Estimate the activity and activity coefficient for $\mathrm{CS}_2$ at $x_{{ }^{C S_2}}=0.722$ using the data in Table for both a Raoult's law and a Henry's law standard state.

(a) A polynomial fit to measurements of the total volume of a binary mixture of A and B is $v=987.93+35.6774 x-0.45923 x^{2}+0.017325 x^{3}$ where $v=V / \mathrm{cm}^{3}, x=n_{\mathrm{B}} / \mathrm{mol},$ and $n_{\mathrm{B}}$ is the amount of B present. Derive an expresion for the partial molar volume of B. (b) A polynomial fit to measurements of the total volume of a binary mixture of A and B is $v=778.55-22.5749 x+0.56892 x^{2}+0.01023 x^{3}+0.00234 x^{4}$ where $v=V / \mathrm{cm}^{3}, x=n_{\mathrm{B}} / \mathrm{mol},$ and $n_{\mathrm{B}}$ is the amount of B present. Derive an expression for the partial molar volume of B.

The volume of an aqueous solution of NaCl at 25 degrees C was measured at a series of molalities b, and it was found that the volume fitted the expression $v=1003+16.62 x+1.77 x^{3 / 2}+0.12 x^{2}$ where $v=V / {cm}^{3}, V$ is the volume of a solution formed from $1.000 {~kg}$ of water, and $x=b / b$. Calculate the partial molar volume of the components in a solution of molality $0.100 {~mol} {~kg}^{-1}$.

A solution is made up of $222.9 \mathrm{~g}$ of ethanol and $130.8 \mathrm{~g}$ of $\mathrm{H}_2 \mathrm{O}$. If the volume of the solution is $403.4 \mathrm{~cm}^3$ and the partial molar volume of $\mathrm{H}_2 \mathrm{O}$ is $17.0 \mathrm{~cm}^3$, what is the partial molar volume of ethanol under these conditions?

In an ideal solution of A and B, $3.00 \mathrm{~mol}$ are in the liquid phase and $5.00 \mathrm{~mol}$ are in the gaseous phase. The overall composition of the system is $Z_A=0.375$ and $x_A=0.250$. Calculate $y_A$.

DNA is capable of forming complex helical structures. An unusual triple-helix structure of poly(dA).2poly(dT) DNA was studied by P. V. Scaria and R. H. Shafer [Journal of Biological Chemistry 266 (1991): 5417] where the intercalation of ethidium bromide was studied using UV absorption and circular dichroism spectroscopy. The representative data were obtained using the results of this study. Using the data, determine $K$ and $N$ for the binding of ethidium bromide to the DNA triple-helical structure.

Assume that 1-bromobutane and 1-chlorobutane form an ideal solution. At $273 \mathrm{~K}, P_{\text {chloro }}^*=3790 \mathrm{~Pa}$ and $P_{\text {bromo }}^*=1394 \mathrm{~Pa}$. When only a trace of liquid is present at $273 \mathrm{~K}, y_{\text {chloro }}=0.750$. a. Calculate the total pressure above the solution. b. Calculate the mole fraction of 1-chlorobutane in the solution. c. What value would $Z_{\text {chloro }}$ have in order for there to be $4.86 \mathrm{~mol}$ of liquid and $3.21 \mathrm{~mol}$ of gas at a total pressure equal to that in part (a)? [Note: This composition is different from that of part (a).]

Freezing point depression is one means of determining the molar mass of a compound. The freezing point depression constant of benzene is $-5.12^{\circ} \mathrm{C} / \mathrm{m}$.
(a) When a $0.503 \mathrm{~g}$ sample of the white crystalline dimer is dissolved in $10.0 \mathrm{~g}$ benzene, the freezing point of benzene is decreased by $0.542^{\circ} \mathrm{C}$. Verify that the molar mass of the dimer is $475 \mathrm{~g} / \mathrm{mol}$ when determined by freezing point depression. Assume no dissociation of the dimer occurs.

Determine the freezing point of a solution that contains 78.8 g of naphthalene (C10H8, molar mass = 128.16 g/mol) dissolved in 722 mL of benzene (d = 0.877 g/mL). Pure benzene has a freezing point of 5.50°C and a freezing point depression constant of 4.90°C/m."

Analysis of a compound gave $39.50 \% \mathrm{C}, 2.21 \% \mathrm{H}$, and $58.30 \% \mathrm{Cl}$. When $0.855 \mathrm{~g}$ of this solid was dissolved in $7.50 \mathrm{~g}$ of naphthalene, the solution had a freezing point of $78.0^{\circ} \mathrm{C}$. The pure solvent freezes at $80.0^{\circ} \mathrm{C}$; its molal freezing point constant is $6.8^{\circ} \mathrm{C} / \mathrm{m}$. a. What is the molecular formula of the compound? b. What is its molecular mass to the nearest $0.1 \mathrm{~g}$ ?

A $0.0182-\mathrm{g}$ sample of an unknown substance was dissolved in $2.135 \mathrm{~g}$ of benzene. The molality of this solution, determined by freezing-point depression, was $0.0698 \mathrm{~m}$. What is the molecular mass of the unknown substance?

What is the Clausius expression of the second law of thermodynamics?

Estimate the mean ionic molality and mean ionic activity of a $0.105 \mathrm{~m} \mathrm{~K}_3 \mathrm{PO}_4$ solution for which the mean ionic activity coefficient is $0.225$.

The vapor pressures of 1-bromobutane and 1-chlorobutane can be expressed in the form

$$\ln \frac{P_{\text {bromo }}}{\mathrm{Pa}}=17.076-\frac{1584.8}{\frac{T}{\mathrm{~K}}-111.88}$$

and

$$\ln \frac{P_{\text {chloro }}}{\mathrm{Pa}}=20.612-\frac{2688.1}{\frac{T}{\mathrm{~K}}-55.725}$$

Assuming ideal solution behavior, calculate $x_{\text {bromo }}$ and $y_{\text {bromo }}$ at $305 \mathrm{~K}$ and a total pressure of $9750 . \mathrm{~Pa}$.