Question

Exposure to high concentration of gaseous ammonia can cause lung damage. The acceptable shortterm ammonia exposure level set by the Occupational Safety and Health Administration (OSHA) is 35 ppm for 15 minutes. Consider a vessel filled with gaseous ammonia at \(30 \mathrm{~mol} / \mathrm{L}\), and a 10 -cm- diameter circular plastic plug with a thickness of \(2 \mathrm{~mm}\) is used to contain the ammonia inside the vessel. The ventilation system is capable of keeping the room safe with fresh air, provided that the rate of ammonia being released is below \(0.2 \mathrm{mg} / \mathrm{s}\). If the diffusion coefficient of ammonia through the plug is \(1.3 \times 10^{-10} \mathrm{~m}^{2} / \mathrm{s}\), determine whether or not the plug can safely contain the ammonia inside the vessel.

Short answer

Based on Fick's first law, we calculated the ammonia diffusion rate through the plastic plug to be 0.26095 mg/s. Since this exceeds the acceptable exposure rate of 0.2 mg/s provided by OSHA and the ventilation system, the plastic plug will not safely contain the ammonia inside the vessel according to the given standard.

Step-by-step solution

Calculate the area of the plastic plug

We'll first find the area of the circular plastic plug: \(A = \pi r^2\) where A is the area, and r is the radius of the circle. Convert the diameter to meters and divide by 2 to get the radius: \( r = 10 \mathrm{\,cm} \cdot(1 \mathrm{~m}/ 100 \mathrm{\,cm}) \cdot (1/2) = 0.05 \mathrm{~m}\) Now, calculate the area: \(A = \pi (0.05 \mathrm{~m})^2 = 0.00785 \mathrm{~m}^2\)

Calculate the concentration gradient

The concentration gradient is the difference in concentration between the inside and the outside of the vessel divided by the thickness of the plug: \(\Delta C = \frac{C_{inside} - C_{outside}}{d}\) Since the outside concentration is negligible, we can assume it is 0 mol/L. Convert the thickness of the plug to meters: \(d = 2 \mathrm{\,mm} \cdot (1 \mathrm{~m} / 1000 \mathrm{\,mm}) = 0.002 \mathrm{~m}\) Now, calculate the concentration gradient: \(\Delta C = \frac{30 \mathrm{~mol} / \mathrm{L}}{0.002 \mathrm{~m}}\) Convert mol/L to mol/m³: \(30 \mathrm{~mol} / \mathrm{L} \cdot (1000 \mathrm{~L} / \mathrm{m}^3) = 30000 \mathrm{~mol} / \mathrm{m}^3\) Calculate the concentration gradient: \(\Delta C = \frac{30000 \mathrm{~mol} / \mathrm{m}^3}{0.002 \mathrm{~m}} = 1.5 \times 10^{7} \mathrm{~mol} / \mathrm{m}^4\)

Apply Fick's first law to calculate the diffusion rate

Fick's first law for diffusion is given by the equation: \(J = -D \cdot A \cdot \Delta C\) where J is the diffusion rate (in mol/s), D is the diffusion coefficient (1.3 x 10⁻¹⁰ m²/s), A is the area of the plug (0.00785 m²), and ΔC is the concentration gradient (1.5 x 10⁷ mol/m⁴). Now, we will calculate the diffusion rate: \(J = -(1.3 \times 10^{-10} \mathrm{~m}^2 / \mathrm{s})(0.00785 \mathrm{~m}^2)(1.5 \times 10^{7} \mathrm{~mol} / \mathrm{m}^4) = -1.535 \times 10^{-5} \mathrm{~ mol/s}\) Since we want only the magnitude of the diffusion rate: \( J = 1.535 \times 10^{-5} \mathrm{~ mol/s}\)

Convert the diffusion rate to mg/s

To determine whether the ventilation system will be sufficient, we need to convert the ammonia diffusion rate from mol/s to mg/s. The molar mass of ammonia (NH3) is approximately 17 g/mol. Convert the diffusion rate to mg/s: \(J = (1.535 \times 10^{-5} \mathrm{~ mol/s}) \cdot (17 \mathrm{~g} / \mathrm{mol}) \cdot (1000 \mathrm{~mg} / \mathrm{g}) = 0.26095 \mathrm{~ mg / s}\)

Compare the calculated rate with the acceptable exposure rate

Now, we compare our calculated diffusion rate (0.26095 mg/s) to the acceptable exposure rate given by OSHA and the ventilation system (0.2 mg/s). Since 0.26095 mg/s > 0.2 mg/s, the plastic plug will not safely contain the ammonia inside the vessel according to the given standard.

Key concepts

Fick's First Law

Fick's First Law of diffusion is a fundamental principle in chemical engineering that describes how particles, such as molecules, spread from areas of high concentration to areas of low concentration. This process is driven by the concentration gradient and is essential for understanding the movement of substances in gases, liquids, and solids.

Fick's First Law is mathematically expressed as:\[ J = -D \cdot abla C \]Where:

• \( J \) is the diffusion flux, representing the amount of substance that goes through a unit area per unit time.
• \( D \) is the diffusion coefficient, which indicates how easily a substance can move through a medium. Its units are square meters per second (\( \mathrm{m}^2/ \mathrm{s} \)).
• \( abla C \) is the concentration gradient, or the rate of change of concentration with respect to distance.

Fick's First Law highlights the relationship between the diffusion rate and the concentration gradient.
It tells us that the rate at which a substance diffuses depends on how sharp the concentration change is across a given space. Understanding this principle helps engineers design and assess equipment used for containing or processing substances, such as in the controlled environment of a chemical vessel.

Concentration Gradient

The concentration gradient plays a crucial role in the diffusion process. It represents the change in concentration of a substance over a specific distance.
In a chemical engineering context, the gradient determines the direction and rate at which molecules move.

A higher concentration gradient means a steeper difference between concentrations, which typically leads to faster diffusion rates.
For example, in the original problem, the concentration difference of ammonia inside and outside the vessel is substantial, which results in a pronounced gradient. This means ammonia molecules will diffuse actively from areas of high concentration inside the vessel to areas outside.

To calculate the concentration gradient, the internal concentration and the thickness of the barrier (the plastic plug in the problem) are considered:

• Formula: \[ \Delta C = \frac{C_{\text{inside}} - C_{\text{outside}}}{d} \]
• Where \( C_{\text{outside}} \) is often negligible, especially if the outside concentration is almost zero.
• \( d \) is the thickness of the medium (in meters), through which diffusion occurs.

The calculated concentration gradient is then used in Fick’s First Law to find out how much substance diffuses over time.

Occupational Safety and Health Administration (OSHA)

The Occupational Safety and Health Administration (OSHA) is a regulatory body that ensures safe working conditions by setting standards and regulations for workplace safety. OSHA's guidelines are crucial in environments where hazardous substances, like ammonia, are present.

To protect workers, OSHA sets permissible exposure limits (PELs) for various chemicals, dictating the maximum concentration allowed in the workplace air over a specified time period. For ammonia, this limit is 35 parts per million (ppm) for short-term exposure, typically over 15 minutes.

These standards guide chemical engineers and safety professionals in assessing risks and designing systems to minimize exposure. In the given exercise, if the ammonia diffusion through the plastic plug exceeds the limit set by OSHA, it indicates a potential safety hazard:

• The calculation of diffusion rate compared against OSHA's permissible rate helps ensure compliance and reflect the effectiveness of containment measures.
• Such assessments can lead to redesign or replacement of materials to improve safety.
• This vigilance helps prevent health implications due to overexposure and ensures a safe working environment.
  

Considering OSHA's standards is not only a legal requirement but also a moral one, safeguarding employee health and well-being.