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Magazine Chapter 7: Problem 41
A cylinder of medical oxygen has a volume of \(35.4 \mathrm{~L}\), and contains \(\mathrm{O}_{2}\) at a pressure of 151 atm and a temperature of \(25^{\circ} \mathrm{C}\). What volume of \(\mathrm{O}_{2}\) does this correspond to at normal body conditions, that is, \(1 \mathrm{~atm}\) and \(37^{\circ} \mathrm{C}\) ?
Short Answer
Expert verified
The volume of \(\mathrm{O}_2\) at body conditions is approximately 5590.8 L.
Step by step solution
01
Understanding the Problem
We have a cylinder containing oxygen gas at a certain pressure and temperature, and we need to find the volume of this gas at different pressure and temperature corresponding to normal body conditions.
02
List Known Values
We know the initial volume (\(V_1 = 35.4 \ \mathrm{L}\)), initial pressure (\(P_1 = 151 \ \mathrm{atm}\)), and initial temperature (\(T_1 = 25^{\circ} \mathrm{C}\)). Body conditions are the final conditions: pressure (\(P_2 = 1 \ \mathrm{atm}\)) and temperature (\(T_2 = 37^{\circ} \mathrm{C}\)).
03
Convert Temperatures to Kelvin
Convert the temperatures from Celsius to Kelvin using the formula: \[T(\mathrm{K}) = T(\mathrm{C}) + 273.15\]Thus, \(T_1 = 25 + 273.15 = 298.15 \ \mathrm{K}\) and \(T_2 = 37 + 273.15 = 310.15 \ \mathrm{K}\).
04
Apply the Combined Gas Law
Use the combined gas law, which states: \[\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\]Rearranged to find \(V_2\): \[V_2 = \frac{P_1 V_1 T_2}{P_2 T_1}\]
05
Substitute Known Values
Substitute the known values into the equation: \[V_2 = \frac{151 \ \mathrm{atm} \times 35.4 \ \mathrm{L} \times 310.15 \ \mathrm{K}}{1 \ \mathrm{atm} \times 298.15 \ \mathrm{K}}\]
06
Calculate the Result
Perform the calculation: \[V_2 = \frac{151 \times 35.4 \times 310.15}{1 \times 298.15} \approx 5590.8 \ \mathrm{L}\]Thus, the volume of \(\mathrm{O}_2\) at body conditions is approximately \(5590.8 \ \mathrm{L}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Gas Laws
Gas laws are essential in understanding the behavior of gases under various conditions of temperature, pressure, and volume. They help explain how gases expand, compress, and react to changes around them.
A few fundamental gas laws include:
A few fundamental gas laws include:
- Boyle's Law: It states that the pressure of a gas is inversely proportional to its volume at constant temperature.
- Charles's Law: It describes how the volume of a gas is directly proportional to its temperature at constant pressure.
- Avogadro's Law: It explains that the volume of a gas is directly proportional to the number of moles of gas, given constant temperature and pressure.
- Combined Gas Law: This is a fusion of Boyle's, Charles's, and Avogadro's laws, showing the relationship between pressure, temperature, and volume.
Temperature Conversion
Temperature is pivotal when exploring gas laws. It's essential to convert temperature readings from Celsius to Kelvin for accurate calculations in gas laws.
The Kelvin scale is an absolute temperature scale where 0 K is absolute zero, the point where all molecular motion stops. Converting from Celsius to Kelvin is straightforward:
The Kelvin scale is an absolute temperature scale where 0 K is absolute zero, the point where all molecular motion stops. Converting from Celsius to Kelvin is straightforward:
- Use the formula: \[T(\mathrm{K}) = T(\mathrm{C}) + 273.15\]
- This ensures that temperatures are always positive, as required by gas law equations.
Pressure-Volume Relationship
The pressure-volume relationship is a central idea in understanding how gases behave. Boyle's Law states:
\[P_1 V_1 = P_2 V_2\]under a constant temperature, highlighting how pressure and volume are inversely related.
\[P_1 V_1 = P_2 V_2\]under a constant temperature, highlighting how pressure and volume are inversely related.
- As pressure on a gas increases, volume decreases, and vice versa.
- This relationship explains how gases expand and contract based on external pressure.
- It's particularly evident in applications like medical oxygen tanks where the pressure variably dictates usage.
Oxygen Cylinders
Oxygen cylinders are medical tools that store oxygen gas under high pressure. These cylinders are essential for healthcare as they provide supplemental oxygen to patients. Understanding how gases work in these cylinders helps manage and utilize them effectively.
Key points about oxygen cylinders:
Key points about oxygen cylinders:
- They store oxygen at high pressures to fit more gas into a smaller volume.
- Effective management involves temperature and pressure monitoring to prevent hazards.
- Regulators adjust the outgoing pressure to safe levels for patient use.
Ideal Gas Law
The Ideal Gas Law is a backbone equation in the study of gases. It expands the combined gas laws into a simple formula:
\[PV = nRT\]Here, \(P\) is pressure, \(V\) is volume, \(n\) represents the number of moles, \(R\) is the ideal gas constant, and \(T\) is temperature in Kelvin.
\[PV = nRT\]Here, \(P\) is pressure, \(V\) is volume, \(n\) represents the number of moles, \(R\) is the ideal gas constant, and \(T\) is temperature in Kelvin.
- This law combines to show how gases respond to changes in conditions.
- It's useful for predicting how a gas will react when subjected to different environmental factors like medical storage situations.
- While ideal, this law assumes there are no interactions between gas molecules, which isn't always true for real gases.
