Problem 43 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

If one mole of a monoatomic gas \((\gamma=5 / 3)\) is mixed with one mole of a diatomic gas \((\gamma=7 / 5)\), the value of \(\gamma\) for the mixture is (a) 1 (b) \(1.5\) (c) 2 (d) \(3.0\)

Step-by-Step Solution

Answer

The value of \(\gamma\) for the mixture is 1.5.

Step 1: Understand the Concept of Heat Capacities

The symbol \(\gamma\) represents the heat capacity ratio, also known as the adiabatic index, which is the ratio of the heat capacity at constant pressure \(C_p\) to the heat capacity at constant volume \(C_v\), i.e., \(\gamma = \frac{C_p}{C_v}\). For a monoatomic gas, \(\gamma = \frac{5}{3}\) and for a diatomic gas, \(\gamma = \frac{7}{5}\). When two gases are mixed, the total heat capacity of the mixture is the sum of the heat capacities of the individual gases.

Step 2: Calculate the Heat Capacities for Each Gas

The number of moles for both gases is the same (one mole each), so we'll calculate heat capacities per mole. For the monoatomic gas, \(C_{v, monoatomic} = \frac{R}{\gamma - 1}\), where \(R\) is the universal gas constant. Similarly, for the diatomic gas, \(C_{v, diatomic} = \frac{R}{\gamma - 1}\).

Step 3: Calculate the Total Heat Capacity at Constant Volume

The total heat capacity at constant volume for the mixture, \(C_{v, mixture}\), is the sum of the heat capacities of the individual gases: \(C_{v, mixture} = C_{v, monoatomic} + C_{v, diatomic}\).

Step 4: Calculate the Total Heat Capacity at Constant Pressure

The total heat capacity at constant pressure for the mixture, \(C_{p, mixture}\), is also the sum of the heat capacities at constant pressure of the individual gases: \(C_{p, mixture} = C_{p, monoatomic} + C_{p, diatomic}\). Since \(C_p = \gamma C_v\), this can be found using the respective \(\gamma\) values.

Step 5: Calculate the Adiabatic Index of the Mixture

The adiabatic index for the mixture is given by \(\gamma_{mixture} = \frac{C_{p, mixture}}{C_{v, mixture}}\). Plugging in the values calculated from the previous steps, we get the \(\gamma\) for the mixture.

Step 6: Determine the Answer

Using \(C_{v, monoatomic} = \frac{3}{2}R\) and \(C_{v, diatomic} = \frac{5}{2}R\), we get \(C_{v, mixture} = (\frac{3}{2} + \frac{5}{2})R = 4R\). Using \(\gamma_{monoatomic} = \frac{5}{3} \Rightarrow C_{p, monoatomic} = \frac{5}{3} C_{v, monoatomic}\) and \(\gamma_{diatomic} = \frac{7}{5} \Rightarrow C_{p, diatomic} = \frac{7}{5} C_{v, diatomic}\), we get \(C_{p, mixture} = (\frac{5}{3} \times \frac{3}{2}R + \frac{7}{5} \times \frac{5}{2}R) = (\frac{5}{2}R + \frac{7}{2}R) = 6R\). Finally, \(\gamma_{mixture} = \frac{C_{p, mixture}}{C_{v, mixture}} = \frac{6R}{4R} = 1.5\). Therefore, the correct answer is (b) 1.5.

Key Concepts

Heat Capacity Ratio

The heat capacity ratio, commonly denoted by the symbol \(\gamma\), is an essential concept in thermodynamics, particularly in the study of adiabatic processes in gases. It is defined as the ratio of the specific heat capacity of a gas at constant pressure \(C_p\) to that at constant volume \(C_v\). Formally, \(\gamma = \frac{C_p}{C_v}\).

The value of \(\gamma\) gives insights into the thermal properties of a substance and affects how the substance behaves when it undergoes an adiabatic process, which is a process where no heat is exchanged with the surroundings. A higher \(\gamma\) indicates that a gas is more adiabatically 'stiff,' meaning that its temperature changes more significantly under compression or expansion without heat exchange.

Understanding the implications of the heat capacity ratio is essential in various applications, ranging from the design of engines and refrigerators to the study of atmospheric phenomena and the behavior of stars in astrophysics. For a monoatomic gas, the typical value of \(\gamma\) is \(\frac{5}{3}\), and for a diatomic gas, it is approximately \(\frac{7}{5}\). These values take into account the degrees of freedom available to each type of molecule based on its atomic structure.

Monoatomic Gas

A monoatomic gas, as its name implies, is composed of single atoms, with noble gases like helium, neon, and argon being prime examples. These gases have the simplest molecular structure among elements, with only one atom per molecule. This simplicity provides them with a characteristic heat capacity ratio (or adiabatic index) of \(\gamma = \frac{5}{3}\).

In thermodynamics, the behavior of monoatomic gases is well described by the kinetic theory of gases. Because they possess three degrees of freedom (movement in three-dimensional space), their behavior can be predicted fairly accurately using simple models. Additionally, monoatomic gases are ideal for studying fundamental thermodynamic processes because their simple structure reduces the influence of factors like vibration or rotation that complicate the behavior of more complex molecules.

Energy Distribution in a Monoatomic Gas

The kinetic energy of a monoatomic gas is equally distributed among its translational degrees of freedom, according to the equipartition theorem. Since there are no chemical bonds to store potential energy, all the internal energy of a monoatomic gas is in the form of kinetic energy.This straightforward energy distribution is why the value of \(\gamma\) for monoatomic gases is higher compared to more complex types of gases, where energy is also distributed in rotational and vibrational forms.

Diatomic Gas

Diatomic gases consist of molecules made up of two atoms. Common examples of diatomic gases include hydrogen \(H_2\), oxygen \(O_2\), and nitrogen \(N_2\). In these gases, the molecules have more degrees of freedom compared to monoatomic gases, including not only translational but also rotational motions.

The heat capacity ratio \(\gamma\) for diatomic gases is usually around \(\frac{7}{5}\) due to these additional degrees of freedom. With two rotating axes (as linear molecules do not rotate about the axis of the bond), plus the translational movement, diatomic gases have a total of five degrees of freedom in classical terms, which is where the numerator of its \(\gamma\) value derives from.

Energy in Rotational Motions

In diatomic gases, part of the internal energy is stored in rotational kinetic energy. This impacts not just the heat capacity but also the way the gas expands or compresses under adiabatic conditions. These gases are less 'stiff' adiabatically than monoatomic gases, which is reflected in their lower \(\gamma\) values.The complexity introduced by additional degrees of freedom in diatomic gases means their physical behavior under different conditions can be more nuanced. When understanding thermodynamic processes involving diatomic gases, considerations around these rotational and vibrational modes become crucial.