Difference Between Cp and Cv in Gases
Specific heat capacities of gases are essential for mastering thermal physics in JEE and NEET. Understanding the nuances between $C_P$, $C_V$, and their applications bridges theory and real problems.
Introduction to Specific Heat Capacities of Gases
The specific heat capacity describes how much heat is needed to raise a unit mass of a gas by one degree Celsius. This property differs for various gases under different conditions.
Gases have unique behaviors because their particles are far apart compared to solids or liquids, affecting how they absorb and release heat during temperature changes.
Real-Life Analogy: Visualising Specific Heat
Imagine inflating a bicycle tire. Pumping air in heats the gas, while the rigid tire maintains nearly constant volume. This resembles measuring specific heat at constant volume, $C_V$.
Alternatively, heating an open balloon lets gas expand as it warms up under constant pressure, analogous to $C_P$, the specific heat at constant pressure, which is always greater than $C_V$.
Defining Specific Heat: Fundamental Concepts
Specific heat is an intensive property, meaning it does not depend on the quantity of gas being considered. It is measured in $J\,kg^{-1}K^{-1}$ or $J\,mol^{-1}K^{-1}$.
Mathematically, specific heat capacity is expressed as $c = \dfrac{\Delta Q}{m\,\Delta T}$, where $\Delta Q$ is heat supplied, $m$ is mass, and $\Delta T$ is temperature change.
Molar Heat Capacity of Gases
Molar heat capacity is critical for gases and is defined as the heat required to increase the temperature of one mole of a substance by one kelvin.
The formula for molar heat capacity is $C = \dfrac{\Delta Q}{n\,\Delta T}$, where $n$ denotes the number of moles. This formulation is central in the Kinetic Theory of Gases.
$C_P$ and $C_V$: Constant Pressure and Constant Volume
For gases, the heat required to raise temperature can occur at constant volume ($C_V$) or constant pressure ($C_P$), each with unique implications due to energy used for expansion.
At constant volume, all heat raises internal energy, so $C_V = \left(\dfrac{\Delta Q}{n\,\Delta T}\right)_V$. For constant pressure, $C_P = \left(\dfrac{\Delta Q}{n\,\Delta T}\right)_P$ and includes work done against external pressure.
Deriving the Relationship: Mayer’s Equation Explained
From the first law of Thermodynamics, $\Delta Q = \Delta U + P\Delta V$, it follows that $C_P = C_V + R$, where $R$ is the universal gas constant.
This fundamental relationship indicates $C_P > C_V$ for all ideal gases and plays a pivotal role in many JEE problems involving heat and work in gases.
| Parameter | Expression |
|---|---|
| Molar heat at constant volume ($C_V$) | $\left(\dfrac{\Delta Q}{n\,\Delta T}\right)_V$ |
| Molar heat at constant pressure ($C_P$) | $\left(\dfrac{\Delta Q}{n\,\Delta T}\right)_P$ |
| Mayer’s Relation | $C_P - C_V = R$ |
Degrees of Freedom and Specific Heat
For ideal gases, specific heat values depend upon the molecule’s degrees of freedom ($f$). Monoatomic, diatomic, and polyatomic gases each have distinct $C_V$ and $C_P$ values.
For a monoatomic ideal gas, $C_V = \dfrac{3}{2}R$, while for a diatomic gas at room temperature (considering rotational energy), $C_V = \dfrac{5}{2}R$.
| Type of Gas | $C_V$ (J mol$^{-1}$ K$^{-1}$) |
|---|---|
| Monoatomic (e.g., He, Ne, Kr) | $1.5\,R$ |
| Diatomic (N$_2$, O$_2$) | $2.5\,R$ |
| Triatomic/Polyatomic | $\geq 3\,R$ |
Illustrative Example: Solving for $C_P$ and $C_V$
Suppose 1 mole of an ideal diatomic gas absorbs $1000$ J while temperature rises by $40$ K at constant pressure. Find $C_P$ and $C_V$ for the gas.
Given: $\Delta Q = 1000$ J, $n = 1$, $\Delta T = 40$ K, gas is diatomic, $R = 8.314$ J/mol K.
So, $C_P = \dfrac{1000}{1 \times 40} = 25$ J/mol K. Since $C_P - C_V = R$, $C_V = C_P - R = 25 - 8.314 = 16.686$ J/mol K approximately.
Practice Question
A sample of helium (monoatomic) is supplied with 498.84 J heat at constant volume, raising its temperature by 40 K. Find the number of moles in the sample.
Common Mistakes Students Make
- Confusing $C_P$ and $C_V$ for gases, especially when conditions are not explicit
- Ignoring the effect of molecular composition on degrees of freedom
- Treating real gases as ideal in all temperature and pressure ranges
Visualization Using a Graph
Plotting temperature along the x-axis with heat supplied per mole along the y-axis, the slope indicates molar specific heat. Steeper slopes correspond to lower specific heat values.
Temperature increases faster for the same heat input in monoatomic gases ($C_V$) than in complex, polyatomic gases, demonstrating energy partition among more degrees of freedom.
JEE Relevance: Application and Strategy
Questions from this chapter are frequent in JEE, especially involving calculating heat, temperature change, and the relationship between $C_P$ and $C_V$ for different gases.
Mastering this topic is essential for thermodynamics, especially when preparing with Thermodynamics Revision Notes and solving applied problems.
Types of Specific Heat Capacities in Gases
- Specific heat at constant volume ($C_V$)
- Specific heat at constant pressure ($C_P$)
- Molar and mass-based specific heat values
Extra Applications: Real-World Connections
Understanding specific heat capacities aids in fields like atmospheric science, engine design, and Heat Pump efficiency analysis, among others important engineering contexts.
Text-Based Related Topics
- Internal energy of gases and first law of thermodynamics
- Distribution of molecular speeds
- Adiabatic processes and work calculations
- Polytropic processes in thermodynamics
- Calorimetry and heat transfer concepts
- Gas constant and universal relationships
- Predicting molecular behavior at extreme conditions
- Thermal equilibrium and energy exchange
- Thermodynamic efficiency and cycles
For comprehensive problem sets, consult Specific Heat Capacities of Gases resources and attempt Thermodynamics Important Questions for practice.
FAQs on Understanding Specific Heat Capacities of Gases
1. What is the specific heat capacity of gases?
Specific heat capacity of gases refers to the amount of heat required to raise the temperature of one mole of a gas by one degree Celsius at a constant pressure or volume. Depending on the thermodynamic conditions, specific heat of gases is measured as:
- Cp (at constant pressure): Heat needed at constant pressure.
- Cv (at constant volume): Heat needed at constant volume.
2. Why do gases have different specific heat capacities at constant pressure and volume?
Gases have different specific heat capacities at constant pressure (Cp) and constant volume (Cv) because extra heat is required at constant pressure to do work by expanding against surrounding pressure.
- Cp > Cv because energy is used for both raising temperature and expansion.
- At constant volume, no work is done by the gas, so heat only goes into raising temperature.
- For most ideal gases: Cp - Cv = R, where R is the universal gas constant.
3. What factors affect the specific heat capacity of a gas?
The specific heat capacity of a gas depends on its molecular nature and temperature. Main factors include:
- Type of gas: Monoatomic, diatomic, or polyatomic gases have different values due to varying degrees of freedom.
- Temperature: At higher temperatures, more molecular motions contribute, affecting specific heat.
- Pressure or Volume: Whether measured at constant pressure or volume changes the value (Cp vs Cv).
4. How are Cp and Cv related for an ideal gas?
For an ideal gas, Cp and Cv are related by the equation:
- Cp - Cv = R, where R is the universal gas constant.
- This means the difference between the specific heat at constant pressure and constant volume is always equal to R for an ideal gas.
5. What is the molar specific heat capacity of a monoatomic gas?
For a monoatomic ideal gas:
- Cv = (3/2)R
- Cp = (5/2)R
6. Why is specific heat capacity of a gas higher at constant pressure than at constant volume?
Specific heat at constant pressure (Cp) is higher than at constant volume (Cv) because, at constant pressure, the gas must do work to expand when heated. Thus:
- Extra energy is used both to raise temperature and to expand the gas against atmospheric pressure.
- At constant volume, all the heat goes into raising the temperature.
7. What is meant by degrees of freedom of a gas molecule?
Degrees of freedom refer to the independent forms of energy a gas molecule can store energy in. For gases:
- Monoatomic: 3 translational degrees.
- Diatomic: 3 translational + 2 rotational (total 5 at low temperatures).
- Polyatomic: Additional rotational and vibrational degrees.
8. State Mayer’s relation and its significance.
Mayer’s relation connects the specific heats of an ideal gas:
- Cp - Cv = R, linking molar heat at constant pressure and volume.
- It shows that Cp is always greater than Cv for an ideal gas by exactly the universal gas constant (R).
9. What is the value of Cp/Cv (gamma) for monoatomic and diatomic gases?
The ratio Cp/Cv is denoted as gamma (γ):
- Monoatomic gases: γ = 5/3 ≈ 1.67
- Diatomic gases: γ = 7/5 = 1.4
10. Why does specific heat capacity of a gas change with temperature?
The specific heat capacity of a gas changes with temperature because, at higher temperatures, more degrees of freedom (like rotational or vibrational modes) become active. This leads to:
- Increase in Cv and Cp with rising temperature for diatomic and polyatomic gases.
- Monoatomic gases show minimal change as most degrees are already active.
11. What are the practical applications of understanding specific heat capacity of gases?
Understanding specific heat capacity of gases is crucial for:
- Engineering thermodynamics: Designing engines, turbines, and refrigerators.
- Weather studies: Predicting atmospheric thermal behavior.
- Thermal calculations in physics and chemistry labs.
- CBSE and competitive exam numericals involving heat transfer.
