Chapter 6.6, Problem 21P

A)

Interpretation Introduction

Interpretation:

To determine the coefficient of thermal expansion and isothermal compressibility for the given condition $P\text{=100bar,}T{\text{=100}}^{\text{o}}\text{C}$.

Concept introduction:

Coefficient of thermal expansion:

The change in length of an object with unit degree increase in temperature at constant pressure is known as coefficient of thermal expansion.

The formula to calculate the coefficient of thermal expansion $\left({\alpha }_V\right)$ is given by the equation:

${\alpha }_V=\frac{1}{\underset{\_}{V}}{\left(\frac{\partial \underset{\_}{V}}{\partial T}\right)}_P$

Here, molar volume is $\underset{\_}{V}$, and change in molar volume and change in temperature at constant pressure is ${\left(\partial \underset{\_}{V}\right)}_P,\text{and}{\left(\partial T\right)}_P$ respectively.

Isothermal compressibility:

Isothermal compressibility is the reciprocal of the bulb modulus.

The formula to calculate the isothermal compressibility $\left({\kappa }_T\right)$ is given by:

${\kappa }_T=-\frac{1}{\underset{\_}{V}}{\left(\frac{\partial \underset{\_}{V}}{\partial P}\right)}_T$

Here, change in molar volume and change in pressure at constant temperature is ${\left(\partial \underset{\_}{V}\right)}_T,\text{and}{\left(\partial P\right)}_T$ respectively.

Explanation of Solution

The formula to calculate the coefficient of thermal expansion $\left({\alpha }_V\right)$ is given by the equation:

${\alpha }_V=\frac{1}{\underset{\_}{V}}{\left(\frac{\partial \underset{\_}{V}}{\partial T}\right)}_P$        (1)

Here, molar volume is $\underset{\_}{V}$, and change in molar volume and change in temperature at constant pressure is ${\left(\partial \underset{\_}{V}\right)}_P,\text{and}{\left(\partial T\right)}_P$ respectively.

The equation (1) can be rewritten as:

${\alpha }_V=\frac{1}{{\underset{\_}{V}}_{givenvalue}}\left(\frac{{\underset{\_}{V}}_2-{\underset{\_}{V}}_1}{T_2-T_1}\right)$        (2)

Here, given value of molar volume is ${\underset{\_}{V}}_{givenvalue}$, final molar volume is ${\underset{\_}{V}}_2$, initial molar volume is ${\underset{\_}{V}}_1$, final temperature is $T_2$, and initial temperature is $T_1$.

The formula to calculate the isothermal compressibility $\left({\kappa }_T\right)$ is given by:

${\kappa }_T=-\frac{1}{\underset{\_}{V}}{\left(\frac{\partial \underset{\_}{V}}{\partial P}\right)}_T$        (3)

Here, change in molar volume and change in pressure at constant temperature is ${\left(\partial \underset{\_}{V}\right)}_T,\text{and}{\left(\partial P\right)}_T$ respectively.

The equation (3) can be rewritten as:

${\kappa }_T=-\frac{1}{{\underset{\_}{V}}_{givenvalue}}\left(\frac{{\underset{\_}{V}}_2-{\underset{\_}{V}}_1}{P_2-P_1}\right)$        (4)

Here, final pressure is $P_2$, and initial pressure is $P_1$.

Take initial and final temperature as $\text{80°C}\text{and}\text{120°C}$ because in the compressed liquid tables, Table A-4, “compressed liquid” the preceding and following temperature of $\text{100°C}$ are $\text{80°}\text{and}\text{120°C}$.

$\begin{array}{l}{T}_1=\text{80°}\\ T_2=\text{120°C}\end{array}$

Referring to appendix Table A-4, “compressed liquid”, the value of initial molar volume $\left({\underset{\_}{V}}_1\right)$ corresponding to initial temperature of $\text{80°C}$ and pressure of $\text{100}\text{bar}$ is obtained as:

$\begin{array}{c}{\underset{\_}{V}}_1=0.001024{\text{m}}^{\text{3}}/\text{kg}\\ =0.001024{\text{m}}^{\text{3}}/\text{kg}\left(0.01802\text{kg}/\text{mol}\right)\\ =1.85\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}\end{array}$

Referring to appendix Table A-4, “compressed liquid”, the value of final molar volume $\left({\underset{\_}{V}}_2\right)$ corresponding to final temperature of $\text{120°C}$ and pressure of $\text{100}\text{bar}$ is obtained as:

$\begin{array}{c}{\underset{\_}{V}}_2=0.001055{\text{m}}^{\text{3}}/\text{kg}\\ =0.001055{\text{m}}^{\text{3}}/\text{kg}\left(0.01802\text{kg}/\text{mol}\right)\\ =1.90\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}\end{array}$

Referring to appendix Table A-4, “compressed liquid”, the value of given value of molar volume $\left({\underset{\_}{V}}_{givenvalue}\right)$ corresponding to given temperature of $\text{100°C}$ and pressure of $\text{100}\text{bar}$ is obtained as:

$\begin{array}{c}{\underset{\_}{V}}_{givenvalue}=0.001038{\text{m}}^{\text{3}}/\text{kg}\\ =0.001038{\text{m}}^{\text{3}}/\text{kg}\left(0.01802\text{kg}/\text{mol}\right)\\ =1.87\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}\end{array}$

Substitute ${\underset{\_}{V}}_{givenvalue}=1.87\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}$, ${\underset{\_}{V}}_2=1.90\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}$, ${\underset{\_}{V}}_1=1.85\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}$, $T_1=\text{80°C}$, and $T_2=\text{120°C}$ in Equation (2).

$\begin{array}{c}{\alpha }_V=\frac{1}{{\underset{\_}{V}}_{givenvalue}}\left(\frac{{\underset{\_}{V}}_2-{\underset{\_}{V}}_1}{T_2-T_1}\right)\\ =\frac{1}{1.87\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}}\left(\frac{1.90\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}-1.85\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}}{\text{120°C}-8\text{0°C}}\right)\\ =6.684\times {10}^{-4}{\text{K}}^{-1}\end{array}$

The coefficient of thermal expansion $\left({\alpha }_V\right)$ is $6.684\times {10}^{-4}{\text{K}}^{-1}$.

Take the initial and final pressure as $\text{50}\text{bar}\text{and}\text{150}\text{bar}$ from the compressed liquid tables, Table A-4, “compressed liquid”.

$\begin{array}{l}{P}_1=\text{50}\text{bar}\\ P_2=1\text{50}\text{bar}\end{array}$

Referring to appendix Table A-4, “compressed liquid”, the value of initial molar volume $\left({\underset{\_}{V}}_1\right)$ corresponding to initial pressure of $\text{50}\text{bar}$ and temperature of $\text{100°C}$ is obtained as:

$\begin{array}{c}{\underset{\_}{V}}_1=0.001041{\text{m}}^{\text{3}}/\text{kg}\\ =0.001041{\text{m}}^{\text{3}}/\text{kg}\left(0.01802\text{kg}/\text{mol}\right)\\ =1.88\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}\end{array}$

Referring to appendix Table A-4, “compressed liquid”, the value of final molar volume $\left({\underset{\_}{V}}_2\right)$ corresponding to final pressure of $\text{150}\text{bar}$ and temperature of $\text{100°C}$ is obtained as:

$\begin{array}{c}{\underset{\_}{V}}_2=0.001036{\text{m}}^{\text{3}}/\text{kg}\\ =0.001036{\text{m}}^{\text{3}}/\text{kg}\left(0.01802\text{kg}/\text{mol}\right)\\ =1.87\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}\end{array}$

Referring to appendix Table A-4, “compressed liquid”, the value of given value of molar volume $\left({\underset{\_}{V}}_{givenvalue}\right)$ corresponding to given pressure of $\text{100}\text{bar}$ and temperature of $\text{100°C}$ is obtained as:

$\begin{array}{c}{\underset{\_}{V}}_{givenvalue}=0.001038{\text{m}}^{\text{3}}/\text{kg}\\ =0.001038{\text{m}}^{\text{3}}/\text{kg}\left(0.01802\text{kg}/\text{mol}\right)\\ =1.87\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}\end{array}$

Substitute ${\underset{\_}{V}}_{givenvalue}=1.87\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}$, ${\underset{\_}{V}}_2=1.87\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}$, ${\underset{\_}{V}}_1=1.88\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}$, $P_1=\text{50}\text{bar}$, and $P_2=\text{150}\text{bar}$ in Equation (4).

$\begin{array}{c}{\kappa }_T=-\frac{1}{{\underset{\_}{V}}_{givenvalue}}\left(\frac{{\underset{\_}{V}}_2-{\underset{\_}{V}}_1}{P_2-P_1}\right)\\ =-\frac{1}{1.87\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}}\left(\frac{1.87\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}-1.88\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}}{\text{150}\text{bar}-\text{50}\text{bar}}\right)\\ =5.348\times {10}^{-5}{\text{bar}}^{-1}\end{array}$

The isothermal compressibility $\left({\kappa }_T\right)$ is $5.348\times {10}^{-5}{\text{bar}}^{-1}$.

B)

Interpretation Introduction

Interpretation:

To determine the coefficient of thermal expansion and isothermal compressibility for the given condition $P\text{=200bar,}T{\text{=100}}^{\text{o}}\text{C}$.

Concept introduction:

Coefficient of thermal expansion:

The change in length of an object with unit degree increase in temperature at constant pressure is known as coefficient of thermal expansion.

The formula to calculate the coefficient of thermal expansion $\left({\alpha }_V\right)$ is given by the equation:

${\alpha }_V=\frac{1}{\underset{\_}{V}}{\left(\frac{\partial \underset{\_}{V}}{\partial T}\right)}_P$

Here, molar volume is $\underset{\_}{V}$, and change in molar volume and change in temperature at constant pressure is ${\left(\partial \underset{\_}{V}\right)}_P,\text{and}{\left(\partial T\right)}_P$ respectively.

Isothermal compressibility:

Isothermal compressibility is the reciprocal of the bulb modulus.

The formula to calculate the isothermal compressibility $\left({\kappa }_T\right)$ is given by:

${\kappa }_T=-\frac{1}{\underset{\_}{V}}{\left(\frac{\partial \underset{\_}{V}}{\partial P}\right)}_T$

Here, change in molar volume and change in pressure at constant temperature is ${\left(\partial \underset{\_}{V}\right)}_T,\text{and}{\left(\partial P\right)}_T$ respectively.

Explanation of Solution

The formula to calculate the coefficient of thermal expansion $\left({\alpha }_V\right)$ is given by the equation:

${\alpha }_V=\frac{1}{\underset{\_}{V}}{\left(\frac{\partial \underset{\_}{V}}{\partial T}\right)}_P$        (1)

Here, molar volume is $\underset{\_}{V}$, and change in molar volume and change in temperature at constant pressure is ${\left(\partial \underset{\_}{V}\right)}_P,\text{and}{\left(\partial T\right)}_P$ respectively.

The equation (1) can be rewritten as:

${\alpha }_V=\frac{1}{{\underset{\_}{V}}_{givenvalue}}\left(\frac{{\underset{\_}{V}}_2-{\underset{\_}{V}}_1}{T_2-T_1}\right)$        (2)

Here, given value of molar volume is ${\underset{\_}{V}}_{givenvalue}$, final molar volume is ${\underset{\_}{V}}_2$, initial molar volume is ${\underset{\_}{V}}_1$, final temperature is $T_2$, and initial temperature is $T_1$.

The formula to calculate the isothermal compressibility $\left({\kappa }_T\right)$ is given by:

${\kappa }_T=-\frac{1}{\underset{\_}{V}}{\left(\frac{\partial \underset{\_}{V}}{\partial P}\right)}_T$        (3)

Here, change in molar volume and change in pressure at constant temperature is ${\left(\partial \underset{\_}{V}\right)}_T,\text{and}{\left(\partial P\right)}_T$ respectively.

The equation (3) can be rewritten as:

${\kappa }_T=-\frac{1}{{\underset{\_}{V}}_{givenvalue}}\left(\frac{{\underset{\_}{V}}_2-{\underset{\_}{V}}_1}{P_2-P_1}\right)$        (4)

Here, final pressure is $P_2$, and initial pressure is $P_1$.

Take initial and final temperature as $\text{80°C}\text{and}\text{120°C}$ because in the compressed liquid tables, Table A-4, “compressed liquid” the preceding and following temperature of $\text{100°C}$ are $\text{80°}\text{and}\text{120°C}$.

$\begin{array}{l}{T}_1=\text{80°}\\ T_2=\text{120°C}\end{array}$

Referring to appendix Table A-4, “compressed liquid”, the value of initial molar volume $\left({\underset{\_}{V}}_1\right)$ corresponding to initial temperature of $\text{80°C}$ and pressure of $\text{200}\text{bar}$ is obtained as:

$\begin{array}{c}{\underset{\_}{V}}_1=0.001020{\text{m}}^{\text{3}}/\text{kg}\\ =0.001020{\text{m}}^{\text{3}}/\text{kg}\left(0.01802\text{kg}/\text{mol}\right)\\ =1.84\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}\end{array}$

Referring to appendix Table A-4, “compressed liquid”, the value of final molar volume $\left({\underset{\_}{V}}_2\right)$ corresponding to final temperature of $\text{120°C}$ and pressure of $\text{200}\text{bar}$ is obtained as:

$\begin{array}{c}{\underset{\_}{V}}_2=0.001050{\text{m}}^{\text{3}}/\text{kg}\\ =0.001050{\text{m}}^{\text{3}}/\text{kg}\left(0.01802\text{kg}/\text{mol}\right)\\ =1.89\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}\end{array}$

Referring to appendix Table A-4, “compressed liquid”, the value of given value of molar volume $\left({\underset{\_}{V}}_{givenvalue}\right)$ corresponding to given temperature of $\text{100°C}$ and pressure of $\text{200}\text{bar}$ is obtained as:

$\begin{array}{c}{\underset{\_}{V}}_{givenvalue}=0.001034{\text{m}}^{\text{3}}/\text{kg}\\ =0.001034{\text{m}}^{\text{3}}/\text{kg}\left(0.01802\text{kg}/\text{mol}\right)\\ =1.86\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}\end{array}$

Substitute ${\underset{\_}{V}}_{givenvalue}=1.86\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}$, ${\underset{\_}{V}}_2=1.89\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}$, ${\underset{\_}{V}}_1=1.84\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}$, $T_1=\text{80°C}$, and $T_2=\text{120°C}$ in Equation (2).

$\begin{array}{c}{\alpha }_V=\frac{1}{{\underset{\_}{V}}_{givenvalue}}\left(\frac{{\underset{\_}{V}}_2-{\underset{\_}{V}}_1}{T_2-T_1}\right)\\ =\frac{1}{1.86\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}}\left(\frac{1.89\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}-1.84\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}}{\text{120°C}-8\text{0°C}}\right)\\ =6.720\times {10}^{-4}{\text{K}}^{-1}\end{array}$

The coefficient of thermal expansion $\left({\alpha }_V\right)$ is $6.720\times {10}^{-4}{\text{K}}^{-1}$.

Take the initial and final pressure as $\text{150}\text{bar}\text{and}2\text{50}\text{bar}$ from the compressed liquid tables, Table A-4, “compressed liquid”.

$\begin{array}{l}{P}_1=1\text{50}\text{bar}\\ P_2=2\text{50}\text{bar}\end{array}$

Referring to appendix Table A-4, “compressed liquid”, the value of initial molar volume $\left({\underset{\_}{V}}_1\right)$ corresponding to initial pressure of $\text{150}\text{bar}$ and temperature of $\text{100°C}$ is obtained as:

$\begin{array}{c}{\underset{\_}{V}}_1=0.001036{\text{m}}^{\text{3}}/\text{kg}\\ =0.001036{\text{m}}^{\text{3}}/\text{kg}\left(0.01802\text{kg}/\text{mol}\right)\\ =1.87\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}\end{array}$

Referring to appendix Table A-4, “compressed liquid”, the value of final molar volume $\left({\underset{\_}{V}}_2\right)$ corresponding to final pressure of $\text{250}\text{bar}$ and temperature of $\text{100°C}$ is obtained as:

$\begin{array}{c}{\underset{\_}{V}}_2=0.001031{\text{m}}^{\text{3}}/\text{kg}\\ =0.001031{\text{m}}^{\text{3}}/\text{kg}\left(0.01802\text{kg}/\text{mol}\right)\\ =1.86\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}\end{array}$

Referring to appendix Table A-4, “compressed liquid”, the value of given value of molar volume $\left({\underset{\_}{V}}_{givenvalue}\right)$ corresponding to given pressure of $\text{200}\text{bar}$ and temperature of $\text{100°C}$ is obtained as:

$\begin{array}{c}{\underset{\_}{V}}_{givenvalue}=0.001034{\text{m}}^{\text{3}}/\text{kg}\\ =0.001034{\text{m}}^{\text{3}}/\text{kg}\left(0.01802\text{kg}/\text{mol}\right)\\ =1.86\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}\end{array}$

Substitute ${\underset{\_}{V}}_{givenvalue}=1.86\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}$, ${\underset{\_}{V}}_2=1.86\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}$, ${\underset{\_}{V}}_1=1.87\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}$, $P_1=\text{150}\text{bar}$, and $P_2=\text{250}\text{bar}$ in Equation (4).

$\begin{array}{c}{\kappa }_T=-\frac{1}{{\underset{\_}{V}}_{givenvalue}}\left(\frac{{\underset{\_}{V}}_2-{\underset{\_}{V}}_1}{P_2-P_1}\right)\\ =-\frac{1}{1.86\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}}\left(\frac{1.86\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}-1.87\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}}{\text{250}\text{bar}-1\text{50}\text{bar}}\right)\\ =5.37\times {10}^{-5}{\text{bar}}^{-1}\end{array}$

The isothermal compressibility $\left({\kappa }_T\right)$ is $5.37\times {10}^{-5}{\text{bar}}^{-1}$.

C)

Interpretation Introduction

Interpretation:

To determine the coefficient of thermal expansion and isothermal compressibility for the given condition $P\text{=100bar,}T{\text{=200}}^{\text{o}}\text{C}$.

Concept introduction:

Coefficient of thermal expansion:

The change in length of an object with unit degree increase in temperature at constant pressure is known as coefficient of thermal expansion.

The formula to calculate the coefficient of thermal expansion $\left({\alpha }_V\right)$ is given by the equation:

${\alpha }_V=\frac{1}{\underset{\_}{V}}{\left(\frac{\partial \underset{\_}{V}}{\partial T}\right)}_P$

Here, molar volume is $\underset{\_}{V}$, and change in molar volume and change in temperature at constant pressure is ${\left(\partial \underset{\_}{V}\right)}_P,\text{and}{\left(\partial T\right)}_P$ respectively.

Isothermal compressibility:

Isothermal compressibility is the reciprocal of the bulb modulus.

The formula to calculate the isothermal compressibility $\left({\kappa }_T\right)$ is given by:

${\kappa }_T=-\frac{1}{\underset{\_}{V}}{\left(\frac{\partial \underset{\_}{V}}{\partial P}\right)}_T$

Here, change in molar volume and change in pressure at constant temperature is ${\left(\partial \underset{\_}{V}\right)}_T,\text{and}{\left(\partial P\right)}_T$ respectively.

Explanation of Solution

The formula to calculate the coefficient of thermal expansion $\left({\alpha }_V\right)$ is given by the equation:

${\alpha }_V=\frac{1}{\underset{\_}{V}}{\left(\frac{\partial \underset{\_}{V}}{\partial T}\right)}_P$        (1)

Here, molar volume is $\underset{\_}{V}$, and change in molar volume and change in temperature at constant pressure is ${\left(\partial \underset{\_}{V}\right)}_P,\text{and}{\left(\partial T\right)}_P$ respectively.

The equation (1) can be rewritten as:

${\alpha }_V=\frac{1}{{\underset{\_}{V}}_{givenvalue}}\left(\frac{{\underset{\_}{V}}_2-{\underset{\_}{V}}_1}{T_2-T_1}\right)$        (2)

Here, given value of molar volume is ${\underset{\_}{V}}_{givenvalue}$, final molar volume is ${\underset{\_}{V}}_2$, initial molar volume is ${\underset{\_}{V}}_1$, final temperature is $T_2$, and initial temperature is $T_1$.

The formula to calculate the isothermal compressibility $\left({\kappa }_T\right)$ is given by:

${\kappa }_T=-\frac{1}{\underset{\_}{V}}{\left(\frac{\partial \underset{\_}{V}}{\partial P}\right)}_T$        (3)

Here, change in molar volume and change in pressure at constant temperature is ${\left(\partial \underset{\_}{V}\right)}_T,\text{and}{\left(\partial P\right)}_T$ respectively.

The equation (3) can be rewritten as:

${\kappa }_T=-\frac{1}{{\underset{\_}{V}}_{givenvalue}}\left(\frac{{\underset{\_}{V}}_2-{\underset{\_}{V}}_1}{P_2-P_1}\right)$        (4)

Here, final pressure is $P_2$, and initial pressure is $P_1$.

Take initial and final temperature as $\text{180°}\text{and}2\text{20°C}$ because in the compressed liquid tables, Table A-4, “compressed liquid” the preceding and following temperature of $\text{200°C}$ are $\text{180°}\text{and}\text{220°C}$.

$\begin{array}{l}{T}_1=1\text{80°}\\ T_2=\text{220°C}\end{array}$

Referring to appendix Table A-4, “compressed liquid”, the value of initial molar volume $\left({\underset{\_}{V}}_1\right)$ corresponding to initial temperature of $\text{180°C}$ and pressure of $\text{100}\text{bar}$ is obtained as:

$\begin{array}{c}{\underset{\_}{V}}_1=0.001120{\text{m}}^{\text{3}}/\text{kg}\\ =0.001120{\text{m}}^{\text{3}}/\text{kg}\left(0.01802\text{kg}/\text{mol}\right)\\ =2.02\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}\end{array}$

Referring to appendix Table A-4, “compressed liquid”, the value of final molar volume $\left({\underset{\_}{V}}_2\right)$ corresponding to final temperature of $\text{220°C}$ and pressure of $\text{100}\text{bar}$ is obtained as:

$\begin{array}{c}{\underset{\_}{V}}_2=0.001081{\text{m}}^{\text{3}}/\text{kg}\\ =0.001081{\text{m}}^{\text{3}}/\text{kg}\left(0.01802\text{kg}/\text{mol}\right)\\ =2.13\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}\end{array}$

Referring to appendix Table A-4, “compressed liquid”, the value of given value of molar volume $\left({\underset{\_}{V}}_{givenvalue}\right)$ corresponding to given temperature of $\text{200°C}$ and pressure of $\text{100}\text{bar}$ is obtained as:

$\begin{array}{c}{\underset{\_}{V}}_{givenvalue}=0.001148{\text{m}}^{\text{3}}/\text{kg}\\ =0.001148{\text{m}}^{\text{3}}/\text{kg}\left(0.01802\text{kg}/\text{mol}\right)\\ =2.07\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}\end{array}$

Substitute ${\underset{\_}{V}}_{givenvalue}=2.07\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}$, ${\underset{\_}{V}}_2=2.13\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}$, ${\underset{\_}{V}}_1=2.02\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}$, $T_1=\text{180°C}$, and $T_2=\text{220°C}$ in Equation (2).

$\begin{array}{c}{\alpha }_V=\frac{1}{{\underset{\_}{V}}_{givenvalue}}\left(\frac{{\underset{\_}{V}}_2-{\underset{\_}{V}}_1}{T_2-T_1}\right)\\ =\frac{1}{2.07\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}}\left(\frac{2.13\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}-2.02\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}}{\text{220°C}-18\text{0°C}}\right)\\ =1.33\times {10}^{-3}{\text{K}}^{-1}\end{array}$

The coefficient of thermal expansion $\left({\alpha }_V\right)$ is $1.33\times {10}^{-3}{\text{K}}^{-1}$.

Take the initial and final pressure as $\text{50}\text{bar}\text{and}1\text{50}\text{bar}$ from the compressed liquid tables, Table A-4, “compressed liquid”.

$\begin{array}{l}{P}_1=\text{50}\text{bar}\\ P_2=1\text{50}\text{bar}\end{array}$

Referring to appendix Table A-4, “compressed liquid”, the value of initial molar volume $\left({\underset{\_}{V}}_1\right)$ corresponding to initial pressure of $\text{50}\text{bar}$ and temperature of $\text{200°C}$ is obtained as:

$\begin{array}{c}{\underset{\_}{V}}_1=0.001153{\text{m}}^{\text{3}}/\text{kg}\\ =0.001153{\text{m}}^{\text{3}}/\text{kg}\left(0.01802\text{kg}/\text{mol}\right)\\ =2.08\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}\end{array}$

Referring to appendix Table A-4, “compressed liquid”, the value of final molar volume $\left({\underset{\_}{V}}_2\right)$ corresponding to final pressure of $\text{150}\text{bar}$ and temperature of $\text{200°C}$ is obtained as:

$\begin{array}{c}{\underset{\_}{V}}_2=0.001144{\text{m}}^{\text{3}}/\text{kg}\\ =0.001144{\text{m}}^{\text{3}}/\text{kg}\left(0.01802\text{kg}/\text{mol}\right)\\ =2.06\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}\end{array}$

Referring to appendix Table A-4, “compressed liquid”, write the value of given value of molar volume corresponding to given pressure of $\text{100}\text{bar}$ and temperature of $\text{200°C}$.

$\begin{array}{c}{\underset{\_}{V}}_{givenvalue}=0.001148{\text{m}}^{\text{3}}/\text{kg}\\ =0.001148{\text{m}}^{\text{3}}/\text{kg}\left(0.01802\text{kg}/\text{mol}\right)\\ =2.07\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}\end{array}$

Substitute $1.87\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}$ for ${\underset{\_}{V}}_{givenvalue}$, $1.86\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}$ for ${\underset{\_}{V}}_2$, $1.87\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}$ for ${\underset{\_}{V}}_1$, $\text{150}\text{bar}$ for $P_1$, and $\text{250}\text{bar}$ for $P_2$ in Equation (4).

$\begin{array}{c}{\kappa }_T=-\frac{1}{{\underset{\_}{V}}_{givenvalue}}\left(\frac{{\underset{\_}{V}}_2-{\underset{\_}{V}}_1}{P_2-P_1}\right)\\ =-\frac{1}{2.07\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}}\left(\frac{2.06\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}-2.08\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}}{\text{150}\text{bar}-\text{50}\text{bar}}\right)\\ =9.66\times {10}^{-5}{\text{bar}}^{-1}\end{array}$

The isothermal compressibility $\left({\kappa }_T\right)$ is $9.66\times {10}^{-5}{\text{bar}}^{-1}$.

D)

Interpretation Introduction

Interpretation:

To determine the coefficient of thermal expansion and isothermal compressibility for the given condition $P\text{=200bar,}T{\text{=200}}^{\text{o}}\text{C}$.

Concept introduction:

Coefficient of thermal expansion:

The change in length of an object with unit degree increase in temperature at constant pressure is known as coefficient of thermal expansion.

The formula to calculate the coefficient of thermal expansion $\left({\alpha }_V\right)$ is given by the equation:

${\alpha }_V=\frac{1}{\underset{\_}{V}}{\left(\frac{\partial \underset{\_}{V}}{\partial T}\right)}_P$

Here, molar volume is $\underset{\_}{V}$, and change in molar volume and change in temperature at constant pressure is ${\left(\partial \underset{\_}{V}\right)}_P,\text{and}{\left(\partial T\right)}_P$ respectively.

Isothermal compressibility:

Isothermal compressibility is the reciprocal of the bulb modulus.

The formula to calculate the isothermal compressibility $\left({\kappa }_T\right)$ is given by:

${\kappa }_T=-\frac{1}{\underset{\_}{V}}{\left(\frac{\partial \underset{\_}{V}}{\partial P}\right)}_T$

Here, change in molar volume and change in pressure at constant temperature is ${\left(\partial \underset{\_}{V}\right)}_T,\text{and}{\left(\partial P\right)}_T$ respectively.

Explanation of Solution

The formula to calculate the coefficient of thermal expansion $\left({\alpha }_V\right)$ is given by the equation:

${\alpha }_V=\frac{1}{\underset{\_}{V}}{\left(\frac{\partial \underset{\_}{V}}{\partial T}\right)}_P$        (1)

Here, molar volume is $\underset{\_}{V}$, and change in molar volume and change in temperature at constant pressure is ${\left(\partial \underset{\_}{V}\right)}_P,\text{and}{\left(\partial T\right)}_P$ respectively.

The equation (1) can be rewritten as:

${\alpha }_V=\frac{1}{{\underset{\_}{V}}_{givenvalue}}\left(\frac{{\underset{\_}{V}}_2-{\underset{\_}{V}}_1}{T_2-T_1}\right)$        (2)

Here, given value of molar volume is ${\underset{\_}{V}}_{givenvalue}$, final molar volume is ${\underset{\_}{V}}_2$, initial molar volume is ${\underset{\_}{V}}_1$, final temperature is $T_2$, and initial temperature is $T_1$.

The formula to calculate the isothermal compressibility $\left({\kappa }_T\right)$ is given by:

${\kappa }_T=-\frac{1}{\underset{\_}{V}}{\left(\frac{\partial \underset{\_}{V}}{\partial P}\right)}_T$        (3)

Here, change in molar volume and change in pressure at constant temperature is ${\left(\partial \underset{\_}{V}\right)}_T,\text{and}{\left(\partial P\right)}_T$ respectively.

The equation (3) can be rewritten as:

${\kappa }_T=-\frac{1}{{\underset{\_}{V}}_{givenvalue}}\left(\frac{{\underset{\_}{V}}_2-{\underset{\_}{V}}_1}{P_2-P_1}\right)$        (4)

Here, final pressure is $P_2$, and initial pressure is $P_1$.

Take initial and final temperature as $\text{180°}\text{and}2\text{20°C}$ because in the compressed liquid tables, Table A-4, “compressed liquid” the preceding and following temperature of $\text{200°C}$ are $\text{180°}\text{and}\text{220°C}$.

$\begin{array}{l}{T}_1=1\text{80°}\\ T_2=\text{220°C}\end{array}$

Referring to appendix Table A-4, “compressed liquid”, the value of initial molar volume $\left({\underset{\_}{V}}_1\right)$ corresponding to initial temperature of $\text{180°C}$ and pressure of $\text{200}\text{bar}$ is obtained as:

$\begin{array}{c}{\underset{\_}{V}}_1=0.001120{\text{m}}^{\text{3}}/\text{kg}\\ =0.001120{\text{m}}^{\text{3}}/\text{kg}\left(0.01802\text{kg}/\text{mol}\right)\\ =2.02\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}\end{array}$

Referring to appendix Table A-4, “compressed liquid”, the value of final molar volume $\left({\underset{\_}{V}}_2\right)$ corresponding to final temperature of $\text{220°C}$ and pressure of $\text{200}\text{bar}$ is obtained as:

$\begin{array}{c}{\underset{\_}{V}}_2=0.001170{\text{m}}^{\text{3}}/\text{kg}\\ =0.001170{\text{m}}^{\text{3}}/\text{kg}\left(0.01802\text{kg}/\text{mol}\right)\\ =2.11\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}\end{array}$

Referring to appendix Table A-4, “compressed liquid”, the value of given molar volume $\left({\underset{\_}{V}}_{givenvalue}\right)$ corresponding to given temperature of $\text{200°C}$ and pressure of $\text{200}\text{bar}$ is obtained as:

$\begin{array}{c}{\underset{\_}{V}}_{givenvalue}=0.001139{\text{m}}^{\text{3}}/\text{kg}\\ =0.001139{\text{m}}^{\text{3}}/\text{kg}\left(0.01802\text{kg}/\text{mol}\right)\\ =2.05\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}\end{array}$

Substitute $2.05\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}$ for ${\underset{\_}{V}}_{givenvalue}$, $2.11\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}$ for ${\underset{\_}{V}}_2$, $2.02\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}$ for ${\underset{\_}{V}}_1$, $\text{180°}$ for $T_1$, and $\text{220°C}$ for $T_2$ in Equation (2).

$\begin{array}{c}{\alpha }_V=\frac{1}{{\underset{\_}{V}}_{givenvalue}}\left(\frac{{\underset{\_}{V}}_2-{\underset{\_}{V}}_1}{T_2-T_1}\right)\\ =\frac{1}{2.05\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}}\left(\frac{2.11\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}-2.02\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}}{\text{220°C}-18\text{0°C}}\right)\\ =1.09\times {10}^{-3}{\text{K}}^{-1}\end{array}$

Hence, the coefficient of thermal expansion $\left({\alpha }_V\right)$ is $1.09\times {10}^{-3}{\text{K}}^{-1}$.

Take the initial and final pressure as $\text{150}\text{bar}\text{and}2\text{50}\text{bar}$ from the compressed liquid tables, Table A-4, “compressed liquid”

$\begin{array}{l}{P}_1=1\text{50}\text{bar}\\ P_2=2\text{50}\text{bar}\end{array}$

Referring to appendix Table A-4, “compressed liquid”, the value of initial molar volume $\left({\underset{\_}{V}}_1\right)$ corresponding to initial pressure of $\text{150}\text{bar}$ and temperature of $\text{200°C}$ is obtained as.

$\begin{array}{c}{\underset{\_}{V}}_1=0.001144{\text{m}}^{\text{3}}/\text{kg}\\ =0.001144{\text{m}}^{\text{3}}/\text{kg}\left(0.01802\text{kg}/\text{mol}\right)\\ =2.06\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}\end{array}$

Referring to appendix Table A-4, “compressed liquid”, the value of final molar volume $\left({\underset{\_}{V}}_2\right)$ corresponding to final pressure of $\text{250}\text{bar}$ and temperature of $\text{200°C}$ is obtained as:

$\begin{array}{c}{\underset{\_}{V}}_2=0.001135{\text{m}}^{\text{3}}/\text{kg}\\ =0.001135{\text{m}}^{\text{3}}/\text{kg}\left(0.01802\text{kg}/\text{mol}\right)\\ =2.04\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}\end{array}$

Referring to appendix Table A-4, “compressed liquid”, the value of given value of molar volume $\left({\underset{\_}{V}}_{givenvalue}\right)$ corresponding to given pressure of $\text{200}\text{bar}$ and temperature of $\text{200°C}$ is obtained as:

$\begin{array}{c}{\underset{\_}{V}}_{givenvalue}=0.001139{\text{m}}^{\text{3}}/\text{kg}\\ =0.001139{\text{m}}^{\text{3}}/\text{kg}\left(0.01802\text{kg}/\text{mol}\right)\\ =2.05\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}\end{array}$

Substitute $2.05\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}$ for ${\underset{\_}{V}}_{givenvalue}$, $2.04\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}$ for ${\underset{\_}{V}}_2$, $2.06\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}$ for ${\underset{\_}{V}}_1$, $\text{150}\text{bar}$ for $P_1$, and $\text{250}\text{bar}$ for $P_2$ in Equation (4).

$\begin{array}{c}{\kappa }_T=-\frac{1}{{\underset{\_}{V}}_{givenvalue}}\left(\frac{{\underset{\_}{V}}_2-{\underset{\_}{V}}_1}{P_2-P_1}\right)\\ =-\frac{1}{2.05\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}}\left(\frac{2.04\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}-2.06\times {10}^{-5}{\text{m}}^{\text{3}}/\text{mol}}{\text{250}\text{bar}-1\text{50}\text{bar}}\right)\\ =9.75\times {10}^{-5}{\text{bar}}^{-1}\end{array}$

Hence, the isothermal compressibility $\left({\kappa }_T\right)$ is $9.75\times {10}^{-5}{\text{bar}}^{-1}$.

E)

Interpretation Introduction

Interpretation:

To comment on the validity to model isothermal compressibility and coefficient of thermal expansion as constants for liquids at high pressure.

Concept introduction:

Coefficient of thermal expansion:

The change in length of an object with unit degree increase in temperature at constant pressure is known as coefficient of thermal expansion.

The formula to calculate the coefficient of thermal expansion $\left({\alpha }_V\right)$ is given by the equation:

${\alpha }_V=\frac{1}{\underset{\_}{V}}{\left(\frac{\partial \underset{\_}{V}}{\partial T}\right)}_P$

Here, molar volume is $\underset{\_}{V}$, and change in molar volume and change in temperature at constant pressure is ${\left(\partial \underset{\_}{V}\right)}_P,\text{and}{\left(\partial T\right)}_P$ respectively.

Isothermal compressibility:

Isothermal compressibility is the reciprocal of the bulb modulus.

The formula to calculate the isothermal compressibility $\left({\kappa }_T\right)$ is given by:

${\kappa }_T=-\frac{1}{\underset{\_}{V}}{\left(\frac{\partial \underset{\_}{V}}{\partial P}\right)}_T$

Here, change in molar volume and change in pressure at constant temperature is ${\left(\partial \underset{\_}{V}\right)}_T,\text{and}{\left(\partial P\right)}_T$ respectively.

Explanation of Solution

Similar to the approximation of heat capacity to a constant, the approximation of isothermal compressibility and coefficient of thermal expansion as a constant is very suited for some instance specifically when the intervals of pressure and/or temperature are small. However from this problem, it can be noted that the variation in both isothermal compressibility and coefficient of thermal expansion are very substantial over higher intervals of temperature and pressure. It can also be noticed that both isothermal compressibility and coefficient of thermal expansion are functions of both temperature and pressure.