Question

For a gas \(\left(R / C_{V}\right)=0.67\). This gas is made up of molecules which are (A) Diatomic (B) monoatomic (C) polyatomic (D) mixture of diatomic and polyatomic molecules

Short answer

The given ratio of \(R / C_{V} = 0.67\) does not match the expected ratios for monoatomic (\(R/C_{V} = 3\)), diatomic (\(R/C_{V} = 1.67\)), or polyatomic (\(R/C_{V} = 1.4\)) gases. Therefore, the gas is likely a mixture of different types of molecules, specifically a mixture of diatomic and polyatomic molecules. The correct answer is (D) mixture of diatomic and polyatomic molecules.

Step-by-step solution

Recall the Ratios for Different Types of Molecules

For monoatomic, diatomic, and polyatomic molecules, the ratios of \(R/C_{V}\) are as follows: 1. Monoatomic: \(R/C_{V} = \frac{3}{3 - 2} = \frac{3}{1} = 3\) 2. Diatomic: \(R/C_{V} = \frac{5}{5 - 2} = \frac{5}{3} = 1.67\) 3. Polyatomic: \(R/C_{V} = \frac{7}{7 - 2} = \frac{7}{5} = 1.4\) Now we will compare the given ratio to these values.

Compare the Given Ratio to Expected Ratios

The given ratio is \(R / C_{V} = 0.67\). Now we will compare this value to the expected ratios for each type of molecule in order to identify the type of molecules present in the gas: 1. Monoatomic (\(R/C_{V}\) = 3): The given ratio is not equal to the expected ratio for a monoatomic gas. 2. Diatomic (\(R/C_{V}\) = 1.67): The given ratio is not equal to the expected ratio for a diatomic gas. 3. Polyatomic (\(R/C_{V}\) = 1.4): The given ratio is not equal to the expected ratio for a polyatomic gas. Since the given ratio doesn't match the expected ratio for any single type of molecule (monoatomic, diatomic, or polyatomic), we can conclude that the gas is likely a mixture of different types of molecules.

Select the Correct Option

Given the comparison of the given ratio to the expected ratios for each type of molecule, we conclude that the gas is made up of molecules that are a mixture of diatomic and polyatomic molecules. So, the correct option is: (D) mixture of diatomic and polyatomic molecules

Key concepts

Specific Heat Ratios

Specific heat ratio is an essential concept in thermodynamics, often represented by the symbol \( \gamma \). It is the ratio of the specific heat at constant pressure \( C_{P} \) to the specific heat at constant volume \( C_{V} \). The general formula is given by \( \gamma = \frac{C_{P}}{C_{V}} \). This value helps us understand how gases will behave under different thermodynamic processes.

For practical purposes, we also discuss \( R/C_{V} \), where \( R \) is the universal gas constant. The context in which you see \( R/C_{V} \) helps determine the molecular structure of a gas. Each type of molecule, whether monoatomic, diatomic, or polyatomic, has a unique specific heat ratio value that reflects its molecular characteristics and complexity.

By comparing given ratios to the standard values, one can deduce the possible types of molecules present in a gas sample, as shown in the original exercise where a non-standard ratio suggested a mixture of molecular types.

Monoatomic Molecules

Monoatomic molecules consist of single atoms. These atoms are not bonded with other atoms, making them the simplest form of particles. Noble gases like helium (He) and neon (Ne) are classic examples of monoatomic gases.

For monoatomic gases, the specific heat ratio \( R/C_{V} \) is theoretically equal to 3. This is due to the fact that monoatomic molecules have three degrees of freedom: motions along the x, y, and z axes. Since there are no molecular bonds to contribute additional energy modes like rotations or vibrations, the calculation for their specific heat capacity is straightforward.

Understanding these fundamental details about monoatomic molecules helps to differentiate them from more complex molecules in calculations related to thermodynamic properties.

Diatomic Molecules

Diatomic molecules are made up of two atoms. These can be the same kind of atom, like the nitrogen molecule \( \text{N}_2 \), or different atoms, as in hydrogen chloride \( \text{HCl} \).

For diatomic gases, the specific heat ratio \( R/C_{V} \) is approximately 1.67. This ratio arises because diatomic molecules have additional degrees of freedom compared to monoatomic molecules. In addition to translational modes, diatomic molecules have rotational energy modes which affect their thermodynamic properties. In some cases, vibrational modes can also contribute, especially at higher temperatures.

These additional modes result in different heat capacity values, which is why diatomic gases have a specific heat ratio different from monoatomic and polyatomic gases. Recognizing these differences is crucial when working through thermodynamic problems with diatomic gases.

Polyatomic Molecules

Polyatomic molecules refer to molecules containing more than two atoms. Examples include water \( \text{H}_2\text{O} \) and carbon dioxide \( \text{CO}_2 \).

The specific heat ratio \( R/C_{V} \) for polyatomic gases typically is around 1.4. This value is influenced by the complex structure of these molecules, which allows for even more degrees of freedom than diatomic molecules. These freedoms include translational, rotational, and multiple vibrational modes.

Due to their complicated nature, polyatomic molecules absorb and distribute energy in a varied manner, accounting for their specific thermodynamic behavior and properties. Understanding the unique specific heat ratio of polyatomic molecules is essential for tackling problems involving complex gas mixtures, as well as identifying these gases through their thermodynamic signatures.