04_1st-law

1. The First Law of Thermodynamics

Quasi-static process: A process that occurs slowly enough that the system remains in thermal equilibrium at all times.
Work (\(W\)):
- Defined as the integral of pressure over volume: \(dW = F ds = P A ds = P dV \implies W = \int P dV\)
- Geometrically, work is the area under the curve on a \(P\)-\(V\) diagram.
- Sign Convention:
  - Work done by the gas: \(dW = P dV\)
  - Work done to the gas: \(dW = -P dV\)
- Path Dependence: The amount of work done depends on the path taken from the initial state (\(i\)) to the final state (\(f\)). For example, expanding at a constant high pressure yields more work than dropping the pressure first and then expanding (\(W^{(a)} < W^{(c)} < W^{(b)}\)).
The First Law of Thermodynamics:
- Formula: \(\Delta U = \Delta Q - W\)
(Where \(\Delta Q\) is heat absorbed, and \(W\) is work done by the system).
- Differential form: \(dU = \delta Q - \delta W\)
  
  (Note: The symbol \(\delta\) indicates an inexact, path-dependent differential).

The integral of an inexact differential is path-dependent.
Example: \(\delta g = dx + \frac{x}{y} dy\) (inexact)
- Evaluating from \(A(1,1)\) to \(B(2,2)\) via Path 1 (\(A \rightarrow C(2,1) \rightarrow B\)) yields: \(1 + 2\ln2\).
- Evaluating via Path 2 (\(A \rightarrow D(1,2) \rightarrow B\)) yields: \(\ln2 + 1\).
Contrast (Exact): \(df = \frac{dx}{x} + \frac{dy}{y}\) is exact. Both paths yield \(2\ln2\).

图示：General physics 1st law3

Work done by the gas:

\[ W = \int_{V_i}^{V_f} P dV = \int_{V_i}^{V_f} \frac{nRT}{V} dV = nRT \ln \frac{V_f}{V_i} \]

Governing equation: \(P V^\gamma = \text{constant}\) (where \(\gamma \equiv \frac{C_P}{C_V}\))

Proof:
1. First law for adiabatic: \(dU = 0 - P dV = -P dV\)
2. Using \(U = n C_V T \implies n C_V dT = -P dV\)
3. Ideal gas law differential: \(P dV + V dP = nR dT\)
4. Substitute (2) into (3): \(-\frac{P}{C_V} dV = \frac{P}{R} dV + \frac{V}{R} dP\)
5. Rearrange: \(P\left(\frac{1}{R} + \frac{1}{C_V}\right) dV + \frac{V}{R} dP = 0\)
6. Multiply by \(R\) and use \(C_P = C_V + R\): \(\frac{C_P}{C_V} \frac{dV}{V} + \frac{dP}{P} = 0\)
7. Integrate (\(\gamma = \frac{C_P}{C_V}\)): \(\gamma \ln V + \ln P = \text{constant} \implies P V^\gamma = \text{constant}\).

\(\Delta Q = 0\) (Adiabatic) and \(W = 0\) (Free expansion into a vacuum).
It is not quasi-static (the system is out of equilibrium except for the initial and final states).
Since \(\Delta U = 0 \implies U_i = U_f\), the temperature remains constant (\(T_i = T_f\)).

Definition: A cyclic device that extracts energy (heat) from the environment and does work (e.g., steam engine, gasoline engine).
Cyclic Process:
- The net change in internal energy over one cycle is zero (\(\Delta U = 0\)).
- Work done per cycle: \(W = \oint P dV\)
- Net heat absorbed: \(\Delta Q = W = \oint P dV\)
Thermal Efficiency (\(\varepsilon\)):

\[ \varepsilon \equiv \frac{W}{Q_H} \]

Since \(W = Q_H - Q_L\) (\(Q_H\): heat from high-T reservoir, \(Q_L\): heat to low-T reservoir):

\[ \varepsilon = 1 - \frac{Q_L}{Q_H} \]

Commonly used in petrol engines.

图示：General physics 1st law5

Four Strokes:
- \(O \rightarrow A\): Intake (often omitted in schematic drawings)
- \(A \rightarrow B\): Adiabatic Compression
- \(B \rightarrow C \rightarrow D\): Combustion/Ignition/Power (\(B \rightarrow C\) is isochoric heat absorption \(Q_H\); \(C \rightarrow D\) is adiabatic expansion)
- \(D \rightarrow A (\rightarrow O)\): Exhaust (\(D \rightarrow A\) is isochoric heat rejection \(Q_L\))
Efficiency Derivation:
- \(Q_L = n C_V (T_D - T_A)\) and \(Q_H = n C_V (T_C - T_B)\)
- \(\varepsilon_{\text{Otto}} = 1 - \frac{T_D - T_A}{T_C - T_B}\)
- Applying \(T V^{\gamma-1} = \text{constant}\) to the adiabatic steps (\(A \rightarrow B\) and \(C \rightarrow D\)):
  
  \[ \left(\frac{V_2}{V_1}\right)^{\gamma-1} = \frac{T_A}{T_B} = \frac{T_D}{T_C} \]
- Final Otto Efficiency: \(\varepsilon_{\text{Otto}} = 1 - \left(\frac{V_2}{V_1}\right)^{\gamma-1}\)

Commonly used in diesel engines.

图示：General physics 1st law6

Key Difference: Heat absorption (\(Q_H\)) occurs via an isobaric process (\(B \rightarrow C\)), while heat rejection (\(Q_L\)) remains isochoric (\(D \rightarrow A\)).
Efficiency Derivation:
- \(T_B = \left(\frac{V_1}{V_2}\right)^{\gamma-1} T_A\) (from adiabatic \(A \rightarrow B\))
- \(T_C = \frac{V_3}{V_2} T_B = \frac{V_3}{V_2} \left(\frac{V_1}{V_2}\right)^{\gamma-1} T_A\) (from isobaric \(B \rightarrow C\))
- \(T_D = \left(\frac{V_3}{V_1}\right)^{\gamma-1} T_C = \left(\frac{V_3}{V_2}\right)^\gamma T_A\) (from adiabatic \(C \rightarrow D\))
- \(Q_H = n C_P (T_C - T_B)\) and \(Q_L = n C_V (T_D - T_A)\)
- Final Diesel Efficiency:
  
  \[ \varepsilon_{\text{diesel}} = 1 - \frac{C_V (T_D - T_A)}{C_P (T_C - T_B)} = 1 - \frac{1}{\gamma} \frac{(V_3/V_2)^\gamma - 1}{(V_3/V_2) - 1} \left(\frac{V_2}{V_1}\right)^{\gamma-1} \]

Commonly used in ship engines.

图示：General physics 1st law7

Cycle structure: Heat addition is split into two sequential steps—an isochoric process followed by an isobaric process—bridged by adiabatic expansion and compression, concluding with an isochoric heat rejection.