Finding the partial pressure of a gas within a mixture is essential for understanding gas behavior and is typically done using one of two primary methods: either applying the Ideal Gas Law to individual components or utilizing the mole fraction of each gas in relation to the total pressure.
Partial pressure refers to the pressure that a single gas in a mixture would exert if it alone occupied the entire volume of the container at the same temperature. Understanding partial pressures is fundamental to gas chemistry, particularly under Dalton's Law of Partial Pressures, which states that the total pressure exerted by a mixture of non-reacting gases is equal to the sum of the partial pressures of the individual gases.
1. Using the Ideal Gas Law (PV = nRT)
One direct way to determine the partial pressure of a gas is to apply the Ideal Gas Law to that specific gas, as if it were the only gas present in the container. This method is highly effective when you know the number of moles of the individual gas, the volume of the container, and the temperature of the gas mixture.
The Ideal Gas Law is expressed as:
$PV = nRT$
Where:
- $P$ = pressure of the gas (in atmospheres, Pascals, or kPa)
- $V$ = volume of the gas (in liters or cubic meters)
- $n$ = number of moles of the gas (amount of substance)
- $R$ = ideal gas constant (a constant value depending on units of P and V, e.g., 0.0821 L·atm/(mol·K) or 8.314 J/(mol·K))
- $T$ = absolute temperature of the gas (in Kelvin)
To find the partial pressure ($P_{gas}$) of a specific gas (e.g., gas A) using this method, you would rearrange the formula:
$P_A = \frac{n_A RT}{V}$
Example:
Imagine a 10.0 L container holds 0.50 moles of oxygen gas ($O_2$) and 0.30 moles of nitrogen gas ($N_2$) at a temperature of 298 K. To find the partial pressure of oxygen:
- $n_{O_2}$ = 0.50 mol
- $V$ = 10.0 L
- $R$ = 0.0821 L·atm/(mol·K)
- $T$ = 298 K
$P_{O2} = \frac{(0.50 \text{ mol})(0.0821 \text{ L·atm/mol·K})(298 \text{ K})}{10.0 \text{ L}}$
$P{O_2} \approx 1.22 \text{ atm}$
This method is particularly useful when you have data for individual components and want to calculate their specific contributions to the total pressure.
2. Using the Mole Fraction
Another common and often simpler method for calculating partial pressure involves using the mole fraction of a gas. The mole fraction represents the proportion of a specific gas's moles relative to the total moles of all gases in the mixture. Once the mole fraction is known, it can be multiplied by the total pressure of the gas mixture to find the partial pressure of that specific gas.
The mole fraction ($X_{gas}$) for a gas (e.g., gas B) is calculated as:
$X_B = \frac{\text{moles of gas B}}{\text{total moles of all gases}}$
Then, the partial pressure ($P_B$) is:
$P_B = XB \times P{\text{total}}$
Where $P_{\text{total}}$ is the total pressure of the gas mixture. This method essentially assigns a percentage of the total pressure to each individual gas based on its proportional presence in the mixture.
Example:
Using the same gas mixture from the previous example (0.50 moles $O_2$ and 0.30 moles $N_2$ in a 10.0 L container at 298 K).
First, calculate the total moles of gas:
Total moles ($n{\text{total}}$) = $n{O2} + n{N_2}$ = 0.50 mol + 0.30 mol = 0.80 mol
Next, find the mole fraction of nitrogen ($N2$):
$X{N_2} = \frac{0.30 \text{ mol}}{0.80 \text{ mol}} = 0.375$
Now, to use this method, you need the total pressure of the gas mixture. You can calculate the total pressure using the Ideal Gas Law for the total moles:
$P{\text{total}} = \frac{n{\text{total}} RT}{V}$
$P{\text{total}} = \frac{(0.80 \text{ mol})(0.0821 \text{ L·atm/mol·K})(298 \text{ K})}{10.0 \text{ L}}$
$P{\text{total}} \approx 1.96 \text{ atm}$
Finally, calculate the partial pressure of nitrogen ($N2$):
$P{N2} = X{N2} \times P{\text{total}}$
$P_{N2} = 0.375 \times 1.96 \text{ atm}$
$P{N_2} \approx 0.735 \text{ atm}$
(As a check, $P_{O2} \approx 1.22 \text{ atm}$ and $P{N_2} \approx 0.735 \text{ atm}$. Their sum is $1.22 + 0.735 = 1.955 \text{ atm}$, which is very close to the calculated total pressure of 1.96 atm, accounting for rounding.)
This approach is highly convenient when the total pressure of the mixture is already known or easily determinable, as it avoids the need for volume and temperature if these are not directly provided for individual components.
Choosing the Right Method
The choice between these two methods often depends on the information available:
- Use the Ideal Gas Law when you know the moles ($n$), volume ($V$), and temperature ($T$) for the individual gas you are interested in.
- Use the Mole Fraction when you know the total pressure of the mixture ($P_{\text{total}}$) and the moles of each individual gas, making it straightforward to calculate the mole fraction.
Both methods are derived from fundamental principles of gas behavior and will yield consistent results when applied correctly.
