Internal Energy: MDCAT Chemistry notes
Internal Energy MDCAT notes: meaning of internal energy, first law sign conventions, heat at constant volume, and how enthalpy relates to it.
What internal energy is
The internal energy ($E$ or $U$) of a system is the sum of all the kinetic and potential energies of its particles:
- Kinetic energy: translational, rotational and vibrational motion of molecules. Temperature measures the average kinetic energy.
- Potential energy: energy stored in chemical bonds and in intermolecular attractions.
The absolute value of $E$ cannot be measured; only the change $\Delta E = E_2 - E_1$ can be. Internal energy is a state function.
What an increase in internal energy can do
- Raise the temperature, by increasing the kinetic energy of the molecules.
- Cause a phase change such as melting or evaporation, by overcoming intermolecular forces.
- Bring about a chemical reaction, if enough energy is supplied to break bonds.
A drop in molecular kinetic energy lowers the temperature; it can never raise it.
First law and internal energy
$$\Delta E = q + w$$
| Process | Sign |
|---|---|
| Heat absorbed by the system | $q$ positive |
| Heat released by the system | $q$ negative |
| Work done on the system (compression) | $w$ positive |
| Work done by the system (expansion) | $w$ negative |
For pressure-volume work against a constant external pressure: $w = -P\Delta V$.
Worked example 1
A system absorbs 250 J of heat and does 90 J of work on the surroundings. $q = +250$, $w = -90$, so $\Delta E = +160\ \mathrm{J}$.
Worked example 2
A system releases 50 kJ of heat while 30 kJ of work is done on it. $\Delta E = -50 + 30 = -20\ \mathrm{kJ}$.
Constant volume
At constant volume $\Delta V = 0$, so no pressure-volume work is done and
$$\Delta E = q_v$$
The heat exchanged at constant volume equals the change in internal energy. It is measured in a bomb calorimeter, a sealed steel vessel.
Relation between enthalpy and internal energy
$$\Delta H = \Delta E + P\Delta V$$
- For gases: $\Delta H = \Delta E + \Delta n\,RT$, where $\Delta n$ is the change in moles of gas.
- For solids and liquids $\Delta V$ is tiny, so $\Delta H \approx \Delta E$.
- When a substance contracts slightly (e.g. ice melting to water), $P\Delta V$ is small and negative, so $\Delta E$ is slightly larger than $\Delta H$.
- Conversion: $1\ \mathrm{dm^3\,atm} = 101.3\ \mathrm{J}$.
Worked example 3
At 1 atm, a process absorbs 5000 J and the volume increases by $0.5\ \mathrm{dm^3}$. $w = -P\Delta V = -0.5\ \mathrm{dm^3\,atm} = -50.7\ \mathrm{J}$. So $\Delta E = 5000 - 50.7 = 4949.3\ \mathrm{J}$.
Key formulas
- $\Delta E = q + w$; $w = -P\Delta V$
- $\Delta E = q_v$; $\Delta H = q_p$
- $\Delta H = \Delta E + P\Delta V = \Delta E + \Delta n RT$
Common MDCAT traps
- Heat absorbed and work done on the system both increase $E$; add them.
- Work done by the system is negative.
- If volume decreases, work is done on the system, so $\Delta E$ is slightly greater than $q_p$.
- Remember to convert $\mathrm{dm^3\,atm}$ to joules before combining with $q$.
Quick revision
- Internal energy = total kinetic + potential energy of particles.
- Only $\Delta E$ is measurable.
- At constant volume, heat supplied equals $\Delta E$.
- Bomb calorimeter measures $\Delta E$.
- For solids and liquids, $\Delta H \approx \Delta E$.