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.
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
Text Solution
AI Generated Solution
The correct Answer is:
To derive the correct expression for the density of an ideal gas mixture of two gases, we can follow these steps:
### Step 1: Understand the Ideal Gas Law
The ideal gas law is given by the equation:
\[ PV = nRT \]
where:
- \( P \) = pressure of the gas
- \( V \) = volume of the gas
- \( n \) = number of moles of the gas
- \( R \) = universal gas constant
- \( T \) = temperature in Kelvin
### Step 2: Define Density
Density (\( d \)) is defined as mass per unit volume:
\[ d = \frac{m}{V} \]
where \( m \) is the mass of the gas and \( V \) is the volume.
### Step 3: Relate Mass to Moles
The mass of the gas can be expressed in terms of its number of moles and molar mass:
\[ m = nM \]
where \( M \) is the molar mass of the gas.
### Step 4: Substitute Mass in Density Equation
Substituting the expression for mass into the density equation gives:
\[ d = \frac{nM}{V} \]
### Step 5: Substitute Volume from Ideal Gas Law
From the ideal gas law, we can express volume \( V \) as:
\[ V = \frac{nRT}{P} \]
Now, substituting this expression for volume into the density equation:
\[ d = \frac{nM}{\frac{nRT}{P}} \]
This simplifies to:
\[ d = \frac{PM}{RT} \]
### Step 6: Consider the Mixture of Two Gases
For a mixture of two gases (gas 1 and gas 2), the total mass and total number of moles can be expressed as:
- Total mass \( m = m_1 + m_2 \)
- Total number of moles \( n = n_1 + n_2 \)
The molar mass of the mixture can be expressed as:
\[ M_{mix} = \frac{m_1 + m_2}{n_1 + n_2} \]
### Step 7: Final Expression for Density of the Mixture
Substituting the expressions for total mass and total number of moles into the density equation gives:
\[ d_{mix} = \frac{P \cdot M_{mix}}{RT} \]
### Conclusion
Thus, the correct expression for the density of an ideal gas mixture of two gases is:
\[ d_{mix} = \frac{P \cdot (m_1 + m_2)}{R \cdot (n_1 + n_2) \cdot T} \]
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