First Law for Control Mass Systems

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Core Idea

For a control mass (closed system), the first law states dE/dt = Q̇ - Ẇ, relating the rate of energy change to heat transfer and work rates. Total energy E = U + KE + PE; applications include pistons, springs, and accumulators where mass is conserved but energy changes through boundary interactions. Process constraints (adiabatic, isothermal) simplify the equation.

Explainer

You already know the first law for a closed system: energy is conserved, and any change in the system's energy equals net heat added minus net work done by the system. Now you apply that principle to concrete physical processes — pistons compressing gas, rigid tanks storing fluid, springs being loaded — and work through what the energy balance predicts in each case.

The general form is dE/dt = Q̇ − Ẇ, where total energy E = U + KE + PE. For most engineering control mass problems, the system itself isn't moving as a whole, so kinetic and potential energy of the system are negligible. The equation simplifies to dU = δQ − δW. The key skill is correctly identifying Q (heat crossing the system boundary, positive when entering) and W (work done by the system, positive when exiting). Sign conventions matter enormously here; always define them before writing the balance.

Boundary work is the work done by a piston face as it moves: W_boundary = ∫P dV. For a constant-pressure (isobaric) process, this simplifies to P ΔV. For a constant-volume (isochoric) process — a rigid tank, a bomb calorimeter — the piston doesn't move, so boundary work is zero and the first law becomes ΔU = Q. All heat input goes directly into raising internal energy. For an adiabatic process, no heat crosses the boundary (Q = 0), so any work done must draw down internal energy: ΔU = −W.

Each process constraint eliminates a term or fixes a relationship between state variables, making the energy balance tractable. For constant-pressure processes in particular, the first law can be rewritten in terms of enthalpy H = U + PV: ΔH = Q_P. This substitution absorbs the boundary work P ΔV into the state property H, making enthalpy the natural energy bookkeeping quantity for constant-pressure processes. This is why engineers track enthalpy — not internal energy — for most practical devices, and why property tables list both U and H values. Recognizing which process constraint applies before writing the energy balance is the central problem-solving discipline of this topic.

Practice Questions 5 questions

Prerequisite Chain

Counting to 10 → Counting to 20 → Understanding Zero → The Number Zero → Counting to Five → One-to-One Correspondence → Combining Small Groups Within 5 → Addition Within 10 → Addition Within 20 → Two-Digit Addition Without Regrouping → Two-Digit Addition with Regrouping → Addition Within 100 → Repeated Addition as Multiplication → Multiplication Facts Within 100 → Division as Equal Sharing → Division as Grouping (Measurement Division) → Division: Grouping (Repeated Subtraction) Model → Division: Fair Sharing Model → Division as Equal Sharing → Division as Grouping → Basic Division Facts → Division Facts Within 100 → Two-Digit by One-Digit Division → Division with Remainders → Remainders and Quotients in Division → Division Word Problems → Introduction to Long Division → Factors and Multiples → Prime and Composite Numbers → Equivalent Fractions → Relating Fractions and Decimals → Decimal Place Value → Reading and Writing Decimals → Comparing and Ordering Decimals → Adding and Subtracting Decimals → Multiplying Decimals → Dividing Decimals → Dividing Fractions → Mixed Number Arithmetic → Order of Operations → Integer Order of Operations → Variable Expressions → Combining Like Terms → One-Step Equations → Two-Step Equations → Solving Multi-Step Equations → Equations with Variables on Both Sides → Angle Pairs: Complementary, Supplementary, and Vertical → Parallel Lines and Transversals → Corresponding Angles → Alternate Interior Angles → Triangle Angle Sum Theorem → Exterior Angle Theorem → Triangle Inequality Theorem → Similar Triangles: AA Similarity → Similar Triangles: SSS and SAS Similarity → Proportions in Similar Triangles → Right Triangle Trigonometry Introduction → Trigonometric Ratios Review → Radian Measure → Converting Between Degrees and Radians → The Unit Circle → Graphing Sine and Cosine → Graphing Tangent and Reciprocal Trigonometric Functions → Derivatives of Trigonometric Functions → Antiderivatives → Iterated Integrals and Fubini's Theorem → Double Integrals in Cartesian Coordinates → Double Integrals over Rectangular Regions → Double Integrals in Polar Coordinates → Double Integrals: Definition and Setup → Iterated Integrals and Fubini's Theorem → Double Integrals over Rectangular Regions → Double Integrals over General Regions → Applications of Double Integrals: Area, Mass, and Moments → Center of Mass → Conservation of Linear Momentum → Elastic Collisions → Inelastic Collisions → Coefficient of Restitution → Collision Analysis and Real-World Applications → Two-Body Collisions in the Center-of-Mass Frame → Reduced Mass and Two-Body Problems → Kinematics in Two Dimensions → Projectile Motion → Circular Motion: Kinematics → Rotational Kinematics → Torque → Moment of Inertia → Rotational Kinetic Energy → The Work-Energy Theorem → Conservation of Mechanical Energy → Work-Energy Principle for Particles → Work-Energy Methods for Systems → Work-Energy Methods for Rigid Bodies → Potential Energy and Conservative Forces → Conservation of Mechanical Energy in Systems → First Law of Thermodynamics for Closed Systems → State Functions and Path Functions in Thermodynamics → First Law for Control Mass Systems

Longest path: 100 steps · 535 total prerequisite topics

Prerequisites (2)

First Law of Thermodynamics for Closed Systemshard State Functions and Path Functions in Thermodynamicssoft

Leads To (1)

First Law for Open Systems and Control Volumessoft