Chapter 3: Problem 10
The value of \(C_{p} / C_{v}\) for inert gases is (a) \(1.33\) (b) \(1.66\) (c) \(2.13\) (d) \(1.99\)
Short Answer
Expert verified
(b) 1.66 is the correct answer.
Step by step solution
01
Understanding the Concept
For an inert (also known as noble or rare) gas, we consider the adiabatic index or gamma. This is the ratio of the specific heat at constant pressure (C_p) to the specific heat at constant volume (C_v). This value is often denoted as γ or kappa. For single atomic gases, which include inert gases like helium, neon etc., this ratio is 5/3.
02
Calculating the Decimal Equivalent of 5/3
The decimal equivalent or the approximate value of 5/3 can be calculated as follows: Divide 5 by 3, which gives the value 1.6666666. This can be approximately written as 1.66.
03
Matching the Calculated Value with the Options
On comparing the calculated value of 1.66 with the given options, it can be seen that the value matches with option (b). Thus, (b) 1.66 is the correct answer.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Specific Heat Ratio
Understanding the specific heat ratio is crucial when studying thermodynamics and the behavior of gases under different conditions. The specific heat ratio, also known as the adiabatic index, is a measure of how a gas responds to changes in pressure and volume without any heat exchange with its surroundings—an adiabatic process.
There are two types of specific heats for gases: at constant volume (\( C_v \) ) and at constant pressure (\( C_p \)). When a gas is heated at a constant volume, all the heat goes into raising the temperature, whereas at constant pressure, some of the heat is used to do work by expanding the gas. Therefore, \( C_p \)) is always greater than \( C_v \). The ratio of these two (\( C_p/C_v \)), is a dimensionless number that has significant implications in various physical phenomena, like the speed of sound in a gas, and the efficiency of various thermodynamic processes.
There are two types of specific heats for gases: at constant volume (\( C_v \) ) and at constant pressure (\( C_p \)). When a gas is heated at a constant volume, all the heat goes into raising the temperature, whereas at constant pressure, some of the heat is used to do work by expanding the gas. Therefore, \( C_p \)) is always greater than \( C_v \). The ratio of these two (\( C_p/C_v \)), is a dimensionless number that has significant implications in various physical phenomena, like the speed of sound in a gas, and the efficiency of various thermodynamic processes.
Cp/Cv Ratio
The \( Cp/Cv \) ratio, often symbolized as \( \gamma \)), is the ratio of the specific heat capacity at constant pressure to the specific heat capacity at constant volume. In simpler terms, this ratio tells you how much more heat energy is required to increase the temperature of a gas when it is allowed to expand compared to when it is confined to a constant volume.
An essential aspect to consider is that this ratio varies depending on the type of gas. For monatomic gases like inert gases, this ratio has a constant value of 5/3 or approximately 1.66. It's because these gases have simple atomic structures with fewer degrees of freedom. In contrast, for diatomic or polyatomic gases, which have more complex structures, this ratio will have different values.
An essential aspect to consider is that this ratio varies depending on the type of gas. For monatomic gases like inert gases, this ratio has a constant value of 5/3 or approximately 1.66. It's because these gases have simple atomic structures with fewer degrees of freedom. In contrast, for diatomic or polyatomic gases, which have more complex structures, this ratio will have different values.
Inert Gases
Inert gases, also known as noble gases, are a group of chemical elements with very similar properties: they are all odorless, colorless, monatomic gases with very low chemical reactivity. The noble gases include helium (He), neon (Ne), argon (Ar), krypton (Kr), xenon (Xe), and radon (Rn).
What's peculiar about these gases is their complete electron shells, which makes them quite stable and inert. Because of this stability, they don't create compounds easily. In terms of their thermal properties, inert gases are often used as a reference point due to their simple molecular structure, which simplifies the calculation of the adiabatic index or \( \gamma \). For this reason, instructors frequently use inert gases to introduce students to concepts of specific heat and adiabatic processes.
What's peculiar about these gases is their complete electron shells, which makes them quite stable and inert. Because of this stability, they don't create compounds easily. In terms of their thermal properties, inert gases are often used as a reference point due to their simple molecular structure, which simplifies the calculation of the adiabatic index or \( \gamma \). For this reason, instructors frequently use inert gases to introduce students to concepts of specific heat and adiabatic processes.
Gamma
Gamma (\( \gamma \)) plays a pivotal role in thermodynamics as it correlates with the physical properties of gases, especially during adiabatic processes. As noted earlier, for inert gases, \( \gamma \) equals 5/3. This value influences many aspects of the gas's behavior, like how its temperature and pressure will change if its volume changes adiabatically.
For example, in adiabatic expansion, where the volume of the gas increases, the pressure drops and so does the temperature. The magnitude of these changes is tightly related to \( \gamma \). The larger the \( \gamma \), the more dramatic the changes will be. Thus, understanding \( \gamma \) is key to predicting the behavior of gases in natural and industrial processes, like the design of engines and exhaust systems, predicting weather patterns, and even in astrophysics.
For example, in adiabatic expansion, where the volume of the gas increases, the pressure drops and so does the temperature. The magnitude of these changes is tightly related to \( \gamma \). The larger the \( \gamma \), the more dramatic the changes will be. Thus, understanding \( \gamma \) is key to predicting the behavior of gases in natural and industrial processes, like the design of engines and exhaust systems, predicting weather patterns, and even in astrophysics.
