On top of one of the peaks in Rocky Mountain National Park the pressure of the atmosphere is 500. torr. Determine the boiling point of water at this location. The heat of vaporization of water is 40.7 kj/mol.

Use the Clausius-Clapeyron equation. Remember that the boiling point is the temperature at which the vapor pressure of the liquid is equal to the atmospheric pressure.

but there aren't enough information to use Clausius-Clapeyron equation

Why do you think I gave you the hint when I responded? You're supposed to remember that vapor pressure of water at 1 atm (760 torr) is 760 torr when it boils at 100 C.

ln(P2/P1) = (DHvap/R)(1/T1 - 1/T2)
P1 = 760 torr; T1 = 100 C
p2 = 500 torr; T2 = ??
DHvap = given in problem.
R is 8.314 and don't forget T must be in kelvin.
Solve for T2.

To determine the boiling point of water at a certain pressure, we can use the Clausius-Clapeyron equation:

ln(P1/P2) = (ΔHvap/R) * (1/T2 - 1/T1)

Where P1 and P2 are the initial and final pressures, ΔHvap is the heat of vaporization, R is the ideal gas constant, and T1 and T2 are the initial and final temperatures.

In this case, P1 is the pressure at sea level, which is typically 760 torr, and P2 is the given pressure of 500 torr. The heat of vaporization (ΔHvap) is given as 40.7 kJ/mol. The ideal gas constant (R) is approximately 0.0821 L*atm/(mol*K).

We need to solve for T2, which represents the boiling point of water at the given pressure of 500 torr.

1. Convert the given pressure to atm:
Given pressure = 500 torr.
Conversion: 1 atm = 760 torr
Pressure at peak = 500 torr / 760 torr/atm = 0.658 atm

2. Substitute the values into the Clausius-Clapeyron equation:
ln(1/0.658) = (40.7 kJ/mol / (0.0821 L*atm/(mol*K))) * (1/T2 - 1/373 K)

3. Solve for T2:
Rearrange the equation:
ln(1/0.658) = (40.7/0.0821) * (1/T2 - 1/373)
Divide both sides by (40.7/0.0821):
ln(1/0.658) / (40.7/0.0821) = 1/T2 - 1/373
Simplify the left side using a calculator to find ln(1/0.658) / (40.7/0.0821) ≈ 0.00786

Substitute the values back in:
0.00786 = 1/T2 - 1/373
Multiply through by 373T2:
0.00786 * 373T2 = 373 - T2
Distribute and rearrange:
2.928T2 - 0.00786T2² = 373
Rearrange in standard quadratic form:
0.00786T2² - 2.928T2 + 373 = 0

4. Solve the quadratic equation for T2:
Using a quadratic equation solver or calculator, we find that T2 ≈ 373.6 K

Therefore, the boiling point of water at the given pressure of 500 torr on top of the peak in Rocky Mountain National Park is approximately 373.6 K.