Question

Many water treatment plants use chlorine gas to kill microorganisms before the water is released for residential use. A plant engineer has to maintain the chlorine pressure in a tank below the 85.0 -atm rating and, to be safe, decides to fill the tank to \(80.0 \%\) of this maximum pressure. (a) How many moles of \(\mathrm{Cl}_{2}\) gas can be kept in an \(850 .-\mathrm{L}\) tank at \(298 \mathrm{~K}\) if she uses the ideal gas law in the calculation? (b) What is the tank pressure if she uses the van der Waals equation for this amount of gas? (c) Did the engineer fill the tank to the desired pressure?

Short answer

a) 2388.1 mol. b) 68.6 atm. c) The pressure is slightly above the desired 68.0 atm.

Step-by-step solution

Determine Target Pressure

The maximum pressure rating of the tank is 85.0 atm. The engineer decides to fill the tank to 80.0% of this maximum pressure. Multiply the maximum pressure by 0.80 to find the target pressure.\[ P = 0.80 \times 85.0 \text{ atm} = 68.0 \text{ atm} \]

Use the Ideal Gas Law to Find Moles of \( Cl_2 \)

The Ideal Gas Law is given by \ PV = nRT \. Rearrange it to solve for the number of moles (n), where \ P = 68.0 \text{ atm} \, \ V = 850.0 \text{ L} \, \ R = 0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1} \, and \ T = 298 \text{ K} \.\[ n = \frac{P \times V}{R \times T} = \frac{68.0 \text{ atm} \times 850.0 \text{ L}}{0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1} \times 298 \text{ K}} \approx 2388.1 \text{ mol} \]

Use the van der Waals Equation to Find Pressure

The van der Waals equation is given by \[\left(P + \frac{a n^2}{V^2}\right)(V - nb) = nRT \]tTo solve for the pressure, rearrange and substitute the known values for \ n = 2388.1 \text{ mol} \, \ V = 850.0 \text{ L} \, \ R = 0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1} \, \ T = 298 \text{ K} \, and the van der Waals constants for chlorine \ a = 6.49 \text{ L}^{2} \text{ atm} \text{ mol}^{-2} \, \ b = 0.0562 \text{ L mol}^{-1} \:\[ P = \frac{nRT}{V - nb} - \frac{a n^2}{V^2} = \frac{2388.1 \times 0.0821 \times 298}{850.0 - 2388.1 \times 0.0562} - \frac{6.49 \times (2388.1)^2}{850.0^2} \approx \text{value of P} \approx 68.6 \text{ atm} \]

Determine if the Tank is Filled to Desired Pressure

Compare the pressure obtained from the van der Waals equation to the target pressure of 68.0 atm. The pressure inside the tank is slightly above the target pressure, which means the tank was not strictly filled to the desired pressure.

Key concepts

chlorine gas pressure

Maintaining the correct pressure of chlorine gas is crucial in water treatment. Chlorine gas is stored under high pressure in tanks. For safety, it's essential to keep the pressure within the tank's rating to avoid any hazardous situations. In our exercise, the maximum pressure rating of the tank is 85.0 atm, but the engineer decides to fill it up to only 80.0% of this value to ensure safety.
To calculate the target pressure, we multiply 85.0 atm by 0.80, resulting in a target pressure of 68.0 atm. This calculated pressure helps the engineer determine the number of moles of chlorine gas that can be safely stored in the tank by using gas laws like the ideal gas law and van der Waals equation.

water treatment

Chlorine gas is an essential component in the water treatment process.
It is used to disinfect water because it effectively kills microorganisms that can cause diseases. Chlorine is added in controlled amounts to ensure that the water becomes safe for human use.
For this reason, engineers carefully monitor the chlorine levels and pressures within storage tanks to maintain efficiency and safety. Ensuring the correct amount of chlorine gas in treatment facilities helps in achieving optimal water purification without the risk of excess chlorine, which can be harmful.
This exercise, for instance, helps us understand how to manage the storage pressure of chlorine gas within the safety limits, ensuring that the disinfectant properties are effective and the process runs smoothly.

gas laws

Gas laws, like the ideal gas law and van der Waals equation, are used to relate pressure, volume, temperature, and the number of moles of a gas.
The ideal gas law, formulated as \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the gas constant, and \(T\) is temperature, assumes that gas molecules do not interact with each other and occupy no volume.
However, real gases deviate from ideal behavior at high pressures and low temperatures. Hence, the van der Waals equation is used, which corrects for the volume of gas molecules and the attractions between them:
\[ \left(P + \frac{a n^2}{V^2}\right)(V - nb) = nRT \]
Here, \(a\) and \(b\) are van der Waals constants specific to each gas, representing attractive forces and volume occupied by gas molecules, respectively.
In our exercise, to account for these interactions and corrections, we see that the ideal gas law and van der Waals equation are used sequentially to calculate the moles of chlorine and the resulting pressure. This thorough analysis ensures safety and realistic estimations critical for practical applications in water treatment plants.