14.1 Energy, Heat, Work & the First Law of Thermodynamics

14.1 Energy, Heat, Work & the First Law of Thermodynamics

Quick Summary: Thermodynamics describes macroscopic energy flow and transformations. The First Law establishes universal energy conservation: $\Delta U_{\text{universe}} = 0$. System internal energy changes follow $\Delta U = q + w$, where $q$ is heat exchanged and $w$ is work performed. State functions ($P, V, T, U, H, S, G$) depend solely on current equilibrium state, whereas heat and work are path functions. Expansion work against constant external pressure is $w = -P_{\text{ext}} \Delta V$ ($1\text{ L}\cdot\text{atm} = 101.325\text{ J}$). Constant-volume heat exchange equals internal energy change ($q_v = \Delta U$); constant-pressure heat exchange equals enthalpy change ($q_p = \Delta H$), related by $\Delta H = \Delta U + \Delta n_{\text{gas}} RT$.

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1. Thermodynamic Definitions: System, Surroundings & Universe

Thermodynamic analysis divides the physical universe into three components:

• System: The specific portion of the universe singled out for study (e.g., reactants in a flask, gas in a piston).
• Surroundings: Everything outside the system that can exchange energy or matter with it (e.g., container walls, solvent, atmosphere).
• Boundary: The real or hypothetical surface separating system from surroundings.
• Universe: Totality of system plus surroundings ($\text{Universe} = \text{System} + \text{Surroundings}$).

Systems are classified by boundary permeability:

1. Open System: Exchanges both matter and energy with surroundings (e.g., an open beaker of boiling water).
2. Closed System: Exchanges energy (heat/work) but no matter with surroundings (e.g., a sealed cylinder with a movable piston).
3. Isolated System: Exchanges neither matter nor energy with surroundings (e.g., an ideal vacuum Dewar flask).

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2. State Functions versus Path Functions

Thermodynamic variables differ in path dependence:

• State Functions: Properties determined solely by current equilibrium state, independent of history or pathway. For any state function $X$, $\Delta X = X_{\text{final}} - X_{\text{initial}}$. Examples include $P, V, T, U, H, S,$ and $G$.
• Path Functions: Quantities depending on the specific transition route between states. Heat ($q$) and work ($w$) describe energy in transit. Systems do not contain $q$ or $w$; they contain internal energy.

Comparison: State versus Path Functions

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3. The First Law of Thermodynamics & Internal Energy ($\Delta U$)

The First Law of Thermodynamics asserts conservation of energy: $\Delta U_{\text{universe}} = 0 \implies \Delta U_{\text{system}} = -\Delta U_{\text{surroundings}}$ Internal energy ($U$) is the sum of microscopic kinetic energies (molecular motion) and potential energies (chemical bonds, intermolecular forces).

IUPAC Sign Conventions

Under standard IUPAC conventions: $\Delta U = q + w$ Energy entering the system is positive ($+$); energy leaving is negative ($-$).

Thermodynamic Sign Conventions

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4. Mechanical Pressure-Volume ($PV$) Work

Work performed against constant external pressure ($P_{\text{ext}}$) during volume changes is: $w = -P_{\text{ext}} \Delta V = -P_{\text{ext}} (V_{\text{final}} - V_{\text{initial}})$

• Expansion ($\Delta V > 0$): Gas pushes on surroundings, doing work ($w < 0$).
• Compression ($\Delta V < 0$): Surroundings push on gas, adding energy ($w > 0$).
• Free Expansion ($P_{\text{ext}} = 0$): Gas expands into vacuum; $w = 0$.

Unit Conversion

$1\text{ L}\cdot\text{atm} = (10^{-3}\text{ m}^3)(101,325\text{ N/m}^2) = 101.325\text{ J}$

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5. Constant Volume ($\Delta U = q_v$) vs. Constant Pressure ($\Delta H = q_p$)

Thermal transfer depends on operational constraints:

• Constant Volume ($\Delta V = 0$): Because $w = -P_{\text{ext}}(0) = 0$: $\Delta U = q_v$ All heat exchanged directly alters internal energy.
• Constant Pressure (Open Vessel): System expands or contracts ($w = -P\Delta V$): $\Delta U = q_p - P\Delta V \implies q_p = \Delta U + P\Delta V$ Enthalpy is defined as $H = U + PV$. At constant pressure: $\Delta H = \Delta U + P\Delta V = q_p$
• Ideal Gas Relationship: $\Delta H = \Delta U + (\Delta n_{\text{gas}})RT$ where $\Delta n_{\text{gas}} = \sum n_{\text{gas, products}} - \sum n_{\text{gas, reactants}}$ and $R = 8.314\text{ J/(mol}\cdot\text{K)}$.

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6. Worked Example: Work and Internal Energy

Problem: A gas absorbs $450\text{ J}$ of heat while expanding from $1.50\text{ L}$ to $4.00\text{ L}$ against a constant pressure of $1.80\text{ atm}$. Find work ($w$) and internal energy change ($\Delta U$).

Step 1: Compute expansion work $\Delta V = 4.00\text{ L} - 1.50\text{ L} = +2.50\text{ L}$ $w = -(1.80\text{ atm})(2.50\text{ L}) = -4.50\text{ L}\cdot\text{atm}$

Step 2: Convert to joules $w = -4.50\text{ L}\cdot\text{atm} \times 101.325\frac{\text{J}}{\text{L}\cdot\text{atm}} = -456\text{ J}$

Step 3: Calculate $\Delta U$ $\Delta U = q + w = (+450\text{ J}) + (-456\text{ J}) = -6\text{ J}$ Conclusion: Internal energy drops by $6\text{ J}$ because expansion work exceeded heat absorbed.