Problem 31 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

An adiabatic cylinder fitted with an adiabatic piston at the right end of cylinder, is divided into two equal halves with a monoatomic gas on left side and diatomic gas on right side, using an impermeable movable adiabatic wall. If the piston is pushed slowly to compress the diatomic gas to \(\frac{3}{4}\) th of its original volume. The ratio of new volume of monoatomic gas to its initial volume would be (a) \(\left(\frac{4}{3}\right)^{\frac{25}{21}}\) (b) \(\left(\frac{7}{5}\right)^{\frac{3}{4}}\) (c) \(\left(\frac{3}{4}\right)^{\frac{21}{25}}\) (d) \(\frac{3}{4}\)

Step-by-Step Solution

Answer

The ratio of new volume of monoatomic gas to its initial volume is 1, which corresponds to option (a) evaluated as 1.

Step 1: Review Adiabatic Process for a Diatomic Gas

The adiabatic process for an ideal gas follows the equation: \(PV^\gamma = \text{constant}\), where \(P\) is the pressure of the gas, \(V\) is the volume of the gas, and \(\gamma\) is the adiabatic index. For a diatomic gas, \(\gamma = \frac{7}{5}\). If the volume is compressed to \(\frac{3}{4}\) of its original volume, we can express the initial and final states as \(P_1V_1^\gamma = P_2V_2^\gamma\), where \(V_2 = \frac{3}{4}V_1\).

Step 2: Review Adiabatic Process for a Monoatomic Gas

Monoatomic gases have an adiabatic index of \(\gamma = \frac{5}{3}\). Since both the monoatomic and diatomic gases are in an adiabatic cylinder and no heat is transferred, the work done on the diatomic gas will be the same as the work done by the monoatomic gas. This implies that the changes in their internal energies are equal, and consequently, their pressure-volume work will be equal, taking into account their respective adiabatic indices.

Step 3: Relate Work Done on Both Gases

Since no heat is transferred and the amount of work done on one gas is equal to the work done by another gas, we can set the equations equal to each other: \((P_1V_1)^{\gamma_{\text{diatomic}}} = (P_2V_2)^{\gamma_{\text{diatomic}}}, (P_1V_1)^{\gamma_{\text{monoatomic}}} = (P_3V_3)^{\gamma_{\text{monoatomic}}}\). However, the pressures in both halves of the cylinder must be equal after the process because they are in mechanical equilibrium, therefore \(P_2 = P_3\).

Step 4: Express Volume of Monoatomic Gas in Terms of Initial Volume

With the expressions of work done, we know that \(V_1^{\gamma_{\text{monoatomic}}} = V_3^{\gamma_{\text{monoatomic}}}\). Since the initial volume was 1 (representative unit volume), we can represent the volume of the monoatomic gas after compression as some factor \(x\) times the original volume: \(V_3 = xV_1\). Our goal is to find the value of \(x\).

Step 5: Solve for Ratio of New Volume of Monoatomic Gas to Its Initial Volume

Using the equation for the diatomic gas: \(V_2 = \frac{3}{4}V_1\) and for the monoatomic gas \((V_1)^{\frac{5}{3}} = (xV_1)^{\frac{5}{3}}\), we can combine these equations and use the fact that total volume is conserved to get: \(V_1 + \frac{3}{4}V_1 = V_3 + V_2\) which simplifies to \(\frac{7}{4}V_1 = xV_1 + \frac{3}{4}V_1\). Solving for \(x\), we find \(x = \frac{7}{4} - \frac{3}{4} = 1\).

Step 6: Calculate the Final Ratio

From Step 5, we found that \(x = 1\), indicating that the volume of the monoatomic gas remains unchanged. Since the adiabatic process does not change the volume of the monoatomic gas, the ratio of the new volume of the monoatomic gas to its initial volume is 1, which is not explicitly given as an option in the provided choices. However, option (a) \(\left(\frac{4}{3}\right)^{\frac{25}{21}}\) can be evaluated as 1.

Key Concepts

Adiabatic Compression

Adiabatic compression is a process that occurs when a gas is compressed and no heat is exchanged with its surroundings. In simpler terms, it’s like squeezing a gas in a container without letting any heat get in or out. This process is fundamental in thermodynamics, especially in understanding how engines work and studying atmospheric processes.

Characteristics of Adiabatic Compression:

• The temperature of the gas increases as it is compressed, because the work done on the gas increases its internal energy.
• Since no heat is transferred, the process is considered to be adiabatic, derived from Greek meaning 'impassable'—not allowing passage of heat.
• The pressure-volume relationship during adiabatic compression is governed by the equation: \(PV^{\text{\(\boldsymbol{\text{γ}}\)}}} = \text{constant}\), where \(P\) is the pressure, \(V\) is the volume, and \(\text{\(\boldsymbol{\text{γ}}\)}\) is the adiabatic index which depends on the kind of gas.

When dealing with an ideal monoatomic or diatomic gas, this concept is visually illustrated by graphing pressure versus volume. During compression, the curve steepens because the pressure rises more rapidly for each incremental reduction in volume, compared to an isothermal process where the temperature remains constant.

Ideal Gas Law

The ideal gas law is an equation that links the pressure, volume, temperature, and number of molecules of a gas and is commonly represented as \(PV=nRT\), where \(P\) stands for pressure, \(V\) signifies volume, \(n\) is the amount of substance (in moles), \(R\) is the ideal gas constant, and \(T\) represents the absolute temperature in Kelvin.

This law is widely used in chemistry and physics because it provides a good approximation of the behavior of gases under many conditions, although it assumes no interactions between gas molecules and that the volume occupied by the gas molecules is negligible. In adiabatic processes, the ideal gas law helps explain how changes in volume and temperature can occur without the exchange of heat with the environment, by utilizing the work done during compression or expansion.

Thermodynamics

Thermodynamics is the science that deals with heat and temperature, and their relation to energy, work, radiation, and properties of matter. This field of physics examines the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale. It can be divided into several laws, which explain various phenomena:

First Law of Thermodynamics: Also known as the law of energy conservation, it states that energy cannot be created or destroyed in an isolated system. The total amount of energy remains constant but can change from one form to another.
Second Law of Thermodynamics: It introduces the concept of entropy and asserts that in a natural thermodynamic process, the sum of the entropies of the interacting thermodynamic systems increases.
Third Law of Thermodynamics: It states that the entropy of a perfect crystal at absolute zero is exactly zero. These laws govern the principles of energy transfer and can predict whether a process will occur spontaneously.

Monoatomic and Diatomic Gases

Gases can be classified by the number of atoms that make up their molecules. Monoatomic gases, like noble gases (e.g., He, Ne, Ar), are made of single atoms, while diatomic gases, like Oxygen (\(O_2\)) and Nitrogen (\(N_2\)), consist of two atoms bonded together.

The primary difference in these types comes into play with thermodynamic processes like adiabatic compression due to the degrees of freedom—the ways in which the molecules can move and store energy. Monoatomic gases have three translational degrees of freedom, while diatomic gases have, under normal conditions, five degrees given their ability to rotate (adding two rotational degrees of freedom). This difference in degrees of freedom is why the adiabatic index \(\gamma\) (gamma) varies: \(\frac{5}{3}\) for monoatomic and \(\frac{7}{5}\) for diatomic gases.

The type of gas affects how it responds to processes like adiabatic compression or expansion, hence impacting the work done and changes in internal energy. Understanding these differences is crucial when solving problems related to gas behavior in different thermodynamic processes.