Question

If at `"71226 Pa"` of pressure, ethoxyethane boils at `25^@ "C"`, then what vapor pressure is needed to boil at `78^@ "C"` if across the temperature range we can assume that `DeltaH_"vap" = "29.1 kJ/mol"`?

Answer 1

`"418897 Pa"`, or `"4.13 atm"`.

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Recall the Clausius-Clapeyron equation (at least the name):

`ln(P_2/P_1) = -(DeltaH_"vap")/(R)[1/T_2 - 1/T_1]`

where:

• `P` is the vapor pressure of the liquid at the particular temperature `T`.
• `DeltaH_"vap"` is the enthalpy for the vaporization process.
• `R = "8.314472 J/mol"cdot"K"` is the universal gas constant.

Basically, this equation describes how to determine vapor pressure at a new temperature. You were given the following info:

• `DeltaH_"vap" = "29.1 kJ/mol"`
• `P_1 = "71226 Pa"` at `T_1 = 25.0^@ "C"`
• `P_2 = ???` at `T_2 = 78.0^@" C"`

The `DeltaH_"vap"` was written as `DeltaH_"vap"^@` because `T_1` happened to be the same temperature as defined for standard thermodynamic temperature and pressure (`25^@ "C"` and `"1 atm"`, or sometimes `"1 bar"` in some newer texts).

Anyways, to calculate the new vapor pressure, we arbitrarily chose `P_1` for the given pressure and are to calculate `P_2`.

`P_2/P_1 = "exp"[-(DeltaH_"vap")/(R)[1/T_2 - 1/T_1]]`

`P_2 = P_1"exp"[-(DeltaH_"vap")/(R)[1/T_2 - 1/T_1]]`

where `"exp"("stuff")` is the function `e^("stuff")` to keep the equation from looking too small.

So, plug stuff in to get:

`color(blue)(P_2) = ("71226 Pa")"exp"[-(29.1 cancel"kJ/mol")/(0.008314472 cancel("kJ/mol")cdotcancel"K")[1/(78.0 + 273.15 cancel"K") - 1/(25.0 + 273.15 cancel"K")]]`

`=` `color(blue)("418897 Pa")`

That means the vapor pressure increased at a higher temperature.

This should make sense because a higher temperature implies a higher average kinetic energy, so the molecules at the surface of the solution can escape the solution more easily, increasing the vapor pressure above the solution.

In `"torr"` this would be `"3141.99 torr"`, which is about `"4.13 atm"`, so this is rather unusual. But it make sense because the boiling point of ethoxyethane is about `34^@ "C"`, and we're well past that.