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Magazine Chapter 2: Problem 48
Is it reasonable to assume that at the given states the substance behaves as an ideal gas? a. Oxygen at \(30^{\circ} \mathrm{C}, 3 \mathrm{MPa}\) b. Methane at \(30^{\circ} \mathrm{C}, 3 \mathrm{MPa}\) c. Water at \(30^{\circ} \mathrm{C}, 3 \mathrm{MPa}\) d. R-410A at \(30^{\circ} \mathrm{C}, 3 \mathrm{MPa}\) e. \(\mathrm{R}-410 \mathrm{~A}\) at \(30^{\circ} \mathrm{C}, 100 \mathrm{kPa}\)
Short Answer
Expert verified
a. Yes, b. Yes, c. No, d. No, e. Yes
Step by step solution
01
Understanding the Ideal Gas Assumption
To determine if a substance behaves as an ideal gas, we typically examine the relationship between pressure (P), temperature (T), and volume (V). The ideal gas law is expressed as \(PV = nRT\), where \(R\) is the ideal gas constant and \(n\) is the number of moles. Substances at high temperature and low pressure generally behave as ideal gases. We will consider the critical temperature and critical pressure for each substance to make our determination.
02
Evaluate Oxygen at 30掳C, 3 MPa
The critical temperature of oxygen is around -118.6掳C and the critical pressure is 5.043 MPa. At 30掳C and 3 MPa, the temperature is well above the critical temperature and the pressure is below the critical pressure. Hence, oxygen can be assumed to behave as an ideal gas at these conditions.
03
Evaluate Methane at 30掳C, 3 MPa
The critical temperature of methane is -82.5掳C and the critical pressure is 4.6 MPa. At 30掳C, the temperature is higher than the critical temperature, and the pressure is under the critical pressure. Therefore, methane behaves approximatively like an ideal gas under these conditions.
04
Evaluate Water at 30掳C, 3 MPa
The critical temperature of water is 374掳C and the critical pressure is 22.064 MPa. At 30掳C, the temperature is significantly below the critical temperature, and the pressure is not near the critical pressure either. Thus, water is in a liquid state and does not behave as an ideal gas at these conditions.
05
Evaluate R-410A at 30掳C, 3 MPa
The critical temperature of R-410A is about 72掳C and its critical pressure is around 4.882 MPa. While the temperature is below the critical temperature, the pressure is still below its critical pressure. This implies R-410A is more likely to be near liquid phase and thus does not behave like an ideal gas.
06
Evaluate R-410A at 30掳C, 100 kPa
At 30掳C and 100 kPa, the pressure is significantly low compared to its critical pressure of about 4.882 MPa. This low-pressure condition can allow R-410A to behave more typically as an ideal gas because there is less deviation at low pressures.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Critical Temperature
Critical temperature is the highest temperature at which a substance can exist as a liquid. It is significant because, above this temperature, no amount of pressure can liquefy the gas. Understanding the critical temperature helps us predict whether a substance will behave as an ideal gas under certain conditions.
Here鈥檚 how:
Here鈥檚 how:
- If the actual temperature is significantly higher than the critical temperature, the gas is more likely to behave ideally because it is further from the liquid phase.
- Conversely, if it鈥檚 closer to or below the critical temperature, interactions between molecules might cause the gas to deviate from ideal behavior.
Critical Pressure
Critical pressure is the pressure required to liquefy a gas at its critical temperature. Like critical temperature, it plays a crucial role in assessing the behavior of gases.
Important observations about critical pressure include:
Important observations about critical pressure include:
- If the system's pressure is notably less than the critical pressure, the gas is likely to behave more ideally, as it is less compressed.
- When pressure is nearer or equals critical pressure, non-ideal interactions may occur, causing deviations from ideal gas behavior.
Ideal Gas Law
The Ideal Gas Law is a fundamental equation in thermodynamics, represented as \(PV = nRT\). This equation describes the relationship between pressure (\(P\)), volume (\(V\)), and temperature (\(T\)) of an ideal gas, with \(n\) representing the number of moles and \(R\) being the gas constant.
Key points to remember about the Ideal Gas Law include:
Key points to remember about the Ideal Gas Law include:
- It assumes gas particles have no volume and do not interact with each other, which is not true for real gases under many conditions.
- The Law is a good approximation at high temperatures and low pressures because gases behave more ideally when particles move freely without strong intermolecular forces.
Thermodynamic Properties
Thermodynamic properties of substances are essential for understanding their states and behaviors under various conditions. These properties include variables such as temperature, pressure, and volume, which are interrelated through the Ideal Gas Law. Additionally, substances have unique critical points defining the boundaries of their phases.
In the context of ideal gases:
In the context of ideal gases:
- High temperature and low pressure increase the likelihood of ideal gas behavior because they minimize real-world particle interactions.
- Understanding these properties gives insight into phase changes, reaction feasibility, and energy changes in thermodynamic processes.
