Two perfect gases at absolute temperature `T_(1) and T_(2)` are mixed. There is no loss of energy. The masses of the molecules are `m_(1) and m_(2)`. The number of molecules in the gases are `n_(1) and n_(2)`. The temperature of the mixture is

Options

A. `(m_(1)T_(1)+m_(2)T_(2))/(m_(1)+m_(2))`
B. `(M_(1)T_(1)+M_(2)T_(2))/(M_(1)+M_(2))`
C. `((m_(1))/(M_(1))T_(1)+(m_(2))/(M_(2))T_(2))/(((m_(1))/(M_(1))+(m_(2))/(M_(2))))`
D. `((M_(1))/(m_(1))T_(1)+(M_(2))/(m_(2))T_(2))/((M_(1))/(m_(1))+(M_(2))/(M_(2)))`

Answer

To find the temperature of the mixture of two perfect gases at absolute temperatures \( T_1 \) and \( T_2 \), we can follow these steps:

### Step 1: Understand the Energy of the Gases
The total energy \( E \) of a gas can be expressed as:
\[ E = \frac{3}{2} NkT \]
where \( N \) is the number of molecules, \( k \) is Boltzmann's constant, and \( T \) is the absolute temperature.

### Step 2: Write the Energy for Both Gases
For the two gases, we can write the total energy as:
\[ E_1 = \frac{3}{2} N_1 k T_1 \]
\[ E_2 = \frac{3}{2} N_2 k T_2 \]

### Step 3: Total Energy of the Mixture
Since there is no loss of energy when the gases are mixed, the total energy of the mixture can be expressed as:
\[ E_{\text{total}} = E_1 + E_2 = \frac{3}{2} N_1 k T_1 + \frac{3}{2} N_2 k T_2 \]

### Step 4: Express Total Energy in Terms of the Mixture Temperature
Let \( N = N_1 + N_2 \) be the total number of molecules in the mixture, and \( T \) be the temperature of the mixture. The total energy can also be expressed as:
\[ E_{\text{total}} = \frac{3}{2} N k T \]

### Step 5: Set the Energies Equal
Equating the two expressions for total energy, we have:
\[ \frac{3}{2} N_1 k T_1 + \frac{3}{2} N_2 k T_2 = \frac{3}{2} N k T \]

### Step 6: Simplify the Equation
Dividing through by \( \frac{3}{2} k \) gives:
\[ N_1 T_1 + N_2 T_2 = N T \]

### Step 7: Substitute for Total Number of Molecules
Substituting \( N = N_1 + N_2 \) into the equation:
\[ N_1 T_1 + N_2 T_2 = (N_1 + N_2) T \]

### Step 8: Solve for Temperature of the Mixture
Rearranging gives:
\[ T = \frac{N_1 T_1 + N_2 T_2}{N_1 + N_2} \]

### Step 9: Substitute for Number of Molecules
If we express the number of molecules in terms of mass and molecular mass:
\[ N_1 = \frac{m_1}{M_1}, \quad N_2 = \frac{m_2}{M_2} \]
where \( m_1 \) and \( m_2 \) are the masses of the gases and \( M_1 \) and \( M_2 \) are their respective molecular masses.

### Step 10: Final Expression for Temperature
Substituting these values into the equation for \( T \):
\[ T = \frac{\frac{m_1}{M_1} T_1 + \frac{m_2}{M_2} T_2}{\frac{m_1}{M_1} + \frac{m_2}{M_2}} \]

### Conclusion
Thus, the temperature of the mixture is given by:
\[ T = \frac{\frac{m_1}{M_1} T_1 + \frac{m_2}{M_2} T_2}{\frac{m_1}{M_1} + \frac{m_2}{M_2}} \]