Chapter 18: Problem 8
A welder using a tank of volume 0.0750 m\(^3\) fills it with oxygen (molar mass 32.0 g/mol) at a gauge pressure of 3.00 \(\times\) 10\({^5}\) Pa and temperature of 37.0\(^\circ\)C. The tank has a small leak, and in time some of the oxygen leaks out. On a day when the temperature is 22.0\(^\circ\)C, the gauge pressure of the oxygen in the tank is 1.80 \(\times\) 10\({^5}\) Pa. Find (a) the initial mass of oxygen and (b) the mass of oxygen that has leaked out.
Short Answer
Step by step solution
Convert Temperatures
Calculate Initial Moles of Oxygen
Calculate Initial Mass of Oxygen
Calculate Final Moles of Oxygen
Calculate Final Mass of Oxygen
Calculate Mass of Oxygen Leaked
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Molar Mass
Molar mass is typically expressed in units of grams per mole (g/mol). For oxygen, the molar mass is 32.0 g/mol. This is because oxygen naturally occurs as a diatomic molecule, O extsubscript{2}, with each oxygen atom having an atomic mass of approximately 16.0 amu.
So, when using the ideal gas law, the molar mass allows us to relate the number of moles to the actual mass of the gas. It acts as a conversion factor to determine the total mass contained in a certain number of moles, helping us calculate how much of the substance is present in given conditions. When solving problems, always ensure that mass and mole relationships are correctly established using the molar mass for precise results.
Gauge Pressure
Here's how it works:
- Gauge pressure is positive when it is greater than atmospheric pressure.
- It is zero when it equals atmospheric pressure.
- When less than atmospheric pressure, the gauge pressure is negative.
In the given problem, the initial gauge pressure of the oxygen in the tank is 3.00 × 10 extsuperscript{5} Pa. To find the true pressure of the gas (known as absolute pressure), one must add the atmospheric pressure (around 1.00 × 10 extsuperscript{5} Pa) to the gauge pressure. Thus, the absolute initial pressure becomes 4.00 × 10 extsuperscript{5} Pa. Understanding gauge pressure is essential for accurately applying the ideal gas law as it directly affects the calculated values of moles and mass of a gas.
Temperature Conversion
Kelvin is used because it is an absolute temperature scale, starting at absolute zero. In contrast, the Celsius scale doesn't have such a clear physical meaning in terms of kinetic energy of particles, which is what temperature fundamentally measures.
To convert Celsius to Kelvin, you add 273.15 to the Celsius temperature. For example:
- The initial temperature in the problem is 37.0°C. Converted to Kelvin: 37.0 + 273.15 = 310.15 K.
- The final temperature is 22.0°C. Converted to Kelvin: 22.0 + 273.15 = 295.15 K.
These conversions ensure that calculations using the ideal gas law are consistent and accurate.
Gas Leaks
In the context of the problem, a leak results in a change in the pressure inside an oxygen tank. Initially, we see a high gauge pressure that drops as some gas escapes due to a leak. This naturally leads to a decrease in the number of moles as well as the total mass of the gas inside.
When dealing with gas leaks:
- It's crucial to monitor changes in pressure, which can inform you about how much gas has escaped.
- Comparing initial and final conditions can help determine the amount of gas that has leaked. In the exercise, we observe a reduction from 3.00 × 10 extsuperscript{5} Pa to 1.80 × 10 extsuperscript{5} Pa in gauge pressure.
This allows one to calculate the amount of gas that is still contained versus what has been lost due to the leak, providing a comprehensive understanding of the situation.