Problem 19 - Chemistry in 30 days

Question

The \(C_{p} / C_{v}\) ratio for noble gases is (a) \(1.99\) (b) \(1.66\) (c) \(1.33\) (d) \(1.0\)

Step-by-Step Solution

Answer

The correct option is (b) 1.66.

Step 1: Understand the Problem

We need to determine the correct value of the ratio \( \frac{C_p}{C_v} \) for noble gases. Here, \( C_p \) is the heat capacity at constant pressure, and \( C_v \) is the heat capacity at constant volume. This ratio is also known as the adiabatic index or gamma \( \gamma \).

Step 2: Recall the Relevant Formula

For a monatomic ideal gas, which includes noble gases, the adiabatic index \( \gamma \) is given by \( \frac{C_p}{C_v} = \frac{5}{3} \). This is derived based on the degrees of freedom and the kinetic theory of gases.

Step 3: Calculate \( \gamma \) Using Known Values

For monatomic gases such as noble gases, the values for \( C_v \) and \( C_p \) are \( \frac{3}{2}R \) and \( \frac{5}{2}R \) respectively, where \( R \) is the universal gas constant. Thus, \( \gamma = \frac{C_p}{C_v} = \frac{\frac{5}{2}R}{\frac{3}{2}R} = \frac{5}{3} = 1.66 \).

Step 4: Match the Calculation to the Options

Comparing the calculated \( \gamma \) value of 1.66 with the given options, we find that option (b) matches with \( 1.66 \).

Key Concepts

Noble Gases

Noble gases are a group of chemical elements that are characterized by their lack of chemical reactivity. They are located in Group 18 of the periodic table and include elements such as helium, neon, argon, krypton, xenon, and radon. These gases are colorless, odorless, and tasteless. Because they have a complete valence electron shell, noble gases tend not to form compounds readily with other elements.
Noble gases are monatomic, meaning that they exist as single atoms rather than molecules. This quality makes them ideal for studying basic gas properties without the complications of intermolecular forces present in other gases.
Their lack of chemical reactivity and monatomic nature simplify the mathematical expressions for thermodynamic properties, such as the adiabatic index.

Cp/Cv Ratio

The \(\frac{C_p}{C_v}\) ratio, also known as the adiabatic index or gamma \(\gamma\), is a key property of gases that describes how a gas responds to changes in pressure and temperature without heat exchange. It is the ratio of heat capacity at constant pressure \(C_p\) to heat capacity at constant volume \(C_v\).
For noble gases, which are typically monatomic, the \(\gamma\) value is particularly important. It reflects the degrees of freedom of gas molecules. In a monatomic ideal gas, the degrees of freedom are solely translational, resulting in a specific \(\gamma\) value of \(\frac{5}{3}\) or \(1.66\). This ratio indicates how efficiently a gas can expand or compress adiabatically.
The \(\frac{C_p}{C_v}\) ratio governs adiabatic processes, which are critical in applications like thermodynamic engines and atmospheric science.

Heat Capacity

Heat capacity is a measure of the amount of heat energy required to raise the temperature of a substance by a certain amount. It's typically expressed in joules per degree Celsius or Kelvin.
There are two primary types of heat capacity in the context of gases:

• Constant Volume Heat Capacity \(C_v\): This is the heat capacity of a gas when its volume is kept constant. For noble gases, \(C_v = \frac{3}{2}R\), where \(R\) is the gas constant.
• Constant Pressure Heat Capacity \(C_p\): This is the heat capacity of a gas when its pressure is kept constant. For noble gases, \(C_p = \frac{5}{2}R\).

The difference between \(C_p\) and \(C_v\) arises from the additional work done by or on the system when volume changes occur at constant pressure. This is why \(C_p\) is always greater than \(C_v\) for any gas.

Kinetic Theory of Gases

The kinetic theory of gases provides a framework for understanding the behavior and properties of gases. It asserts that gases consist of numerous minuscule particles (atoms or molecules) that are in constant random motion. This theory explains several fundamental properties of gases, including pressure, temperature, and volume.
According to the kinetic theory, the pressure exerted by a gas is due to collisions of gas particles with the walls of the container. Temperature is related to the average kinetic energy of the gas particles.
For noble gases, being monatomic and having simple structures, the application of the kinetic theory is straightforward and obeys the ideal gas laws closely. This is why calculations related to the adiabatic index are easier to perform for gases such as helium or neon. In monatomic gases, with each atom having only translational motion, the heat capacities and the \(\gamma\) value can be derived accurately from principles outlined by the kinetic theory.