Question

A gas mixture contains oxygen and argon at partial pressures of \(0.60 \mathrm{~atm}\) and \(425 \mathrm{mmHg}\). If nitrogen gas added to the sample increases the total pressure to 1250 torr, what is the partial pressure, in torr, of the nitrogen added?

Short answer

369 torr

Step-by-step solution

Convert Partial Pressures to Same Units

First, convert each partial pressure to the same unit to ensure consistency. Convert oxygen's pressure from atm to torr: \text{Pressure (torr)} = \text{Pressure (atm)} \times 760 \text{torr/atm}. For oxygen: \(0.60 \text{atm} \times 760 \text{torr/atm} = 456 \text{torr} \).

Sum of Partial Pressures

Add the partial pressures of oxygen and argon to get the initial total pressure (in torr): \( 456 \text{torr} (O_2) + 425 \text{torr} (Ar) = 881 \text{torr} \).

Determine Partial Pressure of Nitrogen Added

Subtract the initial total pressure from the final total pressure to find the partial pressure of the nitrogen added: \( 1250 \text{torr} - 881 \text{torr} = 369 \text{torr} \).

Key concepts

Gas Laws

Gas laws are crucial for understanding how gases behave under various conditions. These laws relate variables such as pressure, volume, and temperature. One of the most fundamental gas laws is Boyle's Law, which states that the pressure of a gas is inversely proportional to its volume when temperature remains constant:
\[ P_1 V_1 = P_2 V_2 \]
Another important law is Charles's Law, which shows that the volume of a gas is directly proportional to its temperature (in Kelvin) when pressure is constant:
\[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]
For our particular problem, Dalton's Law of Partial Pressures is most relevant. According to Dalton's Law:
\[ P_{total} = P_1 + P_2 + P_3 + \text{...} \]
This law helps us understand that the total pressure of a gas mixture is the sum of the partial pressures of each component gas.

Partial Pressure

Partial pressure refers to the pressure that a single gas component in a mixture would exert if it were alone in the container. It is an essential concept when dealing with gas mixtures. You can think of partial pressure as the contribution of each individual gas to the total pressure. According to Dalton’s Law mentioned above, each gas in a mixture adds to the total pressure based on its partial pressure.
For example, in our exercise, the mixture contains oxygen, argon, and nitrogen. The partial pressures of oxygen and argon can be converted, summed up, and finally, the partial pressure of nitrogen can be figured out by subtracting the initial total pressure from the new total pressure.

Pressure Unit Conversion

Pressure unit conversion is a necessary skill to solve many gas law problems. Pressure can be measured in various units such as atmospheres (atm), millimeters of mercury (mmHg), and torr. Sometimes, you may need to convert these units to get a consistent measurement. For example:
- 1 atmosphere (atm) is equal to 760 millimeters of mercury (mmHg).
To solve our problem, it was crucial to convert the oxygen's partial pressure from atm to torr (1 torr = 1 mmHg):\[0.60 \text{ atm} \times 760 \text{ torr/atm} = 456 \text{ torr} \]This conversion ensures all pressures are in the same unit and can be accurately added together.

Total Pressure

Total pressure is the sum of the partial pressures of all gases present in a mixture. Understanding this concept helps solve problems involving gas mixtures. In our exercise, we calculated the total pressure by adding the partial pressures of oxygen and argon.
Starting with the given pressures:
\[ 0.60 \text{ atm} \times 760 (\text{conversion to torr}) = 456 \text{ torr (O}_2\text{)} \]\[ 425 \text{ torr (Ar)} \] Adding these together gives the initial total pressure:
\[ 456 \text{ torr} + 425 \text{ torr} = 881 \text{ torr} \]To find the partial pressure of the nitrogen added, subtract this initial total pressure from the final total pressure:
\[ 1250 \text{ torr (final total pressure)} - 881 \text{ torr (initial total pressure)} = 369 \text{ torr (N}_2\text{)} \]This shows how each gas contributes to the overall pressure in the container.