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Boiling point of pure water at elevated altitudes

This online calculator finds the water boiling temperature given the atmospheric pressure in millimeters of mercury.

After the creation of a pressure calculator Pressure units converter and an atmospheric pressure calculator Altitude difference by barometric formula, I wanted to know how to calculate the boiling point according to the altitude. It is common knowledge that at higher altitudes, water boils at a lower temperature. But what's that temperature?

This task consists of two stages - establish the atmospheric pressure dependence on the altitude and dependence of boiling point to pressure.

Boiling is a phase transition of the first order(water changes its physical state from liquid to gas).

Phase transition of the first order is described by Clapeyron equation:
\frac{dP}{dT}=\frac{q_{12}}{T(v_2-v_1)},
where
q_{12} -the specific heat of the phase transition, which is numerically equal to the amount of heat received by a unit of mass for the phase transition.

T - phase transition temperature
v_2 - v_1 - change of the specific volume in the transition

Clasius simplified the Clapeyron equation for the case of evaporation and sublimation, assuming that

  1. The vapor obeys the law of ideal gas
  2. The specific volume of fluid is much smaller than the specific volume of steam
    It comes from paragraph 1 that the state of vapor can be described by the Mendeleev-Clapeyron equation
    PV=\frac{M}{\mu} RT,
    and from the 2nd paragraph - the specific volume of fluid v_1can be neglected.
    Thus, Clapeyron equation takes the following form
    \frac{dP}{dT}=\frac{q_{12}}{Tv},
    where the specific volume can be expressed through
    v=\frac{V}{M}=\frac{RT}{P\mu},
    and finally
    \frac{dP}{dT}=\frac{q_{12}\mu P}{RT^2}
    separating the variables, we obtain
    \frac{1}{P}dP=\frac{q_{12}\mu }{RT^2}dT

By integrating the left part P_1 to P_2 and the right part from T_1 to T_2 i.e. from one point (P_1,T_1) to another (P_2,T_2), lying on the line liquid-vapor equilibrium, we obtain the following equation
lnP_2-lnP_1=\frac{q_{12}\mu }{R}(\frac{1}{T_1}-\frac{1}{T_2})
called the Clausius-Clapeyron equation.

Actually, that is the desired dependence of the boiling temperature of the pressure.

Here are some more transformations
ln\frac{P_2}{P_1}=\frac{q_{12}\mu }{R}(\frac{1}{T_1}-\frac{1}{T_2})
\frac{P_2}{P_1}=e^{\frac{q_{12}\mu }{R}(\frac{1}{T_1}-\frac{1}{T_2})},
where
\mu - molar mass of the water, 18 gram/mol
R -universal gas constant 8.31 J/(mol K)
q_{12} - specific heat of water vaporisation 2.310^6 J / kg

Now we have to to establish the dependence of the altitude to the atmospheric pressure. Here we will use the barometric formula (we don't have any other anyways):
P=P_0e^{\frac{-\mu gh}{RT}}
or
\frac{P}{P_0}=e^{\frac{-\mu gh}{RT}},
where
\mu - molar mass of the air , 29 gram/mol
R - universal gas constant, 8.31 J/(mol K)
g - acceleration of gravity, 9.81 m/(s s)
T - air temperature

We will mark the value relating to the air with index v and relating to the water with index h.
By equating and getting rid of the exponent, we will get
-\frac{\mu_v gh}{RT_v}=\frac{q_{12}\mu_h }{R}(\frac{1}{T_0}-\frac{1}{T_h})

And the final formula is
T_h=\frac{T_0T_vq_{12}\mu_h}{q_{12}\mu_hT_v+\mu_vghT_0}

Of course, the actual air pressure doesn't follow the barometric formula as that with high altitude difference; air temperature can not be considered permanent. Moreover, the gravitational acceleration depends on the geographical latitude, atmospheric pressure, and water vapor concentration. So we can have only the estimated results with this formula. Due to this, I also included another calculator which finds boiling point temperature given the atmospheric pressure, according to ln\frac{P_2}{P_1}=\frac{q_{12}\mu }{R}(\frac{1}{T_1}-\frac{1}{T_2}) formula.

Calculator to find boiling point temperature from altitude:

PLANETCALC, Boiling point dependence on the altitude above sea level

Boiling point dependence on the altitude above sea level

Digits after the decimal point: 1
Boiling point
 

Calculator to find boiling point temperature from pressure:

PLANETCALC, Boiling point dependence on the atmospheric pressure

Boiling point dependence on the atmospheric pressure

Digits after the decimal point: 1
Boiling temperature
 

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PLANETCALC, Boiling point of pure water at elevated altitudes
Timur2021-03-12 16:56:25

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