First Law and Internal Energy §

Sign convention. The convention above (engineering / physics) takes $W > 0$ when the system does work on the surroundings. IUPAC (chemistry) uses the opposite sign: $dU = \delta Q + \delta W$ where $W$ is work done on the system. Always check the convention.

State function vs. path function. $U$ is a state function: $\Delta U$ depends only on initial and final states. $Q$ and $W$ are path functions: their values depend on the process. The inexact differentials $\delta Q$, $\delta W$ (not $dQ$, $dW$) reflect this.

Perpetual motion of the first kind. A device that produces work with no energy input (no heat, no mass flow) would violate the first law. The first law rules out such a device but does not rule out a 100%-efficient heat engine (that is the second law).

$P\,dV$ work. For a quasi-static (reversible) expansion, the system pressure equals the external pressure at every instant, and the work is the area under the $P$–$V$ curve:

For an irreversible (free) expansion into vacuum, $P_{\text{ext}} = 0$ and $W = 0$, even though $\Delta V \neq 0$.

Other work. Electrical work in a battery: $\delta W_{\text{elec}} = -\mathcal{E}\,dQ_{\text{charge}}$. Shaft work in a turbine: $\delta W_{\text{shaft}} = \tau\,d\theta$. These are not $P\,dV$ but still count as work in the first law.

Ideal gas. For a monatomic ideal gas, $U = \tfrac{3}{2}Nk_B T$. For a diatomic ideal gas (at moderate $T$), $U = \tfrac{5}{2}Nk_B T$. The key point: $U$ depends only on $T$, not on $V$ or $P$.

Real substances. For a real gas or a condensed phase, $U = U(T, V)$. The volume dependence comes from intermolecular potential energy. The equation of state (e.g., van der Waals) connects $P, V, T$ and allows you to compute $(\partial U/\partial V)_T$ from thermodynamic identities.

Reference state. Only $\Delta U$ is physically meaningful. You choose a reference (e.g., $U = 0$ for an ideal gas at 0 K) and work with differences.

Adiabatic relation. For a reversible adiabatic process of an ideal gas:

where $\gamma = C_P/C_V$. For a monatomic gas $\gamma = 5/3$; for diatomic $\gamma = 7/5$.

Isothermal. Since $\Delta U = 0$ for an ideal gas at constant $T$, all heat input becomes work: $Q = W = nRT\ln(V_f/V_i)$. This is the basis of the Carnot isothermal steps.

This equation is the workhorse of turbomachinery (turbines, compressors, nozzles) and heat exchangers. When kinetic and potential energy changes are negligible (common in heat exchangers), it reduces to $\dot{Q} - \dot{W} = \dot{m}(h_{\text{out}} - h_{\text{in}})$.

Isothermal ideal gas: $\Delta U = 0$ (internal energy depends only on $T$).

Work:

First law: $\Delta U = Q - W \Rightarrow Q = W \approx 5480 \text{ J}$.

The gas absorbs 5.48 kJ of heat and does 5.48 kJ of work on the surroundings; its internal energy is unchanged.