Internal energy
Section: Thermodynamics | Syllabus: Cambridge A Level Physics (9702)
Defining Internal Energy (U)
The internal energy of a system is the sum of all the kinetic energy and potential energy that is randomly distributed among the molecules of the system.
Randomly distributed: This means any molecule in any position can have any energy within the total range; there is no specific pattern to individual molecular energies.
Figure 16.1: Molecular Energy TypesDiagram illustrating:
1. Translational KE: Molecule moving in a straight line.
2. Rotational KE: Molecule spinning about an axis.
3. Vibrational KE: Atoms within a molecule oscillating.
4. Potential Energy: Due to intermolecular forces (attraction) and chemical bonds.
Ideal Gas Assumptions
In the kinetic theory of an ideal gas, we assume:
- No Intermolecular Forces: This means the Potential Energy is zero.
- Point Particles: Noble gases like Helium and Argon exist as single atoms, meaning they have only translational kinetic energy.
Internal energy depends strictly on the thermodynamic temperature (T) for a fixed mass of gas.
Increasing and Decreasing Internal Energy
Internal energy can be changed by supplying thermal energy or by doing work:
- Rubbing Hands: When you rub your hands together, the movement against friction means you are doing work on your hands, increasing their internal energy and temperature.
- Expansion Cooling: When fuel gas expands from a cylinder, it does work against the surroundings. Its internal energy decreases, causing the cylinder to cool so much that ice may form on the outside (Figure 16.2).
Worked Examples
Worked Example 1: Helium Balloon (Q1)Question: A balloon contains 1.79 g of helium. The mean KE of atoms is 6.21 × 10^-21 J. Calculate the internal energy. (Molar mass of He = 4.00 g mol^-1).
Solution1. Find moles: n = 1.79 g / 4.00 g mol^-1 = 0.4475 mol.
2. Find atoms: N = n × N_A = 0.4475 × 6.02 × 10^23 = 2.694 × 10^23.
3. Total U = N × mean KE = 2.694 × 10^23 × 6.21 × 10^-21 = 1.67 kJ.
Worked Example 2: Hydrogen Release (Q2)Question: A cylinder holds 12.5 g of hydrogen (U = 19.8 kJ). (a) Calculate the mean KE of a molecule. (b) Explain why U decreases when the valve is opened.
Solution(a) Moles n = 12.5 / 2.0 = 6.25 mol.
Molecules N = 6.25 × 6.02 × 10^23 = 3.7625 × 10^24.
Mean KE = U / N = 19800 / 3.7625 × 10^24 = 5.26 × 10^-21 J.
(b) Internal energy decreases because:
1. Gas escapes, so there are fewer molecules.
2. The escaping gas does work against the atmosphere as it expands, reducing the energy of the remaining molecules.
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