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

The correct ratio is (b) \(5 / 3\).

Step 1: Understanding the Problem

In monoatomic gases, we need to find the ratio of the specific heat at constant pressure, denoted as \(C_p\), to that at constant volume, denoted as \(C_v\). This ratio is known as the adiabatic index or the heat capacity ratio \(\gamma\).

Step 2: Recall the Formula for Adiabatic Index

The adiabatic index \(\gamma\) for an ideal monoatomic gas is determined by the formula \(\gamma = \frac{C_p}{C_v}\). For monoatomic gases, it is derived from theoretical concepts in thermodynamics.

Step 3: Apply the Values for Monoatomic Gases

In monoatomic gases, the specific heat at constant volume \(C_v\) is \(\frac{3}{2}R\), where \(R\) is the gas constant, and the specific heat at constant pressure \(C_p\) is \(\frac{5}{2}R\).

Step 4: Calculate \(\gamma\)

Using the formula \(\gamma = \frac{C_p}{C_v} = \frac{\frac{5}{2}R}{\frac{3}{2}R}\), simplify by canceling out \(R\) and \(\frac{1}{2}\) to get \(\gamma = \frac{5}{3}\).

Step 5: Compare to Answer Options

The calculated \(\gamma = \frac{5}{3}\) matches with option (b), confirming that this is the correct answer for the ratio of specific heat at constant pressure to that at constant volume in a monoatomic gas.

Key Concepts

Specific Heat Ratio

The specific heat ratio, also known as the adiabatic index, is an important concept in thermodynamics, especially when dealing with gases. It is the ratio of the specific heat at constant pressure, denoted as \(C_p\), to the specific heat at constant volume, \(C_v\). This ratio is symbolized by \(\gamma = \frac{C_p}{C_v}\).

• \(C_p\) is the amount of heat required to raise the temperature of a unit mass of gas by one degree Celsius while maintaining constant pressure.
• \(C_v\) is the heat needed to raise the temperature of the same unit mass by one degree Celsius when the volume remains constant.

The specific heat ratio determines how a gas will expand or compress when it is undergoing an adiabatic process, where no heat is exchanged with the surroundings. For different types of gases, \(\gamma\) varies, impacting their thermodynamic behavior.

Monoatomic Gases

Monoatomic gases are composed of individual atoms, such as helium, neon, or argon, rather than molecules. These gases are interesting from a thermodynamic perspective because they have simple behaviors compared to more complex molecules. In an ideal theoretical world, monoatomic gases follow the ideal gas law closely and have specific heat values that can be neatly defined.For these gases:

• The specific heat at constant volume \(C_v\) is \(\frac{3}{2}R\), where \(R\) is the universal gas constant.
• At constant pressure, \(C_p\) for monoatomic gases can be expressed as \(\frac{5}{2}R\).

Knowing these values makes it easier to calculate the adiabatic index \(\gamma\), which for monoatomic gases is always \(\frac{5}{3}\). This value reflects how these gases efficiently change volume and temperature under constant pressure or volume conditions.

Thermodynamics

Thermodynamics is the branch of physics that deals with energy, heat, and their transformations within a system. It uses principles and laws to predict how energy moves and changes form. This science is crucial for understanding processes in gases, including concepts like specific heat and thermodynamic cycles. The laws of thermodynamics help explain:

• Energy conservation and how energy cannot be created or destroyed (First Law).
• The direction of heat transfer and entropy (Second Law).
• The impossible attainment of absolute zero temperature (Third Law).

Understanding thermodynamics enables students to analyze systems, predict how they will respond to changes, and measure efficiencies of engines and other devices. It is the foundation for topics like the ideal gas law and the behavior of substances at different temperatures and pressures.

Ideal Gas Law

The ideal gas law is a cornerstone equation in chemistry and physics that describes how gases behave under different conditions of pressure, volume, and temperature. It states that for a given amount of gas:\[ PV = nRT \]Where:

• \(P\) is the pressure.
• \(V\) represents the volume.
• \(n\) is the number of moles of the gas.
• \(R\) is the ideal gas constant.
• \(T\) is the temperature in Kelvin.

This equation assumes an ideal scenario where gas molecules occupy no volume and exert no forces upon each other besides elastic collisions. Real gases deviate slightly under extreme conditions, but the ideal gas law provides a good approximation for many cases. It connects to other concepts like specific heat and thermodynamic cycles, offering a simpler way to quantify relationships between properties of gases.