Consider a non-reactive binary mixture of two ideal gases A and B in an isothermal horizontal closed tube. Due to the concentration difference at the two ends of the tube, equimolal counterdiffusion occurs along the axial direction. No concentration gradients exist in the radial direction. The profile for partial pressure of component A ($p_A$) as a function of axial distance $z$ is shown in the figure. The partial pressure of component B ($p_B$) approaches zero at $z = 0.8 \text{ m}$. At a location $z = z_1$ the partial pressures of the components A and B are equal. The value of $z_1$ (in m) is _____ (rounded off to two decimal places).
Summary Solution
In this problem, we are analyzing the Equimolal Counterdiffusion (EMCD) of two ideal gases, A and B, in a steady-state isothermal system. For EMCD, the total pressure \( P \) remains constant throughout the tube, and the partial pressure profiles are linear.
1. Data Extraction from the Graph
By observing the provided graph of partial pressure \( p_A \) vs. axial distance \( z \):
- At \( z = 0 \text{ m} \), the partial pressure of A is \( p_A(0) = 0.6 \text{ bar} \).
- At \( z = 0.8 \text{ m} \), the partial pressure of A is \( p_A(0.8) = 2.6 \text{ bar} \).
2. Determination of Total Pressure
The problem states that at \( z = 0.8 \text{ m} \), the partial pressure of component B approaches zero (\( p_B = 0 \)). According to Dalton's Law of partial pressures:
$$ P = p_A + p_B $$
Substituting the values at \( z = 0.8 \text{ m} \):
$$ P = 2.6 \text{ bar} + 0 \text{ bar} = 2.6 \text{ bar} $$
Since the system is undergoing steady-state equimolal counterdiffusion in a closed tube at constant temperature, the total pressure \( P \) is constant at 2.6 bar for all values of \( z \).
3. Deriving the Partial Pressure Profile
For EMCD, the partial pressure gradient is constant, resulting in a linear profile. The equation for \( p_A(z) \) is:
$$ p_A(z) = p_A(0) + \left( \frac{p_A(0.8) - p_A(0)}{0.8 - 0} \right) \cdot z $$ $$ p_A(z) = 0.6 + \left( \frac{2.6 - 0.6}{0.8} \right) \cdot z $$ $$ p_A(z) = 0.6 + 2.5z $$
4. Finding the Location \( z_1 \)
At the location \( z = z_1 \), the partial pressures of A and B are equal (\( p_A = p_B \)). Given that \( p_A + p_B = P \):
$$ p_A + p_A = P \implies 2p_A = 2.6 \text{ bar} $$ $$ p_A = 1.3 \text{ bar} $$
Now, we substitute \( p_A = 1.3 \) into our linear profile equation to find \( z_1 \):
$$ 1.3 = 0.6 + 2.5z_1 $$ $$ 0.7 = 2.5z_1 $$ $$ z_1 = \frac{0.7}{2.5} $$ $$ z_1 = 0.28 \text{ m} $$
Final Answer: The value of \( z_1 \) is 0.28 (rounded to two decimal places).
Important Questions from Molecular Diffusion
- For a binary mixture of components A and B, $N_A$ and $N_B$ denote the total molar fluxes of components A and B, respectively. $J_A$ and $J_B$ are the corresponding molar diffusive fluxes. Which of the following is true for equimolar counter-diffusion in the binary mixture?
- Consider the following two cases for a binary mixture of ideal gases A and B under steady state conditions. In Case 1, the diffusion of A occurs through non-diffusing B. In Case 2, equimolal counter diffusion of A and B occurs. In both the cases, the total pressure is 100 kPa and the partial pressures of A at two points separated by a distance of 10 mm are 10 kPa and 5 kPa. Assume that the Fick's first law of diffusion is applicable. What is the ratio of molar flux of A in Case 1 to that in Case 2?
- In a binary gas-liquid system, $N_{A,EMD}$ is the molar flux of a gas $A$ for equimolar counter diffusion with a liquid $B$. $N_{A,UMD}$ is the molar flux of $A$ for steady one- component diffusion through stagnant $B$. Using the mole fraction of $A$ in the bulk of the gas phase as $0.2$ and that at the gas-liquid interface as $0.1$ for both the modes of diffusion, the ratio of $N_{A,UMD}$ to $N_{A,EMD}$ is equal to ________ (rounded off to two decimal places).
- Consider two stationary spherical pure water droplets of diameters $d_1$ and $2d_1$. $CO_2$ diffuses into the droplets from the surroundings. If the rate of diffusion of $CO_2$ into the smaller droplet is $W_1$ mol $s^{-1}$, the rate of diffusion of $CO_2$ into the larger droplet is
- Consider steady-state diffusion in a binary A-B liquid at constant temperature and pressure. The mole-fraction of A at two different locations is $0.8$ and $0.1$. Let $N_{A1}$ be the diffusive flux of A calculated assuming B to be non-diffusing, and $N_{A2}$ be the diffusive flux of A calculated assuming equimolar counter-diffusion. The quantity $\frac{(N_{A1}-N_{A2})}{N_{A1}} \times 100$ is ________ (rounded off to one decimal place).
