A mixture of `n_1` moles of monoatomic gas and `n_2` moles of diatomic gas has `gamma=1.5`

Options

A. `n_1=n_2`
B. `2n_1=n_2`
C. `n_1=2n_2`
D. `n_1=3n_2`

Answer

To solve the problem, we need to find the relationship between the moles of monoatomic gas (`n_1`) and diatomic gas (`n_2`) given that their combined `gamma` (γ) is 1.5. Here’s a step-by-step solution:

### Step 1: Understand the Definition of Gamma
Gamma (γ) is defined as the ratio of specific heats at constant pressure (Cp) to specific heats at constant volume (Cv):
\[ \gamma = \frac{C_p}{C_v} \]

### Step 2: Write the Expression for Mixture
For a mixture of gases, the gamma can be expressed as:
\[ \gamma_{mix} = \frac{C_{p, mix}}{C_{v, mix}} \]

### Step 3: Calculate the Specific Heats for the Mixture
The specific heat at constant pressure for the mixture is given by:
\[ C_{p, mix} = \frac{n_1 C_{p1} + n_2 C_{p2}}{n_1 + n_2} \]
And the specific heat at constant volume is:
\[ C_{v, mix} = \frac{n_1 C_{v1} + n_2 C_{v2}}{n_1 + n_2} \]

### Step 4: Substitute Values for Monoatomic and Diatomic Gases
For a monoatomic gas:
- \(C_{p1} = \frac{5}{2}R\)
- \(C_{v1} = \frac{3}{2}R\)

For a diatomic gas:
- \(C_{p2} = \frac{7}{2}R\)
- \(C_{v2} = \frac{5}{2}R\)

### Step 5: Substitute into the Gamma Expression
Now substituting these values into the gamma expression:
\[ \gamma_{mix} = \frac{\frac{n_1 \cdot \frac{5}{2}R + n_2 \cdot \frac{7}{2}R}{n_1 + n_2}}{\frac{n_1 \cdot \frac{3}{2}R + n_2 \cdot \frac{5}{2}R}{n_1 + n_2}} \]

### Step 6: Simplify the Expression
Cancelling \(R\) and simplifying:
\[ \gamma_{mix} = \frac{5n_1 + 7n_2}{3n_1 + 5n_2} \]

### Step 7: Set Gamma Equal to 1.5
Given that \(\gamma_{mix} = 1.5\):
\[ 1.5 = \frac{5n_1 + 7n_2}{3n_1 + 5n_2} \]

### Step 8: Cross Multiply to Eliminate the Fraction
Cross multiplying gives:
\[ 1.5(3n_1 + 5n_2) = 5n_1 + 7n_2 \]
Expanding this:
\[ 4.5n_1 + 7.5n_2 = 5n_1 + 7n_2 \]

### Step 9: Rearranging the Equation
Rearranging the equation:
\[ 4.5n_1 + 7.5n_2 - 5n_1 - 7n_2 = 0 \]
This simplifies to:
\[ -0.5n_1 + 0.5n_2 = 0 \]
Thus,
\[ n_1 = n_2 \]

### Conclusion
The relationship between the moles of monoatomic gas and diatomic gas is:
\[ n_1 = n_2 \]