`n_1` mole of a monoatomic gas is mixed with `n_2` mole of a diatomic gas. (Assume gases are ideal). If for the mixture adiabatic contant is found to be 1.5, then `n_1 : n_2` is :

Options

A. `1 : 1`
B. `1 : 3`
C. `2 :3`
D. `3 : 2`

Explanation

To solve the problem of finding the ratio \( n_1 : n_2 \) when \( n_1 \) moles of a monoatomic gas are mixed with \( n_2 \) moles of a diatomic gas and the adiabatic constant \( \gamma \) for the mixture is 1.5, we can follow these steps:

### Step 1: Identify the degrees of freedom
- For a monoatomic gas, the degrees of freedom \( f_1 = 3 \).
- For a diatomic gas, the degrees of freedom \( f_2 = 5 \).

### Step 2: Calculate the molar specific heats
- The molar specific heat at constant volume \( C_{v1} \) for the monoatomic gas is given by:
\[ C_{v1} = \frac{f_1}{2} R = \frac{3}{2} R \]
- The molar specific heat at constant volume \( C_{v2} \) for the diatomic gas is given by:
\[ C_{v2} = \frac{f_2}{2} R = \frac{5}{2} R \]

### Step 3: Calculate the molar specific heats at constant pressure
- The molar specific heat at constant pressure \( C_{p1} \) for the monoatomic gas is:
\[ C_{p1} = C_{v1} + R = \frac{3}{2} R + R = \frac{5}{2} R \]
- The molar specific heat at constant pressure \( C_{p2} \) for the diatomic gas is:
\[ C_{p2} = C_{v2} + R = \frac{5}{2} R + R = \frac{7}{2} R \]

### Step 4: Write the expression for the adiabatic constant \( \gamma \)
The adiabatic constant \( \gamma \) for the mixture is given by:
\[ \gamma = \frac{n_1 C_{p1} + n_2 C_{p2}}{n_1 C_{v1} + n_2 C_{v2}} \]
Substituting the values we calculated:
\[ 1.5 = \frac{n_1 \left(\frac{5}{2} R\right) + n_2 \left(\frac{7}{2} R\right)}{n_1 \left(\frac{3}{2} R\right) + n_2 \left(\frac{5}{2} R\right)} \]

### Step 5: Simplify the equation
Cancel \( R \) from the numerator and denominator:
\[ 1.5 = \frac{5n_1 + 7n_2}{3n_1 + 5n_2} \]

### Step 6: Cross-multiply to eliminate the fraction
\[ 1.5 (3n_1 + 5n_2) = 5n_1 + 7n_2 \]
Expanding this gives:
\[ 4.5n_1 + 7.5n_2 = 5n_1 + 7n_2 \]

### Step 7: Rearranging the equation
Rearranging the terms leads to:
\[ 4.5n_1 + 7.5n_2 - 5n_1 - 7n_2 = 0 \]
This simplifies to:
\[ -0.5n_1 + 0.5n_2 = 0 \]

### Step 8: Solve for the ratio
From the equation \( -0.5n_1 + 0.5n_2 = 0 \), we can deduce:
\[ n_1 = n_2 \]
Thus, the ratio \( n_1 : n_2 = 1 : 1 \).

### Conclusion
The final answer is:
\[ \text{The ratio } n_1 : n_2 = 1 : 1 \]