Correct expression for density of an ideal gas mixture of two gases 1 and 2, where `m_(1)` and `m_(2)` are masses and `n_(1)` and `n_(2)` are moles and `M_(1)` and `M_(2)` are molar masses.

Options

A. `d=((m_(1)+m_(2)))/((M_(1)+M_(2)))`
B. `d=((m_(1)+m_(2)))/((n_(1)+n_(2)))(P)/(RT)`
C. `d=((n_(1)+n_(2)))/((m_(1)+m_(2)))xx(P)/(RT)`
D. None of these

Answer

To derive the correct expression for the density of an ideal gas mixture of two gases (1 and 2), we will follow these steps:

### Step 1: Understand the definition of density
Density (ρ) is defined as mass (m) per unit volume (V):
\[ \rho = \frac{m}{V} \]

### Step 2: Calculate the total mass of the gas mixture
For the mixture of two gases, the total mass (m_total) is the sum of the masses of the individual gases:
\[ m_{\text{total}} = m_1 + m_2 \]

### Step 3: Use the ideal gas law to find the volume
According to the ideal gas law, the relationship between pressure (P), volume (V), number of moles (n), gas constant (R), and temperature (T) is given by:
\[ PV = nRT \]
For the gas mixture, the total number of moles (n_total) is:
\[ n_{\text{total}} = n_1 + n_2 \]
Thus, the volume of the gas mixture can be expressed as:
\[ V = \frac{n_{\text{total}}RT}{P} = \frac{(n_1 + n_2)RT}{P} \]

### Step 4: Substitute the total mass and volume into the density formula
Now we can substitute the expressions for total mass and volume into the density formula:
\[ \rho = \frac{m_{\text{total}}}{V} = \frac{m_1 + m_2}{\frac{(n_1 + n_2)RT}{P}} \]

### Step 5: Simplify the expression
Rearranging the expression gives us:
\[ \rho = \frac{(m_1 + m_2)P}{(n_1 + n_2)RT} \]

### Step 6: Relate mass and moles to molar mass
We can express the masses in terms of moles and molar masses:
\[ m_1 = n_1 M_1 \quad \text{and} \quad m_2 = n_2 M_2 \]
Substituting these into the density equation:
\[ \rho = \frac{(n_1 M_1 + n_2 M_2)P}{(n_1 + n_2)RT} \]

### Final Expression
The final expression for the density of the ideal gas mixture is:
\[ \rho = \frac{(n_1 M_1 + n_2 M_2)P}{(n_1 + n_2)RT} \]