🔹 1. Definition of Internal Energy

Internal Energy (U) is the total energy possessed by a system due to:

1. Kinetic energy of molecules (translation, rotation, vibration)

2. Potential energy of molecules (intermolecular forces, bonds)

It includes:

Motion of molecules

Attractions/repulsions between molecules

Energy stored in chemical bonds

It does not include:

Kinetic energy of the whole system

Potential energy due to position (height)

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🔹 2. Nature of Internal Energy

1. State function

Internal energy depends only on the state of the system, not on the path.

\Delta U = U_2 - U_1

2. Extensive property

Depends on the amount of substance.

3. Absolute value of U cannot be measured

Only change in internal energy (ΔU) can be measured.

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🔹 3. First Law of Thermodynamics (Basis of Internal Energy)

> Energy can neither be created nor destroyed, only converted from one form to another.

Mathematically:

\Delta U = q + w

Where:

 = change in internal energy

 = heat supplied to the system

 = work done on the system

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🔹 4. Derivation of ΔU = q + w

Case: System changes from state A to state B

Let:

Initial internal energy = 

Final internal energy = 

Change in internal energy:

\Delta U = U_2 - U_1

According to First Law:

If heat is supplied to the system

And work is done on the system

Then energy gained by system = heat gained + work gained

\boxed{\Delta U = q + w}

This is the thermodynamic equation of internal energy.

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🔹 5. Work Done in Expansion (Derivation)

For gas expansion against external pressure:

w = -P_{ext} \Delta V

Substitute in first law:

\Delta U = q - P_{ext}\Delta V

This is the general expression used in chemistry.

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🔹 6. Internal Energy Change in Special Processes

(A) Isochoric Process (Volume constant)

\Delta V = 0 \Rightarrow w = 0

So:

\Delta U = q_v

➡ Heat supplied = change in internal energy

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(B) Isothermal Process (Temperature constant, ideal gas)

For ideal gas:

\Delta U = 0

Because internal energy depends only on temperature.

So:

q = -w

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🔹 7. Internal Energy of an Ideal Gas (Derivation)

For an ideal gas:

No intermolecular forces

No potential energy

Only kinetic energy contributes

Average kinetic energy ∝ Temperature

Hence:

U = f(T)

So change in internal energy:

\Delta U = nC_v\Delta T

Where:

 = number of moles

 = molar heat capacity at constant volume

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🔹 8. Sign Convention

Process q w ΔU

Heat absorbed +  

Heat released −  

Work done on system + 

Work done by system − 

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🔹 9. Example (Numerical Concept)

If:

Heat supplied = 200 J

Work done by system = 50 J

\Delta U = q + w = 200 + (-50) = 150 J

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Derivation of Internal Energy Equation (First Law of Thermodynamics)

Class 11 Chemistry – Board exam writing format

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Statement of First Law of Thermodynamics

Energy can neither be created nor destroyed, it can only be transformed from one form into another.

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Derivation

Consider a thermodynamic system which changes from state 1 to state 2.

Let:

Internal energy in state 1 = 

Internal energy in state 2 = 

Change in internal energy:

\Delta U = U_2 - U_1

If:

Heat supplied to the system = 

Work done on the system = 

Then, according to the law of conservation of energy:

> Increase in internal energy of the system = Heat supplied + Work done on the system

\Delta U = q + w

This is the mathematical form of the First Law of Thermodynamics.

\boxed{\Delta U = q + w}

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Derivation including work of expansion

For a gas expanding against external pressure :

Work done:

w = -P_{ext}\Delta V

Substitute in first law equation:

\Delta U = q - P_{ext}\Delta V

This is the general thermodynamic equation used in chemistry.

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Derivation for Constant Volume Process

At constant volume:

\Delta V = 0 \Rightarrow w = 0

So:

\Delta U = q_v

Hence, heat absorbed at constant volume equals change in internal energy.

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Derivation for Ideal Gas

Internal energy of an ideal gas depends only on temperature:

U = f(T)

So change in internal energy:

\Delta U = nC_v\Delta T

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Conclusion

Thus, from the first law of thermodynamics we derive the internal energy equation:

\boxed{\Delta U = q + w}

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✍️ Exam Writing Tips

Write in this order:

1. Statement

2. Assumptions

3. Mathematical steps

4. Final formula in box

5. Special cases

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If you want, I can also write this derivation in: ✅ Very short form (5–6 lines)

✅ Long answer (full 5-mark derivation)

✅ With diagram

✅ In Hindi

Just say: short / long / diagram / Hindi

Mon Feb 16, 2026