Problem 93 - A Complete Resource Book in Chemistry for JEE Main

Question

A monoatomic gas 'A' and a diatomic gas 'B', both initially at the same temperature and pressure are compressed adiabatically from a volume \(\mathrm{V}\) to \(\mathrm{V} / 2 .\) The gas which has higher temperature is (a) \(\mathrm{B}\) (b) \(\mathrm{A}\) (c) both have same temperature (d) can not be said

Step-by-Step Solution

Answer

Gas 'A' (monoatomic) ends up with a higher temperature after adiabatic compression.

Step 1: Identify Key Concepts

We are dealing with an adiabatic process, which means there is no heat exchange with the surroundings. For an adiabatic process, the relationship between pressure, volume, and temperature is different due to the lack of heat exchange.

Step 2: Understand Gamma (d\gamma) for Gases

For a monoatomic gas, the adiabatic index \(\gamma\) is \(\frac{5}{3}\), and for a diatomic gas, it is \(\frac{7}{5}\). This index plays a crucial role in determining the temperature change in an adiabatic process because it affects how pressure and volume changes relate to temperature.

Step 3: Use Adiabatic Process Formula

The formula for temperature change in an adiabatic process is given by \(TV^{\gamma-1}= \text{constant}\). This relationship will help us compare the temperature changes of gases 'A' and 'B'.

Step 4: Apply the Formula to Each Gas

For gas 'A' (monoatomic), substitute \(\gamma = \frac{5}{3}\) into the formula: \(T_A(V/2)^{\frac{5}{3}-1} = TV^{\frac{5}{3}-1}\). For gas 'B' (diatomic), substitute \(\gamma = \frac{7}{5}\): \(T_B(V/2)^{\frac{7}{5}-1} = TV^{\frac{7}{5}-1}\).

Step 5: Simplify the Temperature Relationships

Simplify both gas expressions: For 'A', \(T_A \cdot \frac{V^{\frac{2}{3}}}{(V/2)^{\frac{2}{3}}} = T\). For 'B', \(T_B \cdot \frac{V^{\frac{2}{5}}}{(V/2)^{\frac{2}{5}}} = T\).

Step 6: Compare Temperatures

The expressions show that \(T_A\) is higher than \(T_B\) because the exponent in the denominator for gas 'B' is smaller, making its decrease less significant compared to gas 'A'. This results in a higher final temperature for gas 'A' after compression.

Key Concepts

Monoatomic Gas

A monoatomic gas, as the name suggests, consists of single atoms, such as helium, neon, or argon. These gases do not form molecules at standard conditions.
Understanding their behavior is crucial in thermodynamics, especially in processes like adiabatic ones where no heat is exchanged with the environment.
Some key characteristics of monoatomic gases:

• They have three translational degrees of freedom.
• Have a simpler structure, resulting in specific heat capacities that are generally lower than more complex gases.
• In an adiabatic process, the relationship between pressure, volume, and temperature is dictated without any heat exchange, relying heavily instead on internal energy changes.

For monoatomic gases, the adiabatic index (or gamma, \(\gamma\)) is crucial in determining how those energy changes affect temperature, pressure, and volume.

Diatomic Gas

Diatomic gases, like oxygen (O₂) and nitrogen (N₂), consist of molecules made up of two atoms. Understanding the behavior of diatomic gases aids in comprehending more complex gas dynamics involved in adiabatic processes.
Some significant points about diatomic gases:

• They typically possess additional rotational and vibrational degrees of freedom compared to monoatomic gases.
• This increased complexity influences their specific heat capacities, which are generally higher than in monoatomic gases.
• Diatomic gases also follow the adiabatic process laws with their energy changes related explicitly to degrees of freedom.

These gases exhibit changes in thermodynamic properties under varying conditions, with the adiabatic index for diatomic gases typically being \(\gamma = \frac{7}{5}\). This helps in understanding how compression or expansion influences their temperature compared to monoatomic gases.

Adiabatic Index (Gamma)

The adiabatic index, commonly denoted as gamma (\(\gamma\)), is a vital concept in physics for understanding the behavior of gases during adiabatic processes.
Gamma is the ratio of specific heats of a gas at constant pressure to that at constant volume (\[\gamma = \frac{C_p}{C_v}\] In adiabatic processes, no heat is exchanged with the surroundings, which shows how gamma directly affects the pressure, volume, and temperature of a gas:

• For monoatomic gases, \(\gamma\) is typically \(\frac{5}{3}\).
• For diatomic gases, \(\gamma\) is roughly \(\frac{7}{5}\).

Understanding these ratios helps determine how gases will behave under compression or expansion without heat exchange, allowing predictions on temperature changes, which was precisely useful in the problem solving.

Temperature Change in Gases

Temperature change in gases during adiabatic processes is a crucial topic, as it shows how gases respond to being compressed or expanded without heat interaction. This is particularly impactful in determining outcomes like in our comparative problem with monoatomic and diatomic gases.
In an adiabatic compression, gases heat up because work is done on the gas, raising its internal energy. The formula governing this process is: \[TV^{\gamma-1} = \text{constant}\] This formula helps illustrate how a decrease in volume affects the temperature, alongside pressure and the adiabatic index:

• For monoatomic gases, higher \(\gamma\) value indicates a higher temperature increase upon compression.
• For diatomic gases, the change is less pronounced due to a lower \(\gamma\), meaning the temperature rise is not as steep.

In the given problem, this principle clarifies why the monoatomic gas experiences a greater temperature rise during compression compared to the diatomic gas.