The correct expression of partial pressure in terms of mole fraction is

Options

A. (a) `p_(1) = x_(1) p_("total"), p_(2) = x_(2)P_("total")`
B. (b) `P = x_(1) x_(2) x_(2) P_("total")`
C. (c) `P_("total") = P_(1) x_(1) , P_("total") = P_(2) x_(2)`
D. (d) `P_(1) + P_(2) = x_(1) + x_(2)`

Answer

To derive the correct expression of partial pressure in terms of mole fraction, 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: Consider Two Gases
Assume we have two gases in a container, Gas 1 and Gas 2, with partial pressures \( P_1 \) and \( P_2 \) respectively. The volume of the container is fixed and the temperature is constant.

### Step 3: Write the Ideal Gas Equation for Each Gas
For Gas 1:
\[ P_1 = \frac{n_1RT}{V} \]
For Gas 2:
\[ P_2 = \frac{n_2RT}{V} \]

### Step 4: Apply Dalton's Law of Partial Pressures
According to Dalton's Law, the total pressure \( P_t \) is the sum of the partial pressures:
\[ P_t = P_1 + P_2 \]

### Step 5: Substitute the Expressions for Partial Pressures
Substituting the expressions from Step 3 into the equation from Step 4 gives:
\[ P_t = \frac{n_1RT}{V} + \frac{n_2RT}{V} \]
Factoring out the common terms:
\[ P_t = \frac{RT}{V}(n_1 + n_2) \]

### Step 6: Express Partial Pressure in Terms of Total Pressure
Now, to find the expression for \( P_1 \) in terms of \( P_t \):
\[ \frac{P_1}{P_t} = \frac{\frac{n_1RT}{V}}{\frac{RT}{V}(n_1 + n_2)} \]
The \( RT/V \) terms cancel out:
\[ \frac{P_1}{P_t} = \frac{n_1}{n_1 + n_2} \]

### Step 7: Define Mole Fraction
The mole fraction \( x_1 \) of Gas 1 is defined as:
\[ x_1 = \frac{n_1}{n_1 + n_2} \]

### Step 8: Final Expression for Partial Pressure
Thus, we can express the partial pressure of Gas 1 in terms of the total pressure:
\[ P_1 = x_1 P_t \]
Similarly, for Gas 2:
\[ P_2 = x_2 P_t \]
Where \( x_2 = \frac{n_2}{n_1 + n_2} \).

### Conclusion
The correct expression of partial pressure in terms of mole fraction is:
\[ P_i = x_i P_t \]
Where \( P_i \) is the partial pressure of gas \( i \) and \( x_i \) is the mole fraction of gas \( i \).

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