The First Law of Thermodynamics

The First Law of Thermodynamics is a statement of energy conservation for thermodynamic systems: energy can be transferred as heat or work and can be stored as internal, kinetic, or potential energy. For a closed system (fixed mass) the change in total energy equals heat added to the system minus work done by the system. For control volumes (open systems) the rate form relates heat and work rates to mass flow carrying enthalpy, kinetic and potential energy. Sign conventions must be stated explicitly: here we use the engineering sign convention (Q positive when heat is added to the system, W positive when work is done by the system).

Governing FormulaClosed system (single-process): ΔE = ΔU + ΔKE + ΔPE = Q - W (engineering sign). For processes where ΔKE and ΔPE are negligible: ΔU = Q - W. For ideal gases: ΔU = m c_v ΔT. Reversible PV work for a closed system: W_rev = ∫ p dV (for an ideal gas reversible isothermal: W_rev = m R T ln(V1/V2)). Control-volume steady-flow energy equation (neglecting heat/shaft sign subtleties): Q̇ - Ẇ = ṁ( h_out - h_in + (ke_out - ke_in) + (pe_out - pe_in) ). For a cyclic process ΔU = 0 so Q_net = W_net. Pump approximate work for incompressible fluid: w_pump ≈ v (p2 - p1) per unit mass.

Knowledge Check

10 Questions

1.Which equation correctly expresses the First Law for a closed system using the engineering sign convention (Q positive into system, W positive done by system) and neglecting changes in kinetic and potential energy?

2.A closed system receives 500 J of heat and does 200 J of shaft work on its surroundings. Using the engineering sign convention, what is the change in internal energy ΔU?

3.An ideal gas (treated as a closed system) of mass 2.00 kg is heated at constant volume. Take c_v = 718 J·kg^-1·K^-1 and the temperature rises by 30.0 K. Neglect kinetic and potential energy changes. How much heat was added to the gas (in J)?

4.One kilogram of an ideal gas (R = 287 J·kg^-1·K^-1) at 300 K is compressed reversibly and isothermally to half its initial volume. What is the magnitude of work required (in J) to compress it (work done on the gas)?

5.A steady-flow turbine has an inlet enthalpy of 2800 kJ·kg^-1 and an outlet enthalpy of 2300 kJ·kg^-1. The mass flow rate is 2.00 kg·s^-1. The turbine rejects 200 kW of heat to the surroundings. Neglect kinetic and potential energy changes. Using the convention Q positive into the device, what is the shaft power produced by the turbine (in kW)?

6.An engine undergoes a cyclic process and returns to its initial state; during the cycle it does 300 J of net work on the surroundings. According to the First Law, what is the net heat transferred to the working fluid over the cycle?

7.A closed container containing gas is compressed adiabatically (no heat transfer) and 250 J of mechanical work are done on the gas. Using the engineering sign convention, what is the change in internal energy ΔU of the gas?

8.A pump delivers water (assume incompressible specific volume v = 1.00×10^-3 m^3·kg^-1) at mass flow 0.500 kg·s^-1 from 100 kPa to 4.00 MPa. Neglect heat transfer, kinetic and potential energy changes. What power input (in kW) is required by the pump (assume ideal, isentropic mechanical behavior so pump work ≈ vΔp per unit mass)?

9.An ideal gas initially confined to half of an insulated rigid container suddenly expands into the full container volume by opening a valve to a vacuum (free expansion). Neglect kinetic and potential energy changes. According to the First Law, what happens to the gas temperature (assume ideal gas behavior)?

10.A heat engine absorbs 600 J from a hot reservoir and rejects 420 J to a cold reservoir in one cycle. Using the First Law, what is the thermal efficiency of the engine?