Question

The enthalpy (\(H\), in J\(\cdot\)mol\(^{-1}\)) of a binary liquid system at constant temperature and pressure is given as \[ H=40x_1+60x_2+x_1x_2(4x_1+2x_2), \] where \(x_1\) and \(x_2\) represent the mole fractions of species 1 and 2 in the liquid, respectively. Which one of the following is the CORRECT value of the partial molar enthalpy of species 1 at infinite dilution, \(\overline{H}_1^{\infty}\) (in J\(\cdot\)mol\(^{-1}\))?

• A. 100
• B. 42
• C. 64
• D. 40

Hint:

For any molar property \(M(x_1)\) of a binary mixture: \(\overline{M}_1=M+(1-x_1)\dfrac{dM}{dx_1}\) and \(\overline{M}_2=M-x_1\dfrac{dM}{dx_1}\).
“Infinite dilution of 1’’ means \(x_1\to 0\) (and \(x_2\to 1\)).

Answer 1

The Correct Option is B

Step 1: Eliminate \(x_2\) using \(x_2=1-x_1\): \[ H(x_1)=40x_1+60(1-x_1)+x_1(1-x_1)\big(4x_1+2(1-x_1)\big) =60-18x_1-2x_1^3. \] Step 2: Use the binary relation for a molar property \(H\): \[ \overline{H}_1=H+(1-x_1)\frac{dH}{dx_1}. \] Compute \(\dfrac{dH}{dx_1}=-18-6x_1^2\). Hence \[ \overline{H}_1= \big(60-18x_1-2x_1^3\big) + (1-x_1)\big(-18-6x_1^2\big) =42-6x_1^2+4x_1^3. \] Step 3: At infinite dilution of 1, \(x_1\to 0\): \[ \overline{H}_1^{\infty}=\lim_{x_1\to 0}\overline{H}_1=42\ \text{J mol}^{-1}. \] Thus option (B).