Two moles on ideal gas with `gamma=5/3` is mixed with 3 moles of another ideal non reacting gas with `gamma=7/5` .The value of `(C_p)/(C_v)` for the gasous mixture is closer to :

Options

A. (a)1.36
B. (b)1.48
C. (c)1.63
D. (d)1.42

Explanation

To solve the problem of finding the value of \((C_p)/(C_v)\) for a mixture of two ideal gases, we can follow these steps:

### Step 1: Understand the relationship between \(C_p\), \(C_v\), and \(\gamma\)
For an ideal gas, the ratio of specific heats is given by:
\[ \gamma = \frac{C_p}{C_v} \]
From the problem, we have:
- For gas 1 (2 moles): \(\gamma_1 = \frac{5}{3}\)
- For gas 2 (3 moles): \(\gamma_2 = \frac{7}{5}\)

### Step 2: Calculate \(C_p\) and \(C_v\) for each gas
Using the relationship \(\gamma = \frac{C_p}{C_v}\), we can express \(C_p\) and \(C_v\) in terms of \(R\) (the universal gas constant):
- For gas 1:
\[ C_{p1} = \gamma_1 R = \frac{5}{3} R, \quad C_{v1} = C_{p1} - R = \frac{5}{3} R - R = \frac{2}{3} R \]
- For gas 2:
\[ C_{p2} = \gamma_2 R = \frac{7}{5} R, \quad C_{v2} = C_{p2} - R = \frac{7}{5} R - R = \frac{2}{5} R \]

### Step 3: Calculate the total \(C_p\) and \(C_v\) for the mixture
Using the formula for the equivalent heat capacities of the mixture:
\[ C_{p,\text{mix}} = \frac{n_1 C_{p1} + n_2 C_{p2}}{n_1 + n_2} \]
\[ C_{v,\text{mix}} = \frac{n_1 C_{v1} + n_2 C_{v2}}{n_1 + n_2} \]
Substituting the values:
- For \(C_{p,\text{mix}}\):
\[ C_{p,\text{mix}} = \frac{2 \cdot \frac{5}{3} R + 3 \cdot \frac{7}{5} R}{2 + 3} = \frac{\frac{10}{3} R + \frac{21}{5} R}{5} \]
Finding a common denominator (15):
\[ C_{p,\text{mix}} = \frac{\frac{50}{15} R + \frac{63}{15} R}{5} = \frac{\frac{113}{15} R}{5} = \frac{113}{75} R \]

- For \(C_{v,\text{mix}}\):
\[ C_{v,\text{mix}} = \frac{2 \cdot \frac{2}{3} R + 3 \cdot \frac{2}{5} R}{2 + 3} = \frac{\frac{4}{3} R + \frac{6}{5} R}{5} \]
Finding a common denominator (15):
\[ C_{v,\text{mix}} = \frac{\frac{20}{15} R + \frac{18}{15} R}{5} = \frac{\frac{38}{15} R}{5} = \frac{38}{75} R \]

### Step 4: Calculate the ratio \(\frac{C_p}{C_v}\) for the mixture
Now we can find \(\frac{C_{p,\text{mix}}}{C_{v,\text{mix}}}\):
\[ \frac{C_{p,\text{mix}}}{C_{v,\text{mix}}} = \frac{\frac{113}{75} R}{\frac{38}{75} R} = \frac{113}{38} \approx 2.97 \]

### Step 5: Conclusion
The value of \(\frac{C_p}{C_v}\) for the gaseous mixture is approximately \(2.97\).