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Chapter 14: Problem 4

A He-Xe mixture containing \(0.75\) mole fraction of helium is used for cooling of electronics in an avionics application. At a temperature of \(300 \mathrm{~K}\) and atmospheric pressure, calculate the mass fraction of helium and the mass density, molar concentration, and molecular weight of the mixture. If the cooling system capacity is \(10 \mathrm{~L}\), what is the mass of the coolant?

Short Answer

Expert verified
The mass fraction of helium is \(\omega_{He} = \frac{0.75 \times 4}{(0.75 \times 4) + (0.25 \times 131)} = 0.226\). The molecular weight of the mixture is \(M_{mixture} = 0.75 \times 4 + 0.25 \times 131 = 35.75 \, g/mol\). The mass density of the mixture is \(\rho_{mixture} = M_{mixture} \times \frac{P}{RT} = 1.56 \, kg/m^3\), and the mass of the coolant is \(mass = 1.56 \times 0.01 = 0.0156 \, kg\).

Step by step solution

01

Calculate the mass fraction of helium in the mixture

First, we need to determine the mass fraction of helium in the mixture. We can calculate this by dividing the mass of helium by the total mass of the mixture. We're given the mole fraction of helium, so we can use the molar mass of helium and xenon to find the mass fraction. The mole fraction of helium is 0.75 and xenon is 0.25 (since the mole fractions add up to 1). The molar mass of helium is 4 g/mol, and for xenon, it's 131 g/mol. So, the mass fraction of helium \(\omega_{He}\) can be calculated as: \[\omega_{He} = \frac{0.75 \times 4}{(0.75 \times 4) + (0.25 \times 131)}\]
02

Calculate the molecular weight of the mixture

Next, we need to find the molecular weight of the mixture. We can calculate this by using a weighted sum of the molar masses of helium and xenon, using their mole fractions as weights: \[M_{mixture} = x_{He} \times M_{He} + x_{Xe} \times M_{Xe}\] where \(x_{He}\) and \(x_{Xe}\) are mole fractions of helium and xenon, and \(M_{He}\) and \(M_{Xe}\) are their respective molar masses.
03

Calculate the mass density of the mixture

Now we need to calculate the mass density of the mixture. We can use the ideal gas law to find the molar density and convert it to mass density using the molecular weight we found earlier. The ideal gas law is given by: \[PV = nRT\] Where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature. Here, we're given pressure (1 atm) and temperature (300 K), so we can solve for n/V to get the molar density: \[\frac{n}{V} = \frac{P}{RT}\] We can now convert this to mass density by multiplying by the molecular weight of the mixture: \[\rho_{mixture} = M_{mixture} \times \frac{n}{V}\]
04

Calculate the molar concentration of the mixture

Molar concentration is the same as molar density, which we calculated in the previous step.
05

Calculate the mass of the coolant

Finally, we can find the mass of the coolant by multiplying the mass density of the mixture by the volume of the cooling system (10 L), which we must first convert to cubic meters: \[mass = \rho_{mixture} \times volume\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

He-Xe Mixture
In avionics cooling systems, a mixture of helium (He) and xenon (Xe) is often used due to their unique properties. The helium-xenon (He-Xe) mixture combines the best aspects of both gases. Helium is very light and an excellent conductor of heat, making it ideal for efficient heat transfer. On the other hand, xenon is much denser, providing a good balance to the lightweight helium, allowing the mixture to be evenly distributed and retain heat efficiently.
The mole fraction is the starting point in characterizing the composition of the mixture. For this exercise, we are given that helium has a mole fraction of 0.75, meaning that out of a hypothetical combination of gases, 75% of it is helium, and the remaining 25% is xenon. This balance allows for both efficient conduction and optimal density in a cooling context.
Mass Fraction
The mass fraction of a component in a mixture helps us understand its contribution to the total mass. While mole fraction considers the number of particles, mass fraction gives us a glimpse into the real weight of each element in the blend.
For helium in this mixture, the formula to calculate mass fraction involves its molar mass (4 g/mol) compared to the molar mass of xenon (131 g/mol). To find helium's mass fraction, use:
  • Mass fraction of Helium (He):\[\omega_{He} = \frac{0.75 \times 4}{(0.75 \times 4) + (0.25 \times 131)}\]
The result tells us the proportion of helium's mass relative to the entire mixture's mass. In practice, this helps in determining how much helium contributes to the total cooling capacity efficiency.
Molecular Weight
Molecular weight is crucial in determining the physical and chemical properties of a gas mixture. It influences how the mixture behaves under particular conditions of temperature and pressure. To calculate the molecular weight of the He-Xe mixture, we use the weighted sum of their molar masses. The mole fractions serve as weights for this calculation:
  • Molecular weight of the mixture:\[M_{mixture} = x_{He} \times M_{He} + x_{Xe} \times M_{Xe}\]Where \(x_{He}\) and \(x_{Xe}\) are the mole fractions for helium and xenon.
Understanding the molecular weight is vital for using the ideal gas law, especially when calculating molar or mass density, which directly affects cooling efficiency.
Mass Density
Mass density expresses how much mass is contained in a unit volume of the gas mixture, crucial for evaluating a cooling system's performance. We use the ideal gas law to find the molar density first and then convert it to mass density鈥攁 necessary step in assessing how the mixture will perform under typical operational conditions.
The formula involves gas law constants and the molecular weight of the mixture:
  • Ideal gas law:\[PV = nRT\]
  • Molar density:\[\frac{n}{V} = \frac{P}{RT}\]
  • Mass density calculation:\[\rho_{mixture} = M_{mixture} \times \frac{n}{V}\]
Finally, by understanding the mass density, you can estimate the mass of the coolant in the system. Multiply the mass density by the system's volume to find the coolant mass, ensuring the mixture can effectively manage heat dissipation in avionics.

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