A vessel conrain 1mol of O2 and 2 mol of He .What is the value of Cp /Cv of the misxture

To determine the ratio of specific heats, \( \frac{C_p}{C_v} \), for a mixture of gases, we need to consider the individual properties of the gases involved. In this case, we have a mixture of oxygen (O2) and helium (He). Let's break this down step by step.

Understanding Specific Heats

Specific heat at constant pressure (\( C_p \)) and specific heat at constant volume (\( C_v \)) are important thermodynamic properties of gases. The ratio \( \frac{C_p}{C_v} \) is often denoted as \( \gamma \) (gamma) and is crucial in various applications, including thermodynamics and fluid dynamics.

Individual Gases Properties

For our mixture, we need to know the specific heats of each gas:

• Oxygen (O2):
  • \( C_p = 29.19 \, \text{J/(mol·K)} \)
  • \( C_v = 21.16 \, \text{J/(mol·K)} \)
  • \( \frac{C_p}{C_v} = \frac{29.19}{21.16} \approx 1.38 \)
• Helium (He):
  • \( C_p = 20.79 \, \text{J/(mol·K)} \)
  • \( C_v = 12.47 \, \text{J/(mol·K)} \)
  • \( \frac{C_p}{C_v} = \frac{20.79}{12.47} \approx 1.67 \)

Calculating the Mixture Properties

To find the overall \( \frac{C_p}{C_v} \) for the mixture, we can use the mole fractions of each gas. The total number of moles in the mixture is:

Total moles = 1 mol (O2) + 2 mol (He) = 3 mol

Mole Fractions

The mole fractions for each gas are:

• Mole fraction of O2: \( Y_{O2} = \frac{1}{3} \)
• Mole fraction of He: \( Y_{He} = \frac{2}{3} \)

Weighted Average of Specific Heats

Now, we can calculate the average \( C_p \) and \( C_v \) for the mixture:

• Average \( C_p \):

  \( C_{p, mixture} = Y_{O2} \cdot C_{p, O2} + Y_{He} \cdot C_{p, He} \)

  \( C_{p, mixture} = \frac{1}{3} \cdot 29.19 + \frac{2}{3} \cdot 20.79 \approx 23.92 \, \text{J/(mol·K)} \)

• Average \( C_v \):

  \( C_{v, mixture} = Y_{O2} \cdot C_{v, O2} + Y_{He} \cdot C_{v, He} \)

  \( C_{v, mixture} = \frac{1}{3} \cdot 21.16 + \frac{2}{3} \cdot 12.47 \approx 16.70 \, \text{J/(mol·K)} \)

Final Calculation of \( \frac{C_p}{C_v} \)

Now that we have the average values for \( C_p \) and \( C_v \), we can find the ratio:

\( \frac{C_p}{C_v} = \frac{C_{p, mixture}}{C_{v, mixture}} = \frac{23.92}{16.70} \approx 1.43 \)

In summary, the value of \( \frac{C_p}{C_v} \) for the mixture of 1 mol of O2 and 2 mol of He is approximately 1.43. This ratio reflects the thermodynamic behavior of the gas mixture and is essential for understanding processes involving these gases.