What Is The Pressure (in Atm) In A 5.00 L Tank With 10.00 Grams Of Oxygen Gas At 350 K? R = 0.08206 L
Understanding the pressure of a gas within a confined space is fundamental in chemistry, physics, and engineering. When dealing with gases, the ideal gas law provides a straightforward way to calculate the pressure, given certain parameters such as volume, temperature, amount of gas, and the ideal gas constant. In this article, we will explore how to determine the pressure in atmospheres (atm) of oxygen gas contained in a 5.00-liter tank, with a mass of 10.00 grams at a temperature of 350 Kelvin, utilizing the ideal gas law. This comprehensive guide will cover the theoretical background, step-by-step calculation, and practical applications of the concept.
Understanding the Ideal Gas Law
The ideal gas law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and amount of an ideal gas. It is expressed as:
\[ PV = nRT \]
Where:
- \( P \) = pressure (atm)
- \( V \) = volume (L)
- \( n \) = number of moles of gas (mol)
- \( R \) = ideal gas constant (L·atm/(mol·K))
- \( T \) = temperature (K)
This law assumes that gases behave ideally, meaning their particles do not interact except elastic collisions, and the volume of particles themselves is negligible compared to the volume of the container.
Parameters Given in the Problem
Before proceeding with calculations, let's clearly identify the given data:
- Volume of the tank, \( V \): 5.00 L
- Mass of oxygen gas, \( m \): 10.00 grams
- Temperature, \( T \): 350 K
- Ideal gas constant, \( R \): 0.08206 L·atm/(mol·K)
Our goal: Find the pressure \( P \) in atmospheres.
Step-by-Step Calculation Process
To determine the pressure, we need to follow a systematic approach:
- Convert mass of oxygen to moles (\( n \)).
- Apply the ideal gas law to solve for \( P \).
Let's break down each step:
1. Calculating the Number of Moles of Oxygen (\( n \))
The molar mass of oxygen gas (\( O_2 \)) is approximately:
- Atomic mass of oxygen (O): 16.00 g/mol
- Molecular mass of \( O_2 \): 2 × 16.00 g/mol = 32.00 g/mol
Using the formula:
\[ n = \frac{m}{M} \]
Where:
- \( m \) = mass of oxygen = 10.00 g
- \( M \) = molar mass of \( O_2 \) = 32.00 g/mol
Calculating:
\[ n = \frac{10.00\, \text{g}}{32.00\, \text{g/mol}} = 0.3125\, \text{mol} \]
2. Applying the Ideal Gas Law to Find Pressure (\( P \))
Rearranged ideal gas law:
\[ P = \frac{nRT}{V} \]
Substituting the known values:
\[ P = \frac{0.3125\, \text{mol} \times 0.08206\, \text{L·atm/(mol·K)} \times 350\, \text{K}}{5.00\, \text{L}} \]
Calculating numerator:
\[ 0.3125 \times 0.08206 \times 350 = 8.956 \]
Dividing by volume:
\[ P = \frac{8.956}{5.00} = 1.7912\, \text{atm} \]
Therefore, the pressure of the oxygen gas in the tank is approximately 1.79 atm.
---
Practical Applications of Gas Pressure Calculations
Understanding how to compute gas pressure has numerous real-world applications:
- Industrial Gas Storage: Ensuring containers can withstand the pressures exerted by stored gases.
- Chemical Reactions: Designing reactors where gas pressures influence reaction rates.
- Aerospace Engineering: Calculating cabin pressures in spacecraft and aircraft.
- Medical Applications: Managing oxygen supply systems in hospitals and emergency settings.
- Environmental Science: Modeling atmospheric gas behaviors and pollutant dispersion.
---
Factors Affecting Gas Pressure in Real-World Scenarios
While the ideal gas law provides a good approximation, several factors can influence actual gas pressure:
- Non-ideal behavior: At high pressures or low temperatures, gases deviate from ideality due to intermolecular forces.
- Container elasticity: Real tanks may expand or contract slightly under pressure.
- Gas purity: Impurities can affect the behavior and pressure.
- Temperature fluctuations: Changes in temperature directly impact pressure, according to Gay-Lussac’s law.
---
Additional Considerations for Accurate Calculations
To ensure precise calculations, consider the following:
- Verify units: Always convert measurements to compatible units.
- Use the correct gas constant: R = 0.08206 L·atm/(mol·K) for pressure in atm.
- Account for gas purity: If the gas isn't pure, adjust the molar amount accordingly.
- Check for ideality: At extreme conditions, use real gas equations like Van der Waals if necessary.
---
Summary of the Calculation
In summary, given a 5.00-liter tank containing 10.00 grams of oxygen gas at 350 K, the pressure exerted by the gas can be calculated as follows:
- Convert mass to moles: \( 10.00\, \text{g} / 32.00\, \text{g/mol} = 0.3125\, \text{mol} \)
- Apply the ideal gas law:
\[ P = \frac{nRT}{V} = \frac{0.3125 \times 0.08206 \times 350}{5.00} \approx 1.79\, \text{atm} \]
This result indicates the oxygen gas exerts a pressure of approximately 1.79 atm within the tank under the given conditions.
Conclusion
Calculating the pressure of a gas in a closed container is a fundamental skill in chemistry that combines understanding of the ideal gas law with careful unit conversion and data interpretation. By knowing the mass of gas, temperature, and volume, you can accurately determine the pressure exerted by gases in various contexts. This knowledge is not only academically valuable but also practically essential in industrial processes, environmental science, and safety engineering. Always remember to consider real-world factors that may cause deviations from ideal behavior, and apply appropriate corrections or advanced models when necessary.
---
References:
- Atkins, P., & de Paula, J. (2010). Physical Chemistry (9th ed.). Oxford University Press.
- Zumdahl, S. S., & Zumdahl, S. A. (2014). Chemistry: An Atoms First Approach. Cengage Learning.
- Lyman, J. (2018). Gases and Gas Laws. Chemguide.
---
Keywords: gas pressure calculation, ideal gas law, oxygen gas, molar mass, chemistry calculations, pressure in atm, volume and temperature, gas laws, scientific computation
