- Identify the individual partial pressures provided: \(P_{N_2} = 3.5 \text{ atm}\), \(P_{O_2} = 2.8 \text{ atm}\), \(P_{Ar} = 0.25 \text{ atm}\), and \(P_{He} = 0.15 \text{ atm}\).
- Use Dalton's Law of Partial Pressures, which states that total pressure \(P_T\) is the sum of partial pressures \(P_T = P_1 + P_2 + P_3 + P_4\).
- Calculate the total pressure: \(P_T = 3.5 + 2.8 + 0.25 + 0.15 = 6.7 \text{ atm}\).
- The total pressure inside the cylinder is \(\boxed{6.7 \text{ atm}}\).
Explanation
- Problem Analysis and Law Application
The problem asks us to determine the total pressure inside a gas cylinder containing four non-reacting gases: nitrogen (\$N_2\$), oxygen (\$O_2\$), argon (\$Ar\$), and helium (\$He\$). We are given their individual partial pressures:
- \(P_{N_2} = 3.5 \text{ atm}\)
- \(P_{O_2} = 2.8 \text{ atm}\)
- \(P_{Ar} = 0.25 \text{ atm}\)
- \(P_{He} = 0.15 \text{ atm}\)
According to Dalton's Law of Partial Pressures, the total pressure of a gas mixture is the sum of the partial pressures of the individual gases. Therefore, we can express the total pressure \(P_T\) as:
\[P_T = P_{N_2} + P_{O_2} + P_{Ar} + P_{He}\]
Our objective is to calculate this sum.
- Calculating Total Pressure
To find the total pressure, we sum the given partial pressures:
\[P_T = 3.5 + 2.8 + 0.25 + 0.15\]
First, sum the main components:
\[3.5 + 2.8 = 6.3\]
Next, sum the remaining components:
\[0.25 + 0.15 = 0.40\]
Finally, add these results together:
\[P_T = 6.3 + 0.4 = 6.7 \text{ atm}\]
The total pressure inside the cylinder is \(6.7 \text{ atm}\).
Examples
In real-life scenarios, such as scuba diving, understanding gas mixtures is critical. Divers use tanks filled with breathing gases (like Nitrox or Heliox). Dalton's Law helps engineers and divers calculate the total pressure of the gas mix to ensure it remains safe and effective for breathing at various depths, preventing oxygen toxicity or decompression sickness. Just as we summed these pressures, experts determine the safe threshold for divers by analyzing the total pressure of each gas component in their tank.