Physics and Mathematics
Heat
1. Concept Overview
The heat energy of a body arises due to the motion of its molecules. Every molecule in a substance is in constant motion, and this motion may be of three types:
- Translational Motion
- Rotational Motion
- Vibrational Motion
Hence, the total heat energy (or internal energy) of a body is the sum of kinetic energies of all the molecules due to these three types of motion.
2. Explanation and Mathematical Representation
Each molecule in a substance possesses energy due to motion. The total kinetic energy per molecule can be represented as:
[
E = E_t + E_r + E_v
]
where:
- [E_t] = Kinetic energy due to Translational Motion
- [E_r] = Kinetic energy due to Rotational Motion
- [E_v] = Kinetic energy due to Vibrational Motion
Hence, the total heat energy of the system (U) containing [N] molecules is:
[
U = N(E_t + E_r + E_v)
]
(a) Translational Motion
Translational motion refers to the movement of a molecule from one place to another.
For a gas molecule of mass [m] and velocity [v], its translational kinetic energy is:
[
E_t = \dfrac{1}{2}mv^2
]
In an ideal gas, the average translational kinetic energy per molecule is related to the temperature [T] by:
[
\langle E_t \rangle = \dfrac{3}{2}kT
]
where [k] is the Boltzmann constant ([1.38 \times 10^{-23} J/K]).
(b) Rotational Motion
Rotational motion refers to the spinning of molecules about their own axes.
For a rotating molecule having moment of inertia [I] and angular velocity [\omega], the rotational kinetic energy is:
[
E_r = \dfrac{1}{2}I\omega^2
]
For diatomic and polyatomic gases, rotational motion contributes significantly to the total energy, whereas for monoatomic gases, it is negligible.
(c) Vibrational Motion
Vibrational motion refers to the oscillatory motion of atoms within a molecule about their mean positions.
Each atom vibrates with a certain frequency, and the vibrational kinetic energy is given by:
[
E_v = \dfrac{1}{2}k_s x^2
]
where:
- [k_s] = force constant (bond stiffness)
- [x] = displacement from mean position
Vibrational motion becomes significant at high temperatures, especially in polyatomic molecules.
3. Dimensions and Units
| Quantity | Symbol | Dimensions | SI Unit |
|---|---|---|---|
| Energy | E | [M¹L²T⁻²] | Joule (J) |
| Temperature | T | [Θ] | Kelvin (K) |
| Boltzmann Constant | k | [M¹L²T⁻²Θ⁻¹] | J/K |
4. Key Features
- The heat energy of a body is the sum of kinetic energies due to translational, rotational, and vibrational motions.
- In monoatomic gases, only translational motion contributes significantly.
- In diatomic and polyatomic gases, rotational and vibrational motions also contribute.
- At higher temperatures, vibrational energy becomes prominent.
- Heat energy is directly proportional to temperature.
5. Important Formulas to Remember
| Type of Motion | Expression for Energy | Remarks |
|---|---|---|
| Translational | [E_t = \dfrac{1}{2}mv^2] | For monoatomic gases |
| Average Translational Energy | [\langle E_t \rangle = \dfrac{3}{2}kT] | Proportional to temperature |
| Rotational | [E_r = \dfrac{1}{2}I\omega^2] | Important for diatomic gases |
| Vibrational | [E_v = \dfrac{1}{2}k_s x^2] | Increases at high temperature |
| Total Molecular Energy | [E = E_t + E_r + E_v] | Represents total energy per molecule |
| Total Energy of System | [U = N(E_t + E_r + E_v)] | N = number of molecules |
6. Conceptual Questions with Solutions
1. What are the three types of molecular motions responsible for heat energy?
Translational, Rotational, and Vibrational motions of molecules are responsible for the heat energy of a body.
2. Which type of molecular motion is dominant in monoatomic gases?
Translational motion is dominant in monoatomic gases because they lack internal structure for rotation or vibration.
3. How does temperature affect vibrational energy?
At higher temperatures, vibrational energy increases significantly due to increased molecular oscillations.
4. What is the relation between temperature and average translational energy?
[\langle E_t \rangle = \dfrac{3}{2}kT] — it shows direct proportionality between energy and temperature.
5. Why is rotational motion negligible in monoatomic gases?
Monoatomic gases have only one atom per molecule, hence no axis for rotation about a molecular structure.
6. What determines the moment of inertia of a molecule?
The moment of inertia depends on the mass distribution of atoms and the distance between them.
7. Why does vibrational motion appear only at high temperatures?
Because at low temperatures, the energy available is insufficient to excite vibrational modes.
8. What happens to the total molecular energy when temperature doubles?
The average kinetic energy (and hence total molecular energy) also doubles.
9. What is the physical significance of Boltzmann’s constant?
It relates the average kinetic energy of molecules to the temperature of the gas.
10. How is heat energy different from internal energy?
Heat energy refers to energy transfer, while internal energy is the total energy stored within a system.
11. Which type of motion contributes most to specific heat?
Vibrational motion contributes significantly to specific heat, especially in polyatomic gases.
12. Why do polyatomic gases have higher specific heat capacities?
Because they possess more degrees of freedom — translational, rotational, and vibrational.
13. What happens to molecular motions at absolute zero?
All molecular motions cease; hence kinetic energy becomes zero.
14. Is potential energy included in heat energy?
No, heat energy includes only the kinetic energies due to molecular motions.
15. Can solids also have molecular motion?
Yes, in solids, atoms vibrate about fixed positions — a form of vibrational motion.
7. FAQ / Common Misconceptions
1. Heat energy exists only in gases.
❌ Incorrect. Heat energy exists in solids, liquids, and gases due to molecular motion.
2. Temperature and heat are the same.
❌ Temperature measures average kinetic energy; heat is the energy transfer due to temperature difference.
3. Molecules in solids do not move.
❌ They do — through vibrations around their mean positions.
4. All molecules in a gas move with the same velocity.
❌ No, they move with different velocities; the distribution is given by Maxwell’s law.
5. Heat energy is always equal to internal energy.
❌ Internal energy includes both kinetic and potential energies; heat is only energy in transit.
6. Only translational motion contributes to temperature.
❌ At high temperatures, rotational and vibrational motions also contribute.
7. Boltzmann constant is used only in gases.
❌ It’s a universal constant used in all systems involving temperature and molecular energy.
8. Vibrational motion does not occur in diatomic molecules.
❌ It does occur, though it requires higher energy.
9. Heat and work are state functions.
❌ They are not; they depend on the path taken.
10. Atoms in gases do not vibrate.
❌ In gases, diatomic or polyatomic molecules do vibrate internally.
8. Practice Questions (With Step-by-Step Solutions)
Q1. Find the average translational kinetic energy of a gas molecule at 300 K.
Solution:
[\langle E_t \rangle] [= \dfrac{3}{2}kT] [= \dfrac{3}{2} \times 1.38 \times 10^{-23} \times 300] [= 6.21 \times 10^{-21} J]
Q2. At what temperature will the average kinetic energy of a gas molecule be [1.0 \times 10^{-20} J]?
Solution:
[\langle E_t \rangle] [= \dfrac{3}{2}kT \Rightarrow T] [= \dfrac{2E_t}{3k}] [= \dfrac{2 \times 1.0 \times 10^{-20}}{3 \times 1.38 \times 10^{-23}}] [= 483.1 K]
Q3. A diatomic molecule has a moment of inertia [2.0 \times 10^{-46} kgm^2] and angular speed [5.0 \times 10^{12} rad/s]. Find its rotational kinetic energy.
Solution:
[E_r] [= \dfrac{1}{2}I\omega^2] [= \dfrac{1}{2} \times 2.0 \times 10^{-46} \times (5.0 \times 10^{12})^2] [= 2.5 \times 10^{-21} J]
Q4. If each molecule has [E_t = 6 \times 10^{-21} J], [E_r = 4 \times 10^{-21} J], and [E_v = 2 \times 10^{-21} J], find the total energy per molecule and per mole.
Solution:
[E = E_t + E_r + E_v] [= 6 + 4 + 2] [= 12 \times 10^{-21} J]
Per mole: [E_{mole}] [= E \times N_A] [= 12 \times 10^{-21} \times 6.022 \times 10^{23}] [= 7.23 kJ/mol]
Q5. Explain why vibrational energy is negligible at low temperature.
Solution:
At low temperatures, molecules lack sufficient thermal energy to excite vibrational modes, so their contribution to total energy is negligible.
Sign in with Google