Question

Two parallel metal plates separated by \(1 \mathrm{cm}\) are held at \(300 .\) and \(298 \mathrm{K}\), respectively. The space between the plates is filled with \(\mathrm{N}_{2}\left(\sigma=0.430 \mathrm{nm}^{2} \text { and } C_{V, m}=5 / 2 R\right)\) Determine the heat flux between the two plates in units of \(\mathrm{W} \mathrm{cm}^{-2}\)

Short answer

The heat flux between the two plates can be calculated by first finding the mean free path (\(\lambda\)) and thermal conductivity (k) for Nitrogen gas, and then using Fourier's Law of heat conduction: \(q = -k \cdot \frac{\Delta T}{d}\), where \(\Delta T\) is the temperature difference between the plates, and d is the separation distance. After finding the heat flux in W/m², convert it to W/cm² by dividing the answer by 10000. By following these steps, you should obtain the value for the heat flux in W/cm².

Step-by-step solution

1. Define Known Parameters

\ We are given the following parameters: - Temperature of Plate 1: \(T_1 = 300 K\) - Temperature of Plate 2: \(T_2 = 298 K\) - Separation between the plates: \(d = 1 cm = 0.01 m\) - Collisional cross-sectional area: \(\sigma = 0.430 nm^2 = 0.430 \times 10^{-18} m^2\) - Molar heat capacity at constant volume: \(C_{V,m} = \frac{5}{2}R\) - Gas constant: \(R = 8.314 J/(mol \cdot K)\)

2. Calculate Mean Free Path and Thermal Conductivity of Nitrogen

\ We need to find the mean free path (λ) and thermal conductivity (k) for Nitrogen gas. The formula for the mean free path (λ) is: \(\lambda = \frac{k_B T}{\sqrt{2} \pi \sigma d}\) where \(k_B\) is Boltzmann's constant and \(d\) is the gas density. In this case, since we are given a fixed volume, we can assume an ideal gas, so the gas density can be defined as: \(d = \frac{P \cdot M}{R \cdot T}\) using ideal gas law, where P is pressure, M is molar mass, R is gas constant, and T is the average temperature (T=(T1 + T2)/2). Now we can calculate the mean free path (λ) for Nitrogen gas. The thermal conductivity (k) can be determined using: \(k = \frac{1}{3} C_{v, m} \cdot v \cdot \lambda\) where \(v\) is the average molecular speed; it can be calculated as: \(v = \sqrt{\frac{8R \cdot T}{\pi \cdot M}}\)

3. Calculate Heat Flux using Fourier's Law

\ Now, we can use Fourier's Law of heat conduction to calculate the heat flux (q) between the plates: \(q = -k \cdot \frac{\Delta T}{d}\) where ΔT is the temperature difference between the plates, and d is the separation distance between the plates. We will first find the temperature difference: \(\Delta T = T_1 - T_2 = 300 - 298 = 2K\) Now we have all the necessary parameters to calculate the heat flux between the plates.

4. Calculate Heat Flux in W/cm²

\ After calculating the heat flux in W/m², we can convert it to W/cm² by dividing the answer by 10000 (since 1 m² = 10000 cm²). By following the steps above and solving for the heat flux, you should obtain the value in W/cm².

Key concepts

Thermal Conductivity

Thermal conductivity is a material's ability to conduct heat. It quantifies how easily heat passes through a material when a temperature gradient exists, being a fundamental concept in heat transfer. Specifically, it represents the amount of heat, in watts, that is transmitted over a distance of one meter through a material one meter square and one degree Celsius at a temperature gradient.

For gases like the nitrogen (N₂) in our problem, thermal conductivity also depends on other factors such as temperature, pressure, and the type of gas molecules involved. Nitrogen molecules, due to their size and speed, move and collide in a way that influences how effectively they can transfer energy from the hot side to the cooler side of the plates.

Mean Free Path

The mean free path is the average distance a particle, such as a gas molecule, travels between collisions with other particles. This concept is crucial in understanding gas behavior at the microscopic level.

The mean free path affects how rapidly gases conduct heat because it determines the frequency with which energy-carrying molecules can move without being impeded by other gas particles. In the case of our exercise with nitrogen gas, calculating the mean free path was essential to determine the thermal conductivity, and subsequently, the heat flux. The mean free path for a gas under known conditions can be derived using the gas's properties, temperature, and density.

Fourier's Law

Fourier's Law of heat conduction is a fundamental principle that describes the rate at which heat energy is transferred through materials due to temperature gradients. The law tells us that the heat flux is directly proportional to the temperature difference and inversely proportional to the material's thickness across which the temperature difference exists.

Applied to our exercise, Fourier's Law enables us to connect the calculated thermal conductivity of nitrogen and the observed temperature difference between the metal plates. By doing so, we can determine how much heat will flow from the hotter plate to the cooler plate per unit of time and area.

Heat Transfer in Gases

Heat transfer in gases, such as nitrogen found between our two metal plates, occurs primarily through conduction, where molecular motion transfers kinetic energy from one molecule to another. Factors including the molecular structure, temperature, and the mean free path of the molecules influence how effectively the gas conducts heat.

It's a complex process because, unlike solids, where atoms vibrate in place, gas molecules travel more freely, making the actual path of heat conduction unpredictable. Recognizing the gas properties (in this case, nitrogen) allows us to apply specific calculations, including the ideal gas law and principles of kinetic theory, to estimate the gas's thermal conductivity and determine the resultant heat flux.