if two ideal gases have the same value of gamma then what will be the value of gamma of mixture of above two gases which are mixed in 1:2 ratio of their masses

To determine the value of gamma (γ) for a mixture of two ideal gases with the same gamma value, we need to understand what gamma represents. Gamma is the ratio of specific heats, defined as the ratio of the heat capacity at constant pressure (Cp) to the heat capacity at constant volume (Cv). When mixing gases, the overall gamma of the mixture can be calculated based on the individual properties of the gases and their respective proportions in the mixture.

Understanding Gamma in Ideal Gases

For any ideal gas, gamma is given by the formula:

• γ = Cp / Cv

When two gases have the same gamma value, it indicates that they have similar heat capacities relative to each other. Let's denote the gamma value of both gases as γ1 = γ2 = γ.

Mixing Gases in a Specific Ratio

In your scenario, we are mixing two gases in a mass ratio of 1:2. This means that for every 1 unit of mass of gas A, there are 2 units of mass of gas B. To find the gamma of the mixture, we can use the following approach:

Calculating the Effective Heat Capacities

The effective heat capacities of the mixture can be calculated using the mass-weighted average of the heat capacities of the individual gases. Since both gases have the same gamma, we can express their heat capacities as follows:

• For gas A: CpA = γ * CvA
• For gas B: CpB = γ * CvB

Now, the total heat capacity at constant pressure (Cp) for the mixture can be expressed as:

• Cp(mixture) = (mA * CpA + mB * CpB) / (mA + mB)

Substituting the masses (mA = 1 and mB = 2) and the heat capacities:

• Cp(mixture) = (1 * CpA + 2 * CpB) / (1 + 2)
• Cp(mixture) = (1 * (γ * CvA) + 2 * (γ * CvB)) / 3

Calculating the Effective Cv

Similarly, the total heat capacity at constant volume (Cv) for the mixture can be calculated as:

• Cv(mixture) = (mA * CvA + mB * CvB) / (mA + mB)

Substituting the masses and the heat capacities:

• Cv(mixture) = (1 * CvA + 2 * CvB) / 3

Finding the Gamma of the Mixture

Now that we have expressions for Cp(mixture) and Cv(mixture), we can find the gamma of the mixture:

• γ(mixture) = Cp(mixture) / Cv(mixture)

Since both gases have the same gamma, we can simplify our calculations. The effective gamma of the mixture will still be equal to γ because the ratio of the heat capacities remains constant regardless of the mass ratio, as long as both gases are ideal and have the same specific heat ratio.

Final Result

Thus, the gamma of the mixture of the two ideal gases, when mixed in a 1:2 mass ratio, will also be equal to the gamma of the individual gases:

• γ(mixture) = γ

This outcome highlights an important property of ideal gases: when mixing gases with the same specific heat ratio, the mixture retains that same ratio, irrespective of the proportions in which they are mixed.