The normal boiling point of a liquid is 282 °C. At what temperature (in °C) would the vapor pressure be 0.350 atm? (∆Hvap = 28.5 kJ/mol)

The relationship between variables in equilibrium between phases of one component system e.g liquid and vapor, solid and vapor , solid and liquid can be obtained from a thermodynamic relationship called Clapeyron equation.

Clausius- Clapeyron Equation can be put in a more convenient form applicable to vaporization and sublimation equilibria in which one of the two phases is gaseous.

The equation for Clausius- Clapeyron Equation can be expressed as:

[tex]\mathtt{In \dfrac{P_2}{P_1}= \dfrac{\Delta \ H _{vap}}{R} \begin {pmatrix} \dfrac{T_2 -T_1}{T_2 \ T_1} \end {pmatrix} }[/tex]

where ;

[tex]P_1[/tex] is the vapor pressure at temperature 1

[tex]P_ 2[/tex] is the vapor pressure at temperature 2

∆Hvap = enthalpy of vaporization

R = universal gas constant

Given that:

[tex]P_1[/tex] = 1 atm

[tex]P_ 2[/tex] = 0.350 atm

∆Hvap = 28.5 kJ/mol = 28.5 × 10³ J/mol

[tex]T_1[/tex] = 282 °C = (282 + 273) K = 555 K

R = 8.314 J/mol/k

Substituting the above values into the Clausius - Clapeyron equation, we have:

[tex]\mathtt{In \dfrac{P_2}{P_1}= \dfrac{\Delta \ H _{vap}}{R} \begin {pmatrix} \dfrac{T_2 -T_1}{T_2 \ T_1} \end {pmatrix} }[/tex]

[tex]\mathtt{In \begin {pmatrix} \dfrac{0.350}{1} \end {pmatrix} } = \dfrac{28.5 \times 10^3 }{ 8.314 } \begin {pmatrix} \dfrac{T_2 - 555}{555T_2} \end {pmatrix} }[/tex]

[tex]\mathtt{In \begin {pmatrix} \dfrac{0.350}{1} \end {pmatrix} } = \dfrac{28.5 \times 10^3 }{ 8.314 } \begin {pmatrix} \dfrac{1}{555}- \dfrac{1}{T_2} \end {pmatrix} }[/tex]

[tex]- 1.0498= 3427.953 \begin {pmatrix} \dfrac{1}{555}- \dfrac{1}{T_2} \end {pmatrix} }[/tex]

[tex]\dfrac{- 1.0498}{3427.953}= \begin {pmatrix} \dfrac{1}{555}- \dfrac{1}{T_2} \end {pmatrix} }[/tex]

[tex]- 3.06246906 \times 10^{-4}= \begin {pmatrix} \dfrac{1}{555}- \dfrac{1}{T_2} \end {pmatrix} }[/tex]

[tex]\dfrac{1}{T_2} = \begin {pmatrix} \dfrac{1}{555}+ (3.06246906 \times 10^{-4} ) \end {pmatrix} }[/tex]

[tex]\dfrac{1}{T_2} = 0.002108048708[/tex]

[tex]T_2 = \dfrac{1}{0.002108048708}[/tex]

[tex]\mathbf{T_2 }[/tex] = 474.37 K

To °C ; we have [tex]\mathbf{T_2 }[/tex] = (474.37 - 273)°C

[tex]\mathbf{T_2 }[/tex] = 201.37 °C

Thus, the temperature at which the vapor pressure would be 0.350 atm is 201.37 °C

onyebuchinnaji onyebuchinnaji · Answer 2 · 2021-10-24T03:59:41+02:00

The temperature of the liquid at the given vapor pressure is 201.5 ⁰C.

The given parameters;

boiling point temperature, = 282 ⁰C
vapor pressure, P₂ = 0.35 atm
enthalpy of vaporization, ∆Hvap = 28.5 kJ/mol

The temperature of the liquid will be determined by applying Clausius- Clapeyron Equation;

[tex]ln(\frac{P_2}{P_1} ) = \frac{\Delta H}{R} (\frac{T_2 -T_1}{T_1T_2} )[/tex]

where;

R is ideal gas constant = 8.314 J/mol.k
T₁ is the initial temperature in Kelvin = 282 + 273 = 555 K
P₁ is the initial pressure = 1 atm

[tex]ln(\frac{P_2}{P_1} ) = \frac{\Delta H}{R} (\frac{T_2 -T_1}{T_1T_2} )\\\\ln(\frac{0.35}{1} ) = \frac{28.5 \times 10^3}{8.314} (\frac{T_2 - 555}{555T_2} )\\\\-1.049 = 6.176- \frac{3427.95}{T_2} \\\\\frac{3427.95}{T_2} = 6.176 + 1.049\\\\\frac{3427.95}{T_2} = 7.225\\\\T_2 = \frac{3427.95}{7.225} \\\\T_2 = 474.5 \ K\\\\T_2 = 474.5 - 273 = 201.5 \ ^0C[/tex]

Thus, the temperature of the liquid at the given vapor pressure is 201.5 ⁰C.

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