Asked by Lilliaan
What is the relationship between temperature and kinetic energy?
All Answers 1
Answered by
GPT-5 mini
AI
Temperature is (proportional to) a measure of the average kinetic energy of the microscopic particles in a substance.
Key formulas and points:
- For an ideal monoatomic gas, the average translational kinetic energy per particle is
<KE> = (1/2) m<v^2> = (3/2) kB T,
where kB ≈ 1.3807×10^−23 J/K is Boltzmann’s constant, m is the particle mass, and T is the absolute temperature (K).
- Per mole, the internal energy of a monoatomic ideal gas is
U = (3/2) N_A kB T = (3/2) R T,
where R = N_A kB ≈ 8.314 J/(mol·K).
- Root-mean-square speed: vrms = sqrt(3 kB T / m). So at the same T, heavier particles move more slowly even though their average kinetic energy is the same.
Caveats:
- “Temperature = average kinetic energy” strictly refers to the average energy associated with the degrees of freedom in equilibrium (translational for an ideal gas). By the equipartition theorem, each quadratic degree of freedom contributes (1/2) kB T of energy.
- In solids, liquids, or at low temperatures, quantum effects and non-translational modes (vibrational, rotational, interactions) change how energy is stored, but temperature still reflects the system’s thermal energy per degree of freedom at equilibrium.
Key formulas and points:
- For an ideal monoatomic gas, the average translational kinetic energy per particle is
<KE> = (1/2) m<v^2> = (3/2) kB T,
where kB ≈ 1.3807×10^−23 J/K is Boltzmann’s constant, m is the particle mass, and T is the absolute temperature (K).
- Per mole, the internal energy of a monoatomic ideal gas is
U = (3/2) N_A kB T = (3/2) R T,
where R = N_A kB ≈ 8.314 J/(mol·K).
- Root-mean-square speed: vrms = sqrt(3 kB T / m). So at the same T, heavier particles move more slowly even though their average kinetic energy is the same.
Caveats:
- “Temperature = average kinetic energy” strictly refers to the average energy associated with the degrees of freedom in equilibrium (translational for an ideal gas). By the equipartition theorem, each quadratic degree of freedom contributes (1/2) kB T of energy.
- In solids, liquids, or at low temperatures, quantum effects and non-translational modes (vibrational, rotational, interactions) change how energy is stored, but temperature still reflects the system’s thermal energy per degree of freedom at equilibrium.
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