Three moles of ideal gas A with `(C_(p))/(C_(v))=(4)/(3)` is mixed with two moles of another ideal gas B with `(C _(P))/(C_(v))=(5)/(3)` The `(C_(P))/(C_(v))` of mixture is (Assuming temperature is constant)

Options

A. `1.5`
B. `1.42`
C. `1.7`
D. `1.3`

Answer

To find the \( \frac{C_p}{C_v} \) ratio of the mixture of gases A and B, we can follow these steps:

### Step 1: Understand the given data
We have:
- Gas A: \( n_1 = 3 \) moles, \( \frac{C_p}{C_v} = \frac{4}{3} \)
- Gas B: \( n_2 = 2 \) moles, \( \frac{C_p}{C_v} = \frac{5}{3} \)

### Step 2: Use the relationship between \( C_p \) and \( C_v \)
The relationship between \( C_p \) and \( C_v \) is given by:
\[ C_p = C_v + R \]
From the \( \frac{C_p}{C_v} \) ratios, we can express \( C_p \) in terms of \( C_v \):
- For gas A:
\[ \frac{C_{p1}}{C_{v1}} = \frac{4}{3} \implies C_{p1} = \frac{4}{3} C_{v1} \]
- For gas B:
\[ \frac{C_{p2}}{C_{v2}} = \frac{5}{3} \implies C_{p2} = \frac{5}{3} C_{v2} \]

### Step 3: Express \( R \) in terms of \( C_v \)
Using the relationship \( C_p = C_v + R \):
- For gas A:
\[ C_{p1} = C_{v1} + R \implies \frac{4}{3} C_{v1} = C_{v1} + R \implies R = \frac{4}{3} C_{v1} - C_{v1} = \frac{1}{3} C_{v1} \]
- For gas B:
\[ C_{p2} = C_{v2} + R \implies \frac{5}{3} C_{v2} = C_{v2} + R \implies R = \frac{5}{3} C_{v2} - C_{v2} = \frac{2}{3} C_{v2} \]

### Step 4: Find \( C_{v1} \) and \( C_{v2} \)
From the equations for \( R \):
1. \( R = \frac{1}{3} C_{v1} \)
2. \( R = \frac{2}{3} C_{v2} \)

Equating the two expressions for \( R \):
\[ \frac{1}{3} C_{v1} = \frac{2}{3} C_{v2} \implies C_{v1} = 2 C_{v2} \]

### Step 5: Substitute \( C_{v1} \) in terms of \( C_{v2} \)
Let \( C_{v2} = C_v \). Then \( C_{v1} = 2C_v \).

### Step 6: Calculate the \( C_p \) values
Now we can find \( C_{p1} \) and \( C_{p2} \):
- For gas A:
\[ C_{p1} = \frac{4}{3} C_{v1} = \frac{4}{3} (2C_v) = \frac{8}{3} C_v \]
- For gas B:
\[ C_{p2} = \frac{5}{3} C_{v2} = \frac{5}{3} C_v \]

### Step 7: Calculate the \( \frac{C_p}{C_v} \) for the mixture
Using the formula for the mixture:
\[ \frac{C_{p}}{C_{v}} = \frac{n_1 C_{p1} + n_2 C_{p2}}{n_1 C_{v1} + n_2 C_{v2}} \]
Substituting the values:
\[ \frac{C_{p}}{C_{v}} = \frac{3 \left(\frac{8}{3} C_v\right) + 2 \left(\frac{5}{3} C_v\right)}{3(2C_v) + 2(C_v)} \]
Simplifying:
\[ = \frac{8C_v + \frac{10}{3} C_v}{6C_v + 2C_v} = \frac{\frac{24}{3} C_v + \frac{10}{3} C_v}{8C_v} = \frac{\frac{34}{3} C_v}{8C_v} \]
\[ = \frac{34}{24} = \frac{17}{12} \]

### Final Answer
The \( \frac{C_p}{C_v} \) ratio of the mixture is \( \frac{17}{12} \).