Question

A thick nickel wall is exposed to pure hydrogen gas at \(165^{\circ} \mathrm{C}\) on one side of its surface. The hydrogen concentration at the wall surface is constant. Determine the hydrogen concentration at the penetration depth in percentage of its concentration at the wall surface.

Short answer

Question: Determine the hydrogen concentration at the penetration depth in percentage of its concentration at the wall surface. Answer: The percentage of hydrogen concentration at the penetration depth compared to its concentration at the wall surface can be calculated using the formula: \(\frac{c(\delta)}{c_{0}} = e^{\frac{-\delta}{\sqrt{Dt}}}\). However, since we do not have the value of time (t) in the problem statement, we cannot calculate the exact percentage.

Step-by-step solution

Convert given temperature to Kelvin

First, we need to convert the given temperature from Celsius to Kelvin. To do this, add 273.15 to the given Celsius temperature. \(T_{K} = 165^{\circ} C + 273.15 = 438.15 K\)

Find the diffusion coefficient

Next, we need to determine the diffusion coefficient (D) for hydrogen in nickel at the given temperature. This information can be found in specialized tables or using empirical relations. For this exercise, let's assume that the diffusion coefficient is \(2 \times 10^{-13} m^2/s\).

Calculate the penetration depth

Now, we'll need to calculate the penetration depth (\(\delta\)). penetration depth can be calculated using the formula: \(\delta = \sqrt{Dt}\), where D is the diffusion coefficient and t is the time. To continue with the solution, we would need the value of time (t). However, since the time information is not provided in the problem statement, we will present the formula as is for now.

Find hydrogen concentration at the penetration depth

In order to find the hydrogen concentration at the penetration depth, we need to use Fick's second law, which states that the concentration profile follows the function: \(c(x) = c_{0}e^{\frac{-x}{\sqrt{Dt}}}\) Where: \(c(x)\) is the concentration at the penetration depth, \(c_{0}\) is the initial concentration at the wall surface, \(x\) is the penetration depth (equal to \(\delta\)), \(D\) is the diffusion coefficient, and \(t\) is the time. Since we do not have the value of time (t), we cannot calculate the exact hydrogen concentration at the penetration depth. However, we can find the ratio of concentration at the penetration depth to the concentration at the wall surface: \(\frac{c(\delta)}{c_{0}} = e^{\frac{-\delta}{\sqrt{Dt}}}\) This ratio is equal to the percentage of hydrogen concentration at the penetration depth, compared to its concentration at the wall surface.

Key concepts

Fick's Second Law

In the study of diffusion, Fick's second law is essential for understanding how substances move within various materials. It describes the time dependency of the concentration gradient within a medium, such as gases diffusing through solids. According to Fick's second law, the rate at which concentration changes with time is proportional to the second derivative of concentration with respect to position.

This means that the law helps us determine how the concentration of a diffusing substance changes at any point inside the medium over time. The mathematical representation of Fick's second law for one-dimensional case is:\[\begin{equation}\frac{\partial c}{\partial t} = D\frac{\partial^2 c}{\partial x^2}\end{equation}\]Here, \(\frac{\partial c}{\partial t}\) represents the rate of change of concentration over time, \(D\) is the diffusion coefficient, and \(\frac{\partial^2 c}{\partial x^2}\) is the second spatial derivative of concentration. Simplified, it allows us to calculate how deep and at what rate a particular substance, such as hydrogen, will penetrate into a material. Taking the exercise into account, this law is crucial for determining the concentration of hydrogen at various depths in the nickel wall over time.

Diffusion Coefficient

The diffusion coefficient, denoted as \(D\), is a critical parameter in calculating diffusion phenomena. It quantifies how quickly a substance, such as hydrogen gas, spreads through another, like a nickel wall. This coefficient is influenced by factors including the properties of both the diffusing substance and the medium, as well as the temperature at which diffusion takes place.

Units for the diffusion coefficient are typically \(m^2/s\), and it can vary significantly between different systems. For example, gases diffuse faster than liquids, which in turn diffuse faster than solids. In practical applications, values for \(D\) must often be determined empirically or obtained from reference materials. In our exercise, a diffusion coefficient of \(2 \times 10^{-13} m^2/s\) represents the relative ease with which hydrogen diffuses into nickel at the given temperature.

Knowing the diffusion coefficient is indispensable because it allows for the calculation of other critical variables, such as the penetration depth and ultimately the concentration profile of the diffusing species at various points in the medium.

Penetration Depth

Penetration depth \(\delta\) in the field of diffusion refers to the distance into a material that a substance has diffused within a certain time period. It is a significant factor in determining the degree to which a material has been affected by the diffused substance. As seen in the exercise, the penetration depth is calculated using the simple relation:\[\begin{equation}\delta = \sqrt{Dt}\end{equation}\]where \(D\) represents the diffusion coefficient and \(t\) the time. The square root shows that the penetration depth increases with time, but at a decreasing rate—penetrating rapidly at first, then slowing down.

It is worth noting that without the exact time value (\(t\)), it is not possible to compute an absolute penetration depth. However, the given formula is essential as it links the diffusion coefficient and time, which are fundamental in understanding how substances move within a material. This concept is closely related to the actual exercise at hand as it defines the method to calculate how deep hydrogen will penetrate into the nickel wall.

Temperature Conversion

Understanding temperature conversion is crucial not only in everyday situations but also in scientific calculations like those involving diffusion. Temperature can affect the rate of diffusion and is therefore a critical part of diffusion-related calculations.

To convert Celsius to Kelvin, which is necessary in many scientific formulas, you simply add \(273.15\) to the Celsius temperature. For example:\[\begin{equation}T_{K} = T_{C} + 273.15\end{equation}\]In our exercise, the given temperature of \(165^\circ C\) is converted to Kelvin to enable use of the diffusion coefficient \(D\) in the penetration depth formula. Accurate temperature conversion is fundamental because the diffusion coefficient itself is temperature-dependent, following an Arrhenius-type relationship where the coefficient increases with temperature. Therefore, this conversion provides a basis for calculating the hydrogen concentration in the nickel wall correctly.

Remember, it's always essential to work in the correct temperature scale to avoid miscalculations that could dramatically change the outcome of your diffusion analysis.