(c) The variation in atmospheric pressure P with elevation h above sea level can be expressed as:
P = P0 * e^(-M * g * h / (R * T))
where P0 is the pressure at sea level, e is the base of the natural logarithm, M is the molar mass of air, g is the acceleration due to gravity, R is the ideal gas constant, T is the temperature, and h is the elevation above sea level.
To calculate the atmospheric pressure at the top of Clingmans Dome, we can substitute the given values into the equation:
P0 = 1 atm
T = 20°C = 293.15 K
M = 28.8 * 10^(-3) kg/mol
h = 2025 m
Plugging these values into the equation, we get:
P = 1 atm * e^(-28.8 * 10^(-3) kg/mol * 9.8 m/s^2 * 2025 m / (8.314 J/(mol*K) * 293.15 K))
Simplifying the equation, we get:
P = 1 atm * e^(-0.081 * 2025)
Calculating the exponential term, we get:
P = 1 atm * e^(-164.025)
Finally, we can calculate the atmospheric pressure at the top of Clingmans Dome by evaluating the exponential term and multiplying it by 1 atm:
P ≈ 1 atm * 1.5 * 10^(-71) atm
Therefore, the atmospheric pressure at the top of Clingmans Dome is approximately 1.5 * 10^(-71) atm.