Cp and Cv: Comprehensive NEET Physics Notes

1. Cp and Cv: Definition and Concept

1.1 Introduction to Heat Capacities

Heat capacity is a fundamental concept in thermodynamics that describes the amount of heat required to raise the temperature of a system by a certain amount. For gases, two specific heat capacities are defined:

1. Specific heat at constant volume (Cv): The amount of heat required to raise the temperature of a gas by 1 K while keeping the volume constant. $C_v={\left(\frac{dQ}{dT}\right)}_V$
2. Specific heat at constant pressure (Cp): The amount of heat required to raise the temperature of a gas by 1 K while keeping the pressure constant. $C_p={\left(\frac{dQ}{dT}\right)}_P$

These heat capacities are measured in J/kg·K (specific heat capacities) or J/mol·K (molar heat capacities). In real-world applications, understanding these quantities helps in controlling systems where temperature and pressure variations occur, such as engines and refrigerators.

1.2 Molar Heat Capacities for Ideal Gases

In ideal gases, the difference between $C_p$ and $C_v$ occurs because, at constant pressure, a gas must do work to expand, requiring additional heat compared to heating at constant volume. This is essential when calculating energy transfer in gases, such as air in our atmosphere.

From the first law of thermodynamics: $dQ=dU+PdV$

For an ideal gas, the internal energy change ($dU$) is dependent only on temperature, while work ($PdV$) is performed during expansion or compression.

1.3 Relationship Between Cp and Cv

For an ideal gas, the relationship between the molar heat capacities at constant pressure and constant volume is given by the equation: $C_p-C_v=R$

Where:$PdV=nRdT$

• $R$ is the universal gas constant (8.314 J/mol·K).

The extra heat in the case of constant pressure is used to do work by expanding the gas. This explains why $C_p$ is greater than $C_v$.

Real-life Application: In car engines, the expansion of gas during combustion follows this principle, where the gas does work against the piston. The heat capacities at constant pressure and volume play a crucial role in determining the engine's efficiency.

1.4 Ratio of Specific Heats: $\gamma$

The ratio of specific heats, denoted as $\gamma$ (gamma), is an important parameter in thermodynamics: $\gamma =\frac{C_p}{C_v}$

For a monoatomic gas like helium or neon, $\gamma$ is approximately 1.67, and for diatomic gases like oxygen or nitrogen, it is around 1.4. This ratio plays a vital role in processes such as sound propagation and gas dynamics.

NEET Tip: For NEET, remember that the ratio $\gamma$ for diatomic gases is close to 1.4, while for monoatomic gases, it is around 1.67. These values are frequently used in problem-solving related to thermodynamic processes.

2. Derivation of the Relation $C_p-C_v=R$

2.1 Derivation Using the First Law of Thermodynamics

To derive the relationship between $C_p$ and $C_v$, we use the first law of thermodynamics: $dQ=dU+PdV$

For an ideal gas at constant volume, no work is done, so: $d{Q}_V=dU=C_{v}dT$

At constant pressure, the gas does work by expanding, so: $d{Q}_P=dU+PdV$

Using the ideal gas law ($PV=nRT$), we find: $PdV=nRdT$

Substituting this into the expression for $d{Q}_P$, we get: $d{Q}_P=C_{v}dT+nRdT$

Thus: $C_{p}dT=C_{v}dT+nRdT$

Dividing through by $dT$, we obtain: $C_p=C_v+nR$

For one mole of gas, $n=1$, so: $C_p-C_v=R$

This fundamental relation is essential for understanding heat transfer in gases under various thermodynamic conditions.

Mnemonic: "Comparing Cp with Cv gives R" – Use this phrase to remember the relationship $C_p-C_v=R$.

Real-life Application: In atmospheric science, this relation helps predict how air masses will behave when they rise and cool, affecting weather patterns and cloud formation.

3. Applications of Cp and Cv

3.1 Adiabatic Processes

In an adiabatic process, no heat is exchanged with the surroundings, and the relationship between pressure and volume is given by: $P{V}^{\gamma }=\text{constant}$

This equation helps explain the behavior of gases in insulated systems, such as in the compression phase of a diesel engine. Understanding adiabatic processes is crucial for designing efficient heat engines and refrigerators.

3.2 Isochoric and Isobaric Processes

• Isochoric Process: In this process, the volume remains constant, so no work is done. All the heat supplied changes the internal energy of the system. The heat capacity here is $C_v$.
• Isobaric Process: In this process, the pressure remains constant. The heat supplied is partly used to do work (expand the gas) and partly to increase the internal energy. The heat capacity here is $C_p$.

4. Visualizing Thermodynamic Processes

NEET Problem-Solving Strategy: Always identify the type of process (isobaric, isochoric, adiabatic) before attempting to solve thermodynamic problems. It helps determine which formulae to apply. For adiabatic processes, use the relation $P{V}^{\gamma }=\text{constant}$, while for isochoric processes, focus on changes in internal energy.

Quick Recap

• $C_p$ is the specific heat at constant pressure, and $C_v$ is the specific heat at constant volume.
• The relationship between them is $C_p-C_v=R$, where $R$ is the universal gas constant.
• The ratio of specific heats, $\gamma$, plays a crucial role in adiabatic processes.
• In an adiabatic process, no heat is exchanged, while in isochoric and isobaric processes, work is either done or heat is transferred.

Practice Questions

1. Derive the relationship between $C_p$ and $C_v$ for an ideal gas.
2. A gas expands adiabatically. If the initial temperature is 300 K and $\gamma =1.4$, what will be the final temperature after the gas doubles its volume?
3. Explain why