Using Scale Height

The atmospheric pressure decreases with altitude due to the decreasing weight of the air column above. In a simplified model, assuming the atmosphere is isothermal (constant temperature) and composed of an ideal gas, the pressure ($P$) at a height ($h$) above the ground is related to the pressure at the ground level ($P_0$) by the formula:

\( P(h) = P_0 e^{-h/H} \)

where:

• \(P(h)\) is the pressure at height \(h\).
• \(P_0\) is the pressure at the reference height (usually ground level, \(h=0\)).
• \(e\) is the base of the natural logarithm, approximately 2.71828.
• \(h\) is the height above the reference level.
• \(H\) is the scale height, which is a characteristic height over which the pressure decreases by a factor of \(e\).

In this specific problem, we are given:

• Scale height (\(H\)) = 7 km.
• Height above the ground surface (\(h\)) = 21 km.

We need to find the pressure at a height of 21 km, \(P(21 \text{ km})\). We need to assume a value for the pressure at ground level (\(P_0\)). A standard value for atmospheric pressure at sea level is approximately \(10^5\) Pa (or 1 atm). Let's use this value for \(P_0\).

\(P_0 \approx 1 \times 10^5 \text{ Pa}\)

Now we can substitute the given values into the formula:

\( P(21 \text{ km}) = P_0 e^{-21 \text{ km} / 7 \text{ km}} \)

\( P(21 \text{ km}) = P_0 e^{-3} \)

Next, we need to calculate the value of \(e^{-3}\).

\( e^{-3} = \frac{1}{e^3} \)

Using the approximate value \(e \approx 2.71828\):

\( e^3 \approx (2.71828)^3 \approx 20.0855 \)

\( e^{-3} \approx \frac{1}{20.0855} \approx 0.049787 \)

Now multiply this by \(P_0 = 1 \times 10^5 \text{ Pa}\):

\( P(21 \text{ km}) \approx (1 \times 10^5 \text{ Pa}) \times 0.049787 \)

\( P(21 \text{ km}) \approx 4978.7 \text{ Pa} \)

Rounding this value and expressing it in scientific notation gives:

\( P(21 \text{ km}) \approx 4.98 \times 10^3 \text{ Pa} \)

Looking at the given options:

• Option 1: \(\sim 5 \times 10^3 \text{ Pa}\)
• Option 2: \(\sim 3.3 \times 10^4 \text{ Pa}\)
• Option 3: \(\sim 1.25 \times 10^4 \text{ Pa}\)
• Option 4: \(\sim 1.11 \times 10^4 \text{ Pa}\)

Our calculated value of approximately \(4.98 \times 10^3 \text{ Pa}\) is closest to Option 1, which is \(\sim 5 \times 10^3 \text{ Pa}\). The result is very close to \(5000 \text{ Pa}\).

Revision Table: Key Concepts for Atmospheric Pressure

Additional Information on Scale Height and Atmospheric Pressure

The concept of scale height is a useful approximation for understanding the general behavior of atmospheric pressure with altitude. Here are some important points:

• Assumptions: The formula \(P(h) = P_0 e^{-h/H}\) relies on several assumptions, including an isothermal (constant temperature) atmosphere, constant gravitational acceleration with height, and air behaving as an ideal gas with a uniform composition.
• Real Atmosphere: In reality, the Earth's atmosphere is not isothermal; temperature varies significantly with altitude (e.g., decreasing in the troposphere, increasing in the stratosphere). The composition also changes slightly at very high altitudes, and gravity decreases marginally.
• Variable Scale Height: Because temperature varies with altitude, the actual scale height is not constant. The scale height is larger where the temperature is higher and smaller where the temperature is lower. An average scale height (like the 7 km used in the problem) is often used for simplified calculations or as a representative value for a specific atmospheric layer.
• Pressure Decrease: The exponential formula shows that pressure drops off rapidly with height. For example, at a height equal to the scale height (\(h=H\)), the pressure is \(P_0 e^{-1} \approx P_0 / 2.718\). At twice the scale height (\(h=2H\)), the pressure is \(P_0 e^{-2} \approx P_0 / 7.389\), and at three times the scale height (\(h=3H\)), the pressure is \(P_0 e^{-3} \approx P_0 / 20.085\). In this problem, 21 km is exactly three times the 7 km scale height, so the pressure is approximately \(P_0 / e^3\).
• Impact on Aviation and Weather: The rapid decrease in pressure and density with altitude affects aircraft performance, weather phenomena (most weather occurs in the lower atmosphere), and even mountaineering (requiring acclimatization or oxygen at high altitudes).