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. `(n_1T_1+n_2T_2+n_3T_3)/(n_1+n_2+n_3)`
B. `(n_1T_1^2+n_2T_2^2+n_3T_3^2)/(n_1T_1+n_2T_2+n_3T_3)`
C. `(n_1^2T_1^2+n_2^2T_2^2+n_3^2T_3^2)/(n_1T_1+n_2T_2+n_3T_3)`
D. (T_1+T_2+T_3)/3`

Explanation

To find the final temperature of the 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 Concept of Kinetic Energy
The total kinetic energy of a gas is related to its temperature and can be expressed as:
\[ E = \frac{3}{2} PV \]
where \( P \) is the pressure and \( V \) is the volume of the gas.

### Step 2: Calculate the Initial Energies
For each gas, the initial energy can be expressed as:
- For gas 1:
\[ E_1 = \frac{3}{2} P_1 V_1 \]
- For gas 2:
\[ E_2 = \frac{3}{2} P_2 V_2 \]
- For gas 3:
\[ E_3 = \frac{3}{2} P_3 V_3 \]

### Step 3: Relate Energy to Moles
Using the ideal gas law \( PV = nRT \), we can express the energies in terms of the number of moles \( n \):
- The number of moles for each gas is given by:
\[ n_1 = \frac{N_1}{N_A}, \quad n_2 = \frac{N_2}{N_A}, \quad n_3 = \frac{N_3}{N_A} \]
where \( N_A \) is Avogadro's number.

### Step 4: Write the Total Energy Equation
Assuming no energy loss during mixing, the total initial energy is equal to the final energy:
\[ E_1 + E_2 + E_3 = E_{\text{final}} \]
This can be written as:
\[ \frac{3}{2} P_1 V_1 + \frac{3}{2} P_2 V_2 + \frac{3}{2} P_3 V_3 = \frac{3}{2} P_{\text{final}} V_{\text{final}} \]

### Step 5: Substitute the Ideal Gas Law
Substituting \( PV \) in terms of moles and temperature:
\[ \frac{3}{2} \left( \frac{N_1}{N_A} RT_1 + \frac{N_2}{N_A} RT_2 + \frac{N_3}{N_A} RT_3 \right) = \frac{3}{2} \left( \frac{N_1 + N_2 + N_3}{N_A} RT_{\text{mix}} \right) \]

### Step 6: Cancel Common Terms
Cancel \( \frac{3}{2} \) and \( N_A \) from both sides:
\[ N_1 T_1 + N_2 T_2 + N_3 T_3 = (N_1 + N_2 + N_3) T_{\text{mix}} \]

### Step 7: Solve for Final Temperature
Rearranging gives us:
\[ T_{\text{mix}} = \frac{N_1 T_1 + N_2 T_2 + N_3 T_3}{N_1 + N_2 + N_3} \]

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