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12.6 - SPECIFIC HEAT CAPACITY

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Monatomic Gases

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Teacher
Teacher

Let's start with monatomic gases. Who can tell me what a monatomic gas is?

Student 1
Student 1

Isn't it a gas made of single atoms, like helium or neon?

Teacher
Teacher

Exactly! These gases have only three translational degrees of freedom, which influences their specific heat capacity. Can anyone recall the formula for Cv?

Student 2
Student 2

I think it's Cv = (3/2)R.

Teacher
Teacher

Great memory! Because they have less freedom of movement, they require less energy to change temperature. Now, can someone explain why Cp is greater than Cv?

Student 3
Student 3

It’s because when we heat the gas at constant pressure, part of the added energy does work to expand the gas.

Teacher
Teacher

"Correct! So the molar specific heats are Cv = (3/2)R and Cp = (5/2)R. Let's sum it up:

Diatomic Gases

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Teacher
Teacher

Now, let’s discuss diatomic gases like oxygen. Who can explain the degrees of freedom for these gases?

Student 2
Student 2

I believe they have three translational and two rotational degrees...

Teacher
Teacher

Yes! So that’s a total of five degrees. Therefore, what do we derive for Cv?

Student 1
Student 1

It's Cv = (5/2)R.

Teacher
Teacher

Spot on! And how about Cp?

Student 3
Student 3

Cp should be (7/2)R!

Teacher
Teacher

Excellent! The ratio γ = Cp/Cv is 7/5. That means diatomic gases can store more energy due to their extra degrees of freedom.

Polyatomic Gases

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Teacher
Teacher

Moving on to polyatomic gases. Who can tell me what gives these gases their unique specific heat capacity?

Student 4
Student 4

They have even more degrees of freedom—three translational, three rotational, and possibly more vibrational modes!

Teacher
Teacher

Exactly! This can make their Cv = (3 + f)R, accounting for f vibrational modes. Now, why is it relevant that these calculations align with experimental results?

Student 2
Student 2

It shows that our theoretical models are accurate and help in predicting behaviors of gases.

Teacher
Teacher

"Absolutely! Internal energy prediction accuracy showcases the importance of kinetic theory in thermodynamics. Let’s review:

The Law of Equipartition of Energy

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Teacher
Teacher

We’ve discussed different gases, now let's relate it all back to the law of equipartition of energy. Who can summarize what this law states?

Student 1
Student 1

It says each degree of freedom contributes to the total energy, which helps define Cv and Cp!

Teacher
Teacher

Exactly! Each translational degree contributes 1/2 kBT. Does anyone recall how vibrational modes factor into this?

Student 3
Student 3

Vibrational modes contribute twice, right? For kinetic and potential energy!

Teacher
Teacher

Correct again! So that means in experiments, people observe that specific heats might differ from predictions if they don't account for those vibrational modes.

Conclusion and Summary

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Teacher
Teacher

To finalize our discussions today: What common thread do we see across all three types of gases?

Student 4
Student 4

Their specific heats depend on their molecular structure and degrees of freedom.

Teacher
Teacher

"Exactly! It’s this fundamental understanding that allows us to handle thermal calculations in various engineering applications. Remember:

Introduction & Overview

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Quick Overview

This section discusses the specific heat capacities of gases, detailing the differences between monatomic, diatomic, and polyatomic gases and how the law of equipartition of energy applies to these gas types.

Standard

Specific heat capacity is essential in thermodynamics and varies across gases depending on their molecular structure. Monatomic gases have three translational degrees of freedom; diatomic gases have additional rotational degrees, and polyatomic gases feature both. Utilizing the law of equipartition of energy helps predict the heat capacities of these gases.

Detailed

Overview of Specific Heat Capacity

Specific heat capacity is an important concept in thermodynamics representing the amount of heat required to change the temperature of a unit mass of a substance by one degree Celsius. This property varies among different gases based on their molecular structure, as understood through the law of equipartition of energy.

Monatomic Gases

For monatomic gases like helium, the internal energy at temperature T is derived using the principle of equipartition of energy. Since they possess only three translational degrees of freedom, the molar specific heat at constant volume (Cv) becomes:

Cv = (3/2)R

From this, we can derive the specific heat at constant pressure (Cp) using the relationship:

Cp = Cv + R = (5/2)R

The ratio of specific heats (γ) for monatomic gases is therefore:

γ = Cp/Cv = 5/3.

Diatomic Gases

Diatomic gases like O2 feature additional degrees of freedom due to rotation. Overall, they have five degrees – three translational and two rotational, leading to:

Cv = (5/2)R,
Cp = (7/2)R,
γ = 7/5.

At higher temperatures, if vibrational modes are considered, the specific heats increase, indicating more energy storage capacity.

Polyatomic Gases

Polyatomic gases have both rotational and vibrational modes contributing to even more kinetic energy storage. For these molecules, we find:

Cv = (3 + f)R, where f represents additional vibrational modes.

Conclusion

The specific heats align closely with experimental results, showcasing the relationship between molecular structure and thermal properties of gases.

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Audio Book

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Monatomic Gases

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The molecule of a monatomic gas has only three translational degrees of freedom. Thus, the average energy of a molecule at temperature T is (3/2) kB. The total internal energy of a mole of such a gas is $$ U = \frac{3}{2} k_B T \times N_A = \frac{3}{2} RT $$ The molar specific heat at constant volume \(C_V\), is Cv (monatomic gas) = d dU T =3 2RT For an ideal gas, Cp – Cv = R where Cp is the molar specific heat at constant pressure. Thus, Cp = 5 2 R The ratio of specific heats γ = p v 5 3C C = .

Detailed Explanation

Monatomic gases, such as helium or neon, consist of single atoms not bonded to other atoms. Because they only have three translational degrees of freedom — movement along the x, y, and z axes — each atom can only store energy through kinetic motion. The average energy possessed by each atom at a given temperature (T) is given by the formula (3/2) kB, where kB is the Boltzmann constant. The total internal energy, U, for one mole of gas can be calculated using the equation U = (3/2)RT, where R is the universal gas constant. The specific heat at constant volume (Cv) is derived from the change in internal energy with respect to temperature and is equal to (3/2)R. When we consider the specific heat at constant pressure (Cp), we find that it is greater than Cv by R, leading to Cp = (5/2)R, and the ratio of specific heats (γ) being 5/3.

Examples & Analogies

Imagine heating a large container filled with helium balloons. As you heat the air inside, the gas molecules (the helium atoms) move faster. This increase in speed represents an increase in energy, demonstrating how gases only hold energy through motion due to their simple structure. The specific heat capacities tell us how much energy is required to raise the temperature of the gas: whether it is held at a constant volume or pressure, it requires a different amount of energy due to changes in how the gas expands.

Diatomic Gases

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As explained earlier, a diatomic molecule treated as a rigid rotator, like a dumbbell, has 5 degrees of freedom: 3 translational and 2 rotational. Using the law of equipartition of energy, the total internal energy of a mole of such a gas is 5 5 2 2B A U k T N RT= × = (12.32) The molar specific heats are then given by Cv (rigid diatomic) = 5 2R, Cp = 7 2R (12.33) γ (rigid diatomic) = 7 5(12.34).

Detailed Explanation

Diatomic gases, such as oxygen (O2) and nitrogen (N2), consist of two atoms bonded together. These molecules have complex movements; in addition to moving in three-dimensional space (translational), they can also rotate around their center of mass, giving them a total of five degrees of freedom. Consequently, the total internal energy for a mole of a diatomic gas is derived as (5/2) RT using the law of equipartition of energy. The molar specific heat at constant volume (Cv) for these gases is thus (5/2)R, while at constant pressure (Cp), it becomes (7/2)R. The ratio of specific heats γ for diatomic gases works out to 7/5.

Examples & Analogies

Think of a swinging door in a windy area. The door can move back and forth (translating) and it can also pivot around the hinges (rotating). Similarly, diatomic gas molecules can move freely in space while also rotating around their bond, which allows them to absorb more heat (energy) compared to monatomic gases. This additional energy storage capability changes how these gases behave when heated.

Polyatomic Gases

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In general, a polyatomic molecule has 3 translational, 3 rotational degrees of freedom and a certain number (f) of vibrational modes. According to the law of equipartition of energy, it is easily seen that one mole of such a gas has U = 3 2 kBT + 3 2 kBT + f kBT, leading to Cv = (3 + f) R, Cp = (4 + f) R, and γ = 4 + f 3.

Detailed Explanation

Polyatomic gases, such as carbon dioxide (CO2) or water vapor (H2O), have more complex structures than both monatomic and diatomic gases. These molecules not only have three translational and three rotational degrees of freedom, but they also exhibit vibrational movements due to the bonds between atoms. This added vibrational motion can be quantified as additional degrees of freedom (f). Using the law of equipartition of energy, we can derive the internal energy U for one mole of a polyatomic gas as U = (3/2)kBT + (3/2)kBT + fkBT. The equations for molar specific heats and the specific heat ratio (γ) expand to include these additional degrees of freedom: Cv = (3 + f)R and Cp = (4 + f)R, reflecting the increased complexity of energy storage within these molecules.

Examples & Analogies

Consider a rubber band. Just like a polyatomic gas, it can stretch (vibrate), twist, and stretch further — the way polyatomic gas molecules vibrate in addition to moving and rotating. This is why polyatomic gases possess different heat capacities compared to simpler gases; they can hold onto more energy due to their ability to vibrate, making them behave differently when heated.

Comparison with Experimental Values

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Of course, there are discrepancies between predicted and actual values of specific heats of several other gases (not shown in the table), such as Cl2, C2H6 and many other polyatomic gases. Usually, the experimental values for specific heats of these gases are greater than the predicted values as given in Table 12.1 suggesting that the agreement can be improved by including vibrational modes of motion in the calculation. The law of equipartition of energy is, thus, well verified experimentally at ordinary temperatures.

Detailed Explanation

While the theoretical predictions for specific heat values of gases are generally accurate, certain gases, especially complex polyatomic ones like chlorine (Cl2) and ethane (C2H6), demonstrate discrepancies where their experimental values exceed the predictions. This indicates that the models used to estimate these specific heats may need to take into account additional vibrational degrees of freedom that are active under typical conditions, indicating that real gases can store energy in more ways than the simple models reflect. Hence, while the law of equipartition broadly holds true, refinements to the models can lead to improved predictions of specific heat behaviors observed in experiments.

Examples & Analogies

Think of trying to predict how much heat a group of people in a room will absorb just by knowing their average weight. If you only consider their weight but ignore the effects of how active they are (vibrating or moving around), your prediction may be off. Just as it's important to account for activity when predicting energy absorption in a room, including vibrational modes of motion provides a more accurate depiction of how gases behave when heated.

Specific Heat Capacity of Solids

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We can use the law of equipartition of energy to determine specific heats of solids. Consider a solid of N atoms, each vibrating about its mean position. An oscillation in one dimension has average energy of 2 × ½ kBT = kBT. In three dimensions, the average energy is 3 kBT. For a mole of solid, N = NA, and the total energy is U = 3 kBT × NA = 3 RT. Now at constant pressure ∆Q = ∆U + P∆V = ∆U, since for a solid ∆V is negligible. Hence, 3Q UC RT TΔ Δ= = =Δ Δ(12.37).

Detailed Explanation

Solids have distinct patterns, with atoms vibrating in fixed positions. According to the law of equipartition of energy, each of these vibrations can be considered to contribute to the total energy of the solid. In one dimension, the average energy associated with a vibration is kBT, and since solids vibrate in three dimensions, the total average energy for one mole of a solid is U = 3RT, where R is the universal gas constant. Interestingly, when we consider the heat added to a solid under constant pressure, the increase in energy can be captured as 3Q = CU ΔT, indicating that the specific heat capacity of solids is three times that of a monatomic gas.

Examples & Analogies

Imagine a tightly packed crowd where people are standing still but occasionally bouncing slightly on their feet. Although each individual (atom) seems static, their little movements result in a collective vibrational energy. The amount of energy required to warm up this crowd (solid) can be deduced from how much the temperature rises as people bounce more vigorously (how energy is stored in solids). Just like those people in a crowd, atoms in solids vibrate even though they maintain a fixed structure.

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Specific heat capacity: Amount of heat needed to raise the temperature of a substance.

  • Monatomic gases: Have three degrees of freedom and specific heat of (3/2)R.

  • Diatomic gases: Have five degrees of freedom and specific heats of (5/2)R and (7/2)R.

  • Polyatomic gases: Feature additional vibrational modes affecting their specific heat capacity.

  • Equipartition of energy: Energy is distributed equally across degrees of freedom.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • For a monatomic gas like helium, Cv is (3/2)R.

  • For a diatomic gas like oxygen, Cv is (5/2)R while Cp is (7/2)R.

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎵 Rhymes Time

  • To heat a mono gas, just take it slow, Cv is three halves, R's the way to go.

📖 Fascinating Stories

  • Imagine a party of atoms, helium dancing alone while oxygen pairs show off their moves. Helium's small steps (3/2)R keep it light, while oxygen's twirls (7/2)R give flight!

🧠 Other Memory Gems

  • For monatomic gases, remember 'Three is company (3/2)'. For diatomic, think 'Five and seven fly high (5/2, 7/2)'.

🎯 Super Acronyms

To remember the sequence

  • 'M=mono
  • Di=diatomic
  • Po=polyatomic' all share the heat game
  • each with degrees of freedom to claim!

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Specific heat capacity

    Definition:

    The amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius.

  • Term: Monatomic gas

    Definition:

    A gas that consists of single atoms, such as helium.

  • Term: Diatomic gas

    Definition:

    A gas made up of two atoms, such as oxygen (O2).

  • Term: Polyatomic gas

    Definition:

    A gas containing more than two atoms in its molecules.

  • Term: Degrees of freedom

    Definition:

    The number of independent ways in which a molecule can move or store energy.

  • Term: Equipartition of energy

    Definition:

    A principle stating that energy is distributed equally among all degrees of freedom.