What is Molar Specific Heat of Monoatomic Gas? | Simple Explanation for Class 12

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What is Molar Specific Heat of a Monoatomic Gas?

Grade Level:

Class 12

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Definition

What is it?

Molar specific heat of a monoatomic gas is the amount of heat energy needed to raise the temperature of one mole of that gas by one degree Celsius (or Kelvin). It tells us how much heat a specific amount of a simple gas can hold or absorb. For monoatomic gases like Helium or Neon, this value is fixed at constant volume and constant pressure.

Simple Example

Quick Example

Imagine you have one mole of helium gas in a small cylinder, like the gas used to fill balloons. If you want to make this helium gas just one degree hotter, say from 25°C to 26°C, you need to add a certain amount of heat energy. That specific amount of heat energy is its molar specific heat.

Worked Example

Step-by-Step

Let's find the molar specific heat at constant volume (Cv) for a monoatomic gas.

Step 1: For a monoatomic gas, the number of degrees of freedom (f) is 3 (translation only).
---Step 2: The internal energy (U) of one mole of a gas is given by U = (f/2)RT, where R is the gas constant.
---Step 3: Substitute f = 3 for a monoatomic gas: U = (3/2)RT.
---Step 4: Molar specific heat at constant volume (Cv) is defined as dU/dT.
---Step 5: Differentiate U with respect to T: Cv = d/dT [(3/2)RT] = (3/2)R.
---Step 6: The value of R is approximately 8.314 J/mol·K.
---Step 7: Calculate Cv = (3/2) * 8.314 J/mol·K = 1.5 * 8.314 J/mol·K = 12.471 J/mol·K.

Answer: The molar specific heat at constant volume (Cv) for a monoatomic gas is approximately 12.47 J/mol·K.

Why It Matters

Understanding molar specific heat helps engineers design efficient engines and refrigerators, making our EVs run better and our homes cooler. In space technology, it's crucial for understanding how gases behave in rocket engines and spacecraft. This concept is vital for careers in mechanical engineering, aerospace engineering, and climate science, where controlling temperature and energy is key.

Common Mistakes

MISTAKE: Confusing molar specific heat with specific heat capacity. | CORRECTION: Molar specific heat is for one mole of a substance, while specific heat capacity is for one unit mass (like 1 kg or 1 gram) of a substance.

MISTAKE: Using the wrong value for degrees of freedom for a monoatomic gas. | CORRECTION: For monoatomic gases, degrees of freedom (f) is always 3 (only translational motion). Do not use 5 or 6 which are for diatomic or polyatomic gases.

MISTAKE: Forgetting the units or using incorrect units. | CORRECTION: Molar specific heat is measured in Joules per mole per Kelvin (J/mol·K) or Joules per mole per degree Celsius (J/mol·°C). Always include correct units.

Practice Questions

Try It Yourself

QUESTION: What is the molar specific heat at constant volume (Cv) for a monoatomic gas if the gas constant R = 8.314 J/mol·K? | ANSWER: 12.471 J/mol·K

QUESTION: If the molar specific heat at constant volume (Cv) for a monoatomic gas is 12.471 J/mol·K, what is the molar specific heat at constant pressure (Cp) for the same gas? (Hint: Cp - Cv = R) | ANSWER: Cp = Cv + R = 12.471 J/mol·K + 8.314 J/mol·K = 20.785 J/mol·K

QUESTION: You have 2 moles of a monoatomic gas. How much heat energy is required to raise its temperature by 10 Kelvin at constant volume? (Given R = 8.314 J/mol·K) | ANSWER: First, find Cv = (3/2)R = 1.5 * 8.314 = 12.471 J/mol·K. Then, Heat (Q) = n * Cv * ΔT = 2 moles * 12.471 J/mol·K * 10 K = 249.42 Joules.

MCQ

Quick Quiz

Which of the following gases is considered monoatomic?

Oxygen (O2)

Nitrogen (N2)

Helium (He)

Carbon Dioxide (CO2)

The Correct Answer Is:

Helium (He) exists as single atoms, making it a monoatomic gas. Oxygen, Nitrogen, and Carbon Dioxide are diatomic or polyatomic, meaning they consist of two or more atoms bonded together.

Real World Connection

In the Real World

This concept is vital in designing efficient air conditioners and refrigerators for our homes, ensuring they cool our rooms using less electricity. Companies like Blue Star or Voltas use these principles to create energy-saving models. It's also key in understanding how gases behave in industrial processes and even in the functioning of airbags in cars, where rapid gas expansion is critical.

Key Vocabulary

Key Terms

MONOATOMIC GAS: A gas whose molecules consist of a single atom, like Helium or Neon. | DEGREES OF FREEDOM: The number of independent ways an atom or molecule can move or store energy. | GAS CONSTANT (R): A fundamental physical constant relating pressure, volume, and temperature for an ideal gas. | INTERNAL ENERGY: The total energy contained within a thermodynamic system, excluding kinetic and potential energy of the system as a whole.

What's Next

What to Learn Next

Great job understanding molar specific heat for monoatomic gases! Next, you should explore 'Molar Specific Heat of Diatomic and Polyatomic Gases.' This will help you see how adding more atoms to a gas molecule changes its ability to store heat, building on what you've learned today.