A mixture contains `1` mole of helium `(C_p = 2.5 R, C_v= 1.5 R. )` and `1` mole of hydrogen `(C_p = 3.5 R, C_v = 2.5 R, )` . Calculate the values of `C_p , C_v and gamma` for the mixture.

Explanation

To solve the problem, we need to calculate the values of \( C_p \), \( C_v \), and \( \gamma \) for a mixture of 1 mole of helium and 1 mole of hydrogen.

### Step 1: Write down the given values
For helium:
- \( C_{p, He} = 2.5 R \)
- \( C_{v, He} = 1.5 R \)

For hydrogen:
- \( C_{p, H_2} = 3.5 R \)
- \( C_{v, H_2} = 2.5 R \)

### Step 2: Calculate the total \( C_v \) for the mixture
The total \( C_v \) for the mixture can be calculated using the formula:
\[ C_v = \frac{C_{v, He} + C_{v, H_2}}{n_{total}} \]
where \( n_{total} \) is the total number of moles in the mixture. Since we have 1 mole of helium and 1 mole of hydrogen, \( n_{total} = 1 + 1 = 2 \).

Substituting the values:
\[ C_v = \frac{1.5 R + 2.5 R}{2} = \frac{4 R}{2} = 2 R \]

### Step 3: Calculate the total \( C_p \) for the mixture
Using the formula:
\[ C_p = \frac{C_{p, He} + C_{p, H_2}}{n_{total}} \]
Substituting the values:
\[ C_p = \frac{2.5 R + 3.5 R}{2} = \frac{6 R}{2} = 3 R \]

### Step 4: Calculate \( \gamma \)
The ratio \( \gamma \) is given by:
\[ \gamma = \frac{C_p}{C_v} \]
Substituting the values we found:
\[ \gamma = \frac{3 R}{2 R} = 1.5 \]

### Final Results
- \( C_p = 3 R \)
- \( C_v = 2 R \)
- \( \gamma = 1.5 \)

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