Answer: To calculate the height of the mountain, you can use the fact that atmospheric pressure decreases with increasing altitude. This relationship is described by the equation:
P = P0 * e^(-h/H)
where P is the atmospheric pressure at a given altitude, P0 is the atmospheric pressure at sea level, h is the altitude, and H is a constant known as the scale height.
Since the atmospheric pressure at the top of the mountain is known (760mmHg), and the atmospheric pressure at sea level is a standard value (approximately 101325 Pa), we can rearrange the above equation to solve for h:
h = -H * ln(P/P0)
Plugging in the known values, we have:
h = -H * ln(760mmHg / 101325 Pa)
= -H * ln(0.00750062)
The scale height of the atmosphere can be calculated using the ideal gas law:
H = RT / g
where R is the universal gas constant, T is the temperature of the air, and g is the acceleration due to gravity.
Plugging in the known values, we have:
H = (8.31 J/mol*K) * (288 K) / (9.81 m/s^2)
= 7478.9 m
Substituting this value back into the equation for h gives:
h = -7478.9 m * ln(0.00750062)
= -7478.9 m * (-3.876)
= 28934.7 m
Therefore, the height of the mountain is approximately 28,934 meters.