Three perfect gases at absolute temperature `T_(1), T_(2)` and `T_(3)` are mixed. The masses f molecules are `m_(1), m_(2)` and `m_(3)` and the number of molecules are `n_(1), n_(2)` and `n_(3)` respectively. Assuming no loss of energy, the final temperature of the mixture is

Options

A. `((T_1 + T_2 + T_3))/3`
B. `(n_1 T_1 + n_2 T_2 + n_3 T_3)/(n_1 + n_2 + n_3)`
C. `(n_1 T_1^2 + n_2 T_2^2 + n_3 T_3^2)/(n_1T_1+ n_2 T_2 + n_3 T_3)`
D. `(n_1^2 T_1^2 + n_2^2T_2^2 + n_3^2 T_^3^2)/(n_1 T_1 + n_2 T_2 + n_3T_3)`

Explanation

To find the final temperature of the mixture of three perfect gases, we can use the principle of conservation of energy. Here’s the step-by-step solution:

### Step 1: Define the number of moles of each gas
The number of moles of each gas can be defined as follows:
- For gas 1: \( n_1 = \frac{N_1}{N_A} \)
- For gas 2: \( n_2 = \frac{N_2}{N_A} \)
- For gas 3: \( n_3 = \frac{N_3}{N_A} \)

Where \( N_A \) is Avogadro's number.

### Step 2: Write the energy conservation equation
Since there is no loss of energy, the total energy before mixing is equal to the total energy after mixing. The energy of each gas can be expressed as:
- Energy of gas 1: \( E_1 = n_1 R T_1 \)
- Energy of gas 2: \( E_2 = n_2 R T_2 \)
- Energy of gas 3: \( E_3 = n_3 R T_3 \)

Thus, the total energy before mixing is:
\[ E_{total} = E_1 + E_2 + E_3 = n_1 R T_1 + n_2 R T_2 + n_3 R T_3 \]

### Step 3: Express the total energy after mixing
After mixing, the total number of moles is \( n_{total} = n_1 + n_2 + n_3 \). The energy of the mixture can be expressed as:
\[ E_{mix} = n_{total} R T_{mix} \]

### Step 4: Set the total energy before mixing equal to the total energy after mixing
Setting the total energy before mixing equal to the total energy after mixing gives us:
\[ n_1 R T_1 + n_2 R T_2 + n_3 R T_3 = (n_1 + n_2 + n_3) R T_{mix} \]

### Step 5: Simplify the equation
We can cancel \( R \) from both sides (assuming \( R \neq 0 \)):
\[ n_1 T_1 + n_2 T_2 + n_3 T_3 = (n_1 + n_2 + n_3) T_{mix} \]

### Step 6: Solve for \( T_{mix} \)
Rearranging the equation to solve for \( T_{mix} \) gives:
\[ T_{mix} = \frac{n_1 T_1 + n_2 T_2 + n_3 T_3}{n_1 + n_2 + n_3} \]

### Final Answer
The final temperature of the mixture is:
\[ T_{mix} = \frac{n_1 T_1 + n_2 T_2 + n_3 T_3}{n_1 + n_2 + n_3} \]

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