Two mole of argon are mixed with one mole of hydrogen, then `C_p/C_V` for the mixture is nearly

Options

A. 1,2
B. 1,3
C. 1,4
D. 1,54

Answer

To find the ratio \( \frac{C_p}{C_V} \) (denoted as \( \gamma \)) for a mixture of gases, we can use the following steps:

### Step 1: Identify the gases and their properties
- We have 2 moles of Argon (Ar) which is a monoatomic gas.
- We have 1 mole of Hydrogen (H₂) which is a diatomic gas.

### Step 2: Determine the values of \( \gamma \) for each gas
- For Argon (monoatomic gas), \( \gamma_1 = \frac{C_p}{C_V} = \frac{5}{3} \).
- For Hydrogen (diatomic gas), \( \gamma_2 = \frac{C_p}{C_V} = 1.4 \).

### Step 3: Use the formula for the mixture
The formula for the ratio \( \gamma \) of a mixture of two gases is given by:

\[ \gamma_{mixture} = \frac{n_1 \gamma_1 + n_2 \gamma_2}{n_1 + n_2} \]

Where:
- \( n_1 \) = number of moles of Argon = 2
- \( n_2 \) = number of moles of Hydrogen = 1
- \( \gamma_1 \) = \( \frac{5}{3} \)
- \( \gamma_2 \) = 1.4

### Step 4: Substitute the values into the formula
Now substituting the values into the formula:

\[ \gamma_{mixture} = \frac{(2 \times \frac{5}{3}) + (1 \times 1.4)}{2 + 1} \]

### Step 5: Calculate the numerator
Calculating the numerator:

\[ 2 \times \frac{5}{3} = \frac{10}{3} \]
\[ 1 \times 1.4 = 1.4 \]

Now, convert 1.4 to a fraction:

\[ 1.4 = \frac{14}{10} = \frac{7}{5} \]

Finding a common denominator (15):

\[ \frac{10}{3} = \frac{50}{15} \]
\[ \frac{7}{5} = \frac{21}{15} \]

Adding these:

\[ \frac{50}{15} + \frac{21}{15} = \frac{71}{15} \]

### Step 6: Calculate the denominator
The denominator is:

\[ n_1 + n_2 = 2 + 1 = 3 \]

### Step 7: Final calculation of \( \gamma_{mixture} \)
Now substitute back into the equation:

\[ \gamma_{mixture} = \frac{\frac{71}{15}}{3} = \frac{71}{45} \approx 1.577 \]

### Step 8: Round to two decimal places
Rounding gives us:

\[ \gamma_{mixture} \approx 1.54 \]

### Conclusion
Thus, the ratio \( \frac{C_p}{C_V} \) for the mixture is nearly \( 1.54 \).

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