Question

Adsorption in gas masks. You have a gas mask that adsorbs a certain toxic compound with adsorption constant \(K=1000 \mathrm{~atm}^{-1}\) at \(T=25^{\circ} \mathrm{C}\), and binding energy \(w=-5 \mathrm{kcal} \mathrm{mol}^{-1}\). In the desert, where the temperature is \(T=35^{\circ} \mathrm{C}\), what is the adsorption constant?

Short answer

The adsorption constant at \( T = 35^{\text{ \footmat}\text{ K }} \) is \( K_2 = 760 \text{ atm} ^{-1} \).

Step-by-step solution

Understand the Relationship Between Adsorption Constant and Temperature

The adsorption constant for a gas at different temperatures can be calculated using the Van't Hoff equation: \[ \frac{K_2}{K_1} = \frac{\theta_2}{\theta_1} = e^{-\frac{\triangle H}{R}(\frac{1}{T_2} - \frac{1}{T_1})} \] , where: - \(K_1\) and \(K_2\) are the adsorption constants at temperatures \(T_1\) and \(T_2\) respectively. - \(\triangle H\) is the binding energy (in this case, given as \( w \)). - \(R\) is the universal gas constant (R = 1.987\text{ cal/mol/K}orR = 8.314\text{ J/mol/K}, depending on the unit system used).Convert all temperatures to Kelvin by adding 273.15 to the Celsius values.

Convert Temperatures to Kelvin

Convert the given temperatures from Celsius to Kelvin: \[ T_1 = 25 + 273.15 = 298.15 \text{ K} \] \[ T_2 = 35 + 273.15 = 308.15 \text{ K} \]

Identify and Plug in the Values

Identify the necessary values to plug into the Van't Hoff equation: - \( K_1 = 1000 \text{ atm}^{-1} \) - \( T_1 = 298.15 \text{ K} \) - \( T_2 = 308.15 \text{ K} \) - \( w = - 5 \text{ kcal/mol} = -5000 \text{ cal/mol} \)

Use the Van't Hoff Equation

Plug in all values into the Van't Hoff equation and solve for \( K_2 \) : \[ \frac{K_2}{1000} = e^{\frac{-(-5000)}{1.987}(\frac{1}{308.15} - \frac{1}{298.15})} = e^{\frac{5000}{1.987}(0.003247 - 0.003356)} = e^{\frac{5000}{1.987}(-0.000109)} = e^{-0.274} \]

Calculate \( K_2 \)

Solve for the adsorption constant \( K_2 \): \[ K_2 = 1000 \times e^{-0.274} = 1000 \times 0.760 = 760 \text{ atm}^{-1} \]

Key concepts

Van't Hoff equation

The Van't Hoff equation is fundamental in understanding how the equilibrium constant for a reaction changes with temperature. Specifically, this equation relates the adsorption constant at different temperatures to the binding energy of the process. Here's the equation: ewline ewline \ bf{Van't Hoff Equation} \[ \frac{K_2}{K_1} = e^{-\frac{\triangle H}{R}\big(\frac{1}{T_2} - \frac{1}{T_1}\big)} \] ewline ewline Where: ewline ewline • \(K_1\) and \(K_2\) are the adsorption constants at temperatures \(T_1\) and \(T_2\) respectively ewline • \(\triangle H\) is the binding energy, the energy change when the compound adsorbs onto the surface ewline • \(R\) is the universal gas constant ewline Note how the equation shows the exponential relationship between the temperature and adsorption constant. A higher binding energy means a more sensitive reaction to temperature changes.

Temperature Conversion

Converting temperature from Celsius to Kelvin is essential in many thermodynamic calculations because many equations, including the Van’t Hoff equation, use the Kelvin scale. Here’s the simple conversion formula: ewline ewline \ \[{T (K)} = T (\text{°C}) + 273.15\]ewline ewline For example, in this exercise, the temperatures were converted as follows: ewline • From 25°C to 298.15 K ewline • From 35°C to 308.15 Kewline By converting the temperatures to Kelvin, we ensure consistency and accuracy in our calculations. The Kelvin scale starts at absolute zero, making it ideal for thermodynamic calculations.

Binding Energy

Binding energy is the energy required to adsorb a compound onto a surface. It is a crucial parameter because it indicates how strongly a molecule is held on the adsorbent. In this problem, the binding energy is given as -5 kcal/mol or -5000 cal/mol. A negative value signifies exothermic adsorption—energy is released when the compound adheres to the surface. ewline ewline • Binding energy influences how the adsorption constant changes with temperature. ewline ewline When plugging into the Van't Hoff equation, ensure consistency of units by matching the binding energy with the gas constant's units. Here we use: ewline • \( \triangle H = -5000 \text{ cal/mol} \)ewline • \( R = 1.987 \text{ cal/mol/K} \) ewline Thus, the resulting calculation shows how much the adsorption coefficient changes with temperature due to the binding energy's effect. Understanding binding energy can also help design better adsorbents with optimal performance under different conditions.