Chapter 7, Problem 7.84E

Interpretation Introduction

Interpretation:

The equation 7.49 is to be derived.

Concept introduction:

When a non-volatile solute is added to a solvent, the boiling point of the solution gets raised in comparison to the pure solvent. This phenomenon is known as elevation in boiling point. Elevation in boiling point is a colligative property.

Answer to Problem 7.84E

The equation 7.41 can be represented for the vaporization process as shown below.

$\ln x_{\text{solvent}}=-\frac{{\Delta }_{\text{vap}}H}{R}\left(\frac{1}{T}-\frac{1}{T_{\text{BP}}}\right)$

Where,

• $x_{\text{solvent}}$ is the mole fraction of the solvent in the solution.

• ${\Delta }_{\text{vap}}H$ is the enthalpy change for vaporization process.

• $T_{\text{BP}}$ is the boiling point of solvent.

The $x_{\text{solvent}}$ can be written in the form of solute as $x_{\text{solvent}}=\left(1-x_{\text{solute}}\right)$. Substitute the value of $x_{\text{solvent}}$ with $\left(1-x_{\text{solute}}\right)$ as shown below,

$\ln \left(1-x_{\text{solute}}\right)=-\frac{{\Delta }_{\text{vap}}H}{R}\left(\frac{1}{T}-\frac{1}{T_{\text{BP}}}\right)$

Applying the Taylor-series expansion on $\ln \left(1-x\right)\approx -x$ and substitute the $\ln \left(1-x_{\text{solute}}\right)$ with $-x_{\text{solute}}$ as shown below.

$x_{\text{solute}}\approx \frac{{\Delta }_{\text{vap}}H}{R}\left(\frac{1}{T}-\frac{1}{T_{\text{BP}}}\right)$

Rearrange the temperature terms as shown below.

$x_{\text{solute}}\approx \frac{{\Delta }_{\text{vap}}H}{R}\left(\frac{T_{\text{BP}}-T}{T.T_{\text{BP}}}\right)$

The term $T.T_{\text{BP}}$ can be approximated to $T_{\text{BP}}{}^2$ as dilute solution are considered in this case and the temperature at equilibrium is not much different from boiling point. Also $T_{\text{BP}}-T$ can be rewritten as $\Delta T_{\text{BP}}$ as shown below.

$x_{\text{solute}}\approx \frac{{\Delta }_{\text{vap}}H}{R}\left(\frac{\Delta T_{\text{BP}}}{T_{\text{BP}}{}^2}\right)$

The relationship between molality and mole fraction is,

$m_{\text{solute}}=\frac{1000\cdot x_{\text{solute}}}{x_{\text{solvent}}\cdot M_{\text{solvent}}}$

Where,

• $m_{\text{solute}}$ is molality of solute.

• $x_{\text{solute}}$and $x_{\text{solvent}}$ is mole fraction of solute and solvent.

• $M_{\text{solvent}}$ is the molecular weight of solvent.

Substitute the value of $x_{\text{solute}}$ in the given equation and rearrange for the value of $\Delta T_{\text{BP}}$ as shown below.

$\Delta T_{\text{BP}}\approx \left(\frac{M_{\text{solvent}}R{T}_{\text{BP}}{}^2}{{\Delta }_{\text{vap}}H\cdot 1000}\right)m_{\text{solute}}$

Explanation of Solution

The equation 7.41 can be represented for the vaporization process as shown below.

$\ln x_{\text{solvent}}=-\frac{{\Delta }_{\text{vap}}H}{R}\left(\frac{1}{T}-\frac{1}{T_{\text{BP}}}\right)$

Where,

• $x_{\text{solvent}}$ is the mole fraction of the solvent in the solution.

• ${\Delta }_{\text{vap}}H$ is the enthalpy change for vaporization process.

• $T_{\text{BP}}$ is the boiling point of solvent.

The $x_{\text{solvent}}$ can be written in the form of solute as $x_{\text{solvent}}=\left(1-x_{\text{solute}}\right)$. Substitute the value of $x_{\text{solvent}}$ with $\left(1-x_{\text{solute}}\right)$ as shown below,

$\ln \left(1-x_{\text{solute}}\right)=-\frac{{\Delta }_{\text{vap}}H}{R}\left(\frac{1}{T}-\frac{1}{T_{\text{BP}}}\right)$

Applying the Taylor-series expansion on $\ln \left(1-x\right)\approx -x$ and substitute the $\ln \left(1-x_{\text{solute}}\right)$ with $-x_{\text{solute}}$ as shown below.

$x_{\text{solute}}\approx \frac{{\Delta }_{\text{vap}}H}{R}\left(\frac{1}{T}-\frac{1}{T_{\text{BP}}}\right)$

Rearrange the temperature terms as shown below.

$x_{\text{solute}}\approx \frac{{\Delta }_{\text{vap}}H}{R}\left(\frac{T_{\text{BP}}-T}{T.T_{\text{BP}}}\right)$

The term $T.T_{\text{BP}}$ can be approximated to $T_{\text{BP}}{}^2$ as dilute solution are considered in this case and the temperature at equilibrium is not much different from boiling point. Also $T_{\text{BP}}-T$ can be rewritten as $\Delta T_{\text{BP}}$ as shown below.

$x_{\text{solute}}\approx \frac{{\Delta }_{\text{vap}}H}{R}\left(\frac{\Delta T_{\text{BP}}}{T_{\text{BP}}{}^2}\right)$

The relationship between molality and mole fraction is,

$m_{\text{solute}}=\frac{1000\cdot x_{\text{solute}}}{x_{\text{solvent}}\cdot M_{\text{solvent}}}$

Where,

• $m_{\text{solute}}$ is molality of solute.

• $x_{\text{solute}}$and $x_{\text{solvent}}$ is mole fraction of solute and solvent.

• $M_{\text{solvent}}$ is the molecular weight of solvent.

Substitute the value of $x_{\text{solute}}$ in the given equation and rearrange for the value of $\Delta T_{\text{BP}}$ as shown below.

$\Delta T_{\text{BP}}\approx \left(\frac{M_{\text{solvent}}R{T}_{\text{BP}}{}^2}{{\Delta }_{\text{vap}}H\cdot 1000}\right)m_{\text{solute}}$

Conclusion

The derivation of equation of 7.49 has been rightfully stated.