Mt. McKinley in Alaska has an altitude of 20,320 ft. Water (enthalpy of vaporization=40.7kJ/mol) boils at 77C atop Mt. McKinley. What is the normal atmospheric pressure at the summit?
Use the Clausius-Clapeyron equation.
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To find the normal atmospheric pressure at the summit of Mt. McKinley, we can use the relationship between boiling point and atmospheric pressure. The boiling point of a substance depends on the pressure exerted on it. As the altitude increases, atmospheric pressure decreases, leading to a lower boiling point for water.
We can use the Clausius-Clapeyron equation to determine the boiling point of water at a given altitude:
ln(P₂/P₁) = (ΔH_vap/R) * ((1/T₁) - (1/T₂))
Where:
P₁ = initial pressure (at sea level) = 1 atm
P₂ = final pressure (at the summit of Mt. McKinley)
ΔH_vap = enthalpy of vaporization of water = 40.7 kJ/mol
R = ideal gas constant = 0.008314 J/(mol·K)
T₁ = initial temperature (boiling point at sea level) = 100°C
T₂ = final temperature (boiling point at Mt. McKinley) = 77°C
Let's plug in the values and solve for P₂:
ln(P₂/1 atm) = (40.7 kJ/mol / 0.008314 J/(mol·K)) * ((1/373 K) - (1/350 K))
Simplifying:
ln(P₂/1 atm) = 1627.9 - 1451.8
ln(P₂/1 atm) = 176.1
Now, we can take the exponential of both sides to solve for P₂:
P₂/1 atm = e^(176.1)
P₂ = e^(176.1) atm
Using a calculator:
P₂ ≈ 3.54 × 10^76 atm
It is important to note that the final value of atmospheric pressure at the summit of Mt. McKinley seems very high. This could be due to a mistake in the calculations or an extreme approximation. It is always good practice to double-check the numbers and consider the units used.