Pakistan's K2 is the world's second-tallest mountain, with an altitude of 28,251 ft. Its base camp, where climbers stop to acclimate, is located about 16,400 ft above sea level. (a) Approximate atmospheric pressure P at different altitudes is given by the equation P = e^-h/7000, where P is in atmospheres and h is the altitude in meters. What is the approximate atmospheric pressure in mm Hg at K2 base camp? (b) What is the atmospheric pressure in mm Hg at the summit of K2?

Verified step by step guidance

1. Convert the altitude of K2 base camp from feet to meters. Since 1 ft is approximately 0.3048 meters, multiply the altitude in feet by 0.3048 to get the altitude in meters.
2. Use the given atmospheric pressure equation P = e^-h/7000, where P is the atmospheric pressure in atmospheres and h is the altitude in meters. Substitute the altitude in meters for h in the equation.
3. Calculate the value of P by evaluating the expression e^-h/7000 using the altitude in meters.
4. Convert the atmospheric pressure from atmospheres to millimeters of mercury (mm Hg). Since 1 atmosphere is approximately equal to 760 mm Hg, multiply the pressure in atmospheres by 760 to get the pressure in mm Hg.
5. The result from the previous step gives the approximate atmospheric pressure at K2 base camp in mm Hg.

Key Concepts

Atmospheric Pressure

Atmospheric pressure is the force exerted by the weight of air above a given point. It decreases with increasing altitude due to the lower density of air at higher elevations. At sea level, standard atmospheric pressure is approximately 101.3 kPa or 760 mm Hg. Understanding how pressure changes with altitude is crucial for calculations involving high-altitude environments like K2.

Exponential Decay

The equation P = e^(-h/7000) represents an exponential decay function, where P is the atmospheric pressure in atmospheres and h is the altitude in meters. This model indicates that as altitude increases, the pressure decreases exponentially. This concept is essential for predicting how pressure changes with height, particularly in mountainous regions.

Unit Conversion

Unit conversion is the process of converting a quantity expressed in one set of units to another. In this context, it is necessary to convert atmospheric pressure from atmospheres to mm Hg for the final answer. Knowing the conversion factor (1 atm = 760 mm Hg) allows for accurate calculations and comparisons of pressure at different altitudes.