Three perfect gases at absolute temperature `T_1,T_2, and T_3` are mixed. The masses of molecules are `n_1,n_2 and n_3` respectively. Assuming to loss of energy, the final temperature of the mixture is:

Options

A. `(T_1 + T_2 +T_3)/(3)`
B. `(n_(1)^(2) T_(1) + n_(2)^(2) T_(2) + n_(3)^(2) T_(3))/(n_1 + n_2 + n_3)`
C. `(n_1 T_1 + n_2 T_2 + n_3 T_3)/(n_1 + n_2 + n_3)`
D. `(T_1 + T_2 + T_3)/(n_1 + n_2 + n_3)`

Answer

To find the final temperature of a mixture of three perfect gases at absolute temperatures \( T_1, T_2, \) and \( T_3 \) with respective masses of molecules \( n_1, n_2, \) and \( n_3 \), we can follow these steps:

### Step 1: Understand the Kinetic Energy of Gases
The average kinetic energy \( E \) of a single molecule of a gas is given by the formula:
\[ E = \frac{f}{2} k T \]
where \( f \) is the degrees of freedom, \( k \) is the Boltzmann constant, and \( T \) is the absolute temperature.

### Step 2: Total Kinetic Energy of Each Gas
For \( n_1 \) molecules of the first gas at temperature \( T_1 \), the total kinetic energy \( E_1 \) is:
\[ E_1 = n_1 \cdot \frac{f}{2} k T_1 \]
Similarly, for the second gas with \( n_2 \) molecules at temperature \( T_2 \):
\[ E_2 = n_2 \cdot \frac{f}{2} k T_2 \]
And for the third gas with \( n_3 \) molecules at temperature \( T_3 \):
\[ E_3 = n_3 \cdot \frac{f}{2} k T_3 \]

### Step 3: Total Initial Energy of the Mixture
The total initial energy \( E_{\text{initial}} \) of the mixture is the sum of the energies of the three gases:
\[ E_{\text{initial}} = E_1 + E_2 + E_3 = n_1 \cdot \frac{f}{2} k T_1 + n_2 \cdot \frac{f}{2} k T_2 + n_3 \cdot \frac{f}{2} k T_3 \]

### Step 4: Total Number of Molecules
The total number of molecules in the mixture is:
\[ N = n_1 + n_2 + n_3 \]

### Step 5: Final Energy of the Mixture
Assuming no loss of energy, the final energy \( E_{\text{final}} \) of the mixture can be expressed as:
\[ E_{\text{final}} = N \cdot \frac{f}{2} k T_e \]
where \( T_e \) is the final temperature of the mixture.

### Step 6: Equating Initial and Final Energy
By the conservation of energy, we set the total initial energy equal to the total final energy:
\[ n_1 \cdot \frac{f}{2} k T_1 + n_2 \cdot \frac{f}{2} k T_2 + n_3 \cdot \frac{f}{2} k T_3 = (n_1 + n_2 + n_3) \cdot \frac{f}{2} k T_e \]

### Step 7: Simplifying the Equation
We can cancel \( \frac{f}{2} k \) from both sides:
\[ n_1 T_1 + n_2 T_2 + n_3 T_3 = (n_1 + n_2 + n_3) T_e \]

### Step 8: Solving for Final Temperature
Now, we can solve for \( T_e \):
\[ T_e = \frac{n_1 T_1 + n_2 T_2 + n_3 T_3}{n_1 + n_2 + n_3} \]

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