Two perfect gases (1) and (2) are at temperatures `T_(1)` and `T_(2)` . If the number of molecules of the two gases are `n_(1)` and `n_(2)` and the masses of the molecules `m_(1)` and `m_(2)`, find the resulting temperature when the two gases are mixed. Assume that there is no loss of energy on mixing the two gases.

Explanation

To find the resulting temperature when two perfect gases are mixed, we can follow these steps:

### Step 1: Understand the energy of the gases
The energy of a perfect gas can be expressed using the formula:
\[ E = \frac{3}{2} n k_B T \]
where:
- \( E \) is the energy,
- \( n \) is the number of moles,
- \( k_B \) is the Boltzmann constant,
- \( T \) is the temperature.

### Step 2: Write the energy equations for both gases
For gas (1) at temperature \( T_1 \):
\[ E_1 = \frac{3}{2} n_1 k_B T_1 \]

For gas (2) at temperature \( T_2 \):
\[ E_2 = \frac{3}{2} n_2 k_B T_2 \]

### Step 3: Write the total energy of the mixed gases
When the two gases are mixed, the total energy \( E \) is the sum of the energies of the two gases:
\[ E = E_1 + E_2 \]
Thus,
\[ E = \frac{3}{2} n_1 k_B T_1 + \frac{3}{2} n_2 k_B T_2 \]

### Step 4: Express the total energy in terms of the resulting temperature
The total energy can also be expressed in terms of the resulting temperature \( T \) of the mixture:
\[ E = \frac{3}{2} (n_1 + n_2) k_B T \]

### Step 5: Set the two expressions for total energy equal to each other
Equating the two expressions for total energy:
\[ \frac{3}{2} (n_1 + n_2) k_B T = \frac{3}{2} n_1 k_B T_1 + \frac{3}{2} n_2 k_B T_2 \]

### Step 6: Simplify the equation
We can cancel out \( \frac{3}{2} k_B \) from both sides:
\[ (n_1 + n_2) T = n_1 T_1 + n_2 T_2 \]

### Step 7: Solve for the resulting temperature \( T \)
Rearranging the equation gives:
\[ T = \frac{n_1 T_1 + n_2 T_2}{n_1 + n_2} \]

This is the resulting temperature when the two gases are mixed.

### Final Result:
The resulting temperature \( T \) when the two perfect gases are mixed is:
\[ T = \frac{n_1 T_1 + n_2 T_2}{n_1 + n_2} \]

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