Let `n_(1)` and `n_(2)` moles of two different ideal gases be mixed. If adiabatic coeefiecient of the two gases are `gamma_(1)` and `gamma_(2)` respectively, then adiabatic coefficient `gamma` of the mixture is given through the relation

Options

A. `(n_(1)+n_(2)) gamma =n_(1)gamma_(1)+n_(2)gamma_(2)`
B. `((n_(1)+n_(2)))/(gamma-1) = (n_(1))/(gamma_(1)-1) +(n_(2))/(gamma_(2)-1)`
C. `(n_(1)+n_(2)) (gamma)/(gamma-1)= n_(1) (gamma_(1))/(gamma_(1)-1) +n_(2) (gamma_(2))/(gamma_(2)-1)`
D. `(n_(1) +n_(2)) (gamma -1) = n_(1) (gamma_(1)-1) +n_(2) (gamma_(2) -1)`

Answer

To derive the adiabatic coefficient \( \gamma \) of a mixture of two ideal gases, we start with the known relationships for the specific heats at constant volume \( C_v \) and the adiabatic coefficients \( \gamma_1 \) and \( \gamma_2 \) of the individual gases.

### Step-by-Step Solution:

1. **Define Specific Heat at Constant Volume**:
For an ideal gas, the specific heat at constant volume \( C_v \) is related to the adiabatic coefficient \( \gamma \) by the formula:
\[ C_v = \frac{R}{\gamma - 1} \]
where \( R \) is the universal gas constant.

2. **Write \( C_{v1} \) and \( C_{v2} \)**:
For the two gases, we can write:
\[ C_{v1} = \frac{R}{\gamma_1 - 1} \quad \text{and} \quad C_{v2} = \frac{R}{\gamma_2 - 1} \]

3. **Calculate the Specific Heat of the Mixture**:
The specific heat at constant volume for the mixture \( C_{v \text{ mix}} \) can be expressed as:
\[ C_{v \text{ mix}} = \frac{n_1 C_{v1} + n_2 C_{v2}}{n_1 + n_2} \]

4. **Substituting \( C_{v1} \) and \( C_{v2} \)**:
Substitute the expressions for \( C_{v1} \) and \( C_{v2} \):
\[ C_{v \text{ mix}} = \frac{n_1 \left(\frac{R}{\gamma_1 - 1}\right) + n_2 \left(\frac{R}{\gamma_2 - 1}\right)}{n_1 + n_2} \]

5. **Factor Out \( R \)**:
The \( R \) can be factored out:
\[ C_{v \text{ mix}} = \frac{R \left( \frac{n_1}{\gamma_1 - 1} + \frac{n_2}{\gamma_2 - 1} \right)}{n_1 + n_2} \]

6. **Express \( \gamma \) for the Mixture**:
The adiabatic coefficient \( \gamma \) for the mixture can be expressed as:
\[ \gamma = \frac{C_p}{C_v} \]
where \( C_p \) is the specific heat at constant pressure.

7. **Using the Relation for \( C_p \)**:
We know that:
\[ C_p = C_v + R \]
Therefore, for the mixture:
\[ C_{p \text{ mix}} = \frac{n_1 C_{p1} + n_2 C_{p2}}{n_1 + n_2} \]

8. **Substituting \( C_{p1} \) and \( C_{p2} \)**:
We can express \( C_{p1} \) and \( C_{p2} \) in terms of \( \gamma_1 \) and \( \gamma_2 \):
\[ C_{p1} = C_{v1} + R = \frac{R}{\gamma_1 - 1} + R = \frac{R\gamma_1}{\gamma_1 - 1} \]
\[ C_{p2} = C_{v2} + R = \frac{R}{\gamma_2 - 1} + R = \frac{R\gamma_2}{\gamma_2 - 1} \]

9. **Final Expression for \( \gamma \)**:
Substitute \( C_{p1} \) and \( C_{p2} \) into the equation for \( \gamma \):
\[ \gamma = \frac{n_1 \frac{R\gamma_1}{\gamma_1 - 1} + n_2 \frac{R\gamma_2}{\gamma_2 - 1}}{n_1 \frac{R}{\gamma_1 - 1} + n_2 \frac{R}{\gamma_2 - 1}} \]
Simplifying this gives:
\[ \gamma = \frac{n_1 \gamma_1 + n_2 \gamma_2}{n_1 + n_2} \]

### Final Result:
The adiabatic coefficient \( \gamma \) of the mixture is given by:
\[ \frac{n_1}{\gamma_1 - 1} + \frac{n_2}{\gamma_2 - 1} = \frac{n_1 + n_2}{\gamma - 1} \]