Problem 88 - A Complete Resource Book in Chemistry for JEE Main

Question

In monoatomic gases, ratio of specific heat at constant pressure to that at constant volume is (a) \(3 / 5\) (b) \(5 / 3\) (c) \(7 / 5\) (d) \(4 / 5\)

Step-by-Step Solution

Answer

For monoatomic gases, the ratio of specific heats \(\gamma\) is \(\frac{5}{3}\); hence, the correct answer is (b).

Step 1: Understanding the Ratio of Specific Heats

In thermodynamics, the ratio of specific heat at constant pressure (cp ) to specific heat at constant volume (cv ) for gases is known as the heat capacity ratio, also represented by the symbol \(\gamma\). This ratio is defined as \(\gamma = \frac{C_p}{C_v}\).

Step 2: Knowing the Value for Monoatomic Gases

For monoatomic gases, which are gases consisting of single atoms like noble gases, this heat capacity ratio \(\gamma\) has a standard value. Most commonly, the value of \(\gamma\) for monoatomic gases is \(\frac{5}{3}\). This should be memorized as part of basic thermodynamic properties of gases.

Step 3: Choosing the Correct Answer

Given that we know \(\gamma = \frac{C_p}{C_v} = \frac{5}{3}\) for monoatomic gases, we can look at the options provided: (a) \(\frac{3}{5}\), (b) \(\frac{5}{3}\), (c) \(\frac{7}{5}\), and (d) \(\frac{4}{5}\). The correct answer from the given options is (b) \(\frac{5}{3}\).

Key Concepts

Specific Heat

Specific heat is a fundamental concept in thermodynamics that refers to the amount of energy required to raise the temperature of one unit of mass of a substance by one degree Kelvin.The specific heat can vary depending on whether the process occurs at constant pressure or constant volume.For constant pressure, this value is denoted as \( C_p \) and for constant volume, it’s denoted as \( C_v \). Each of these measurements plays a crucial role in understanding how different substances react to heat.

• For constant volume, the volume of the gas does not change, allowing all the added heat to increase the temperature.
• For constant pressure, the gas can expand, doing work as it absorbs heat.

It's important to be comfortable with these concepts when studying thermodynamic processes or solving related problems, as they help in analyzing energy transfer and work done during various processes.

Monoatomic Gases

Monoatomic gases are composed of single atoms and are exemplified by noble gases such as helium, neon, and argon. These gases are unique as they do not form molecules but exist as single atoms. In the context of thermodynamics, these gases are often treated as ideal because of their simple structure. They follow the ideal gas law, and their behavior is easier to predict and understand than more complex molecular gases.

• Monoatomic gases have three translational degrees of freedom, corresponding to movement in three-dimensional space.
• Because they consist of individual atoms, they can lack rotational and vibrational movements that are typical in diatomic or polyatomic gases.

Understanding how monoatomic gases work helps in comprehending more advanced studies of gas behaviors and thermodynamic cycles.

Heat Capacity Ratio

The heat capacity ratio, often symbolized by \( \gamma \), is a critical concept in thermodynamics when analyzing the behavior of gases. It is defined as the ratio of the specific heat at constant pressure \( C_p \) to the specific heat at constant volume \( C_v \).For monoatomic gases, this ratio of specific heats is particularly simple and is generally remembered as \( \frac{5}{3} \). This value arises from the theory of equipartition of energy, which deals with how energy is distributed among various degrees of freedom.

• \( \gamma = \frac{C_p}{C_v} \) helps determine the relationship between pressure, volume, and temperature in adiabatic processes.
• It plays a vital role in understanding and calculating performance and efficiency in engines, refrigeration cycles, and other thermodynamic systems.

By grasping the heat capacity ratio, students can better handle problems involving gas compression or expansion under different conditions.