Application of the First Law to Closed Systems

Introduction and scope

A closed system is a thermodynamic system that permits energy transfer across its boundary but does not permit mass transfer across the boundary. The First Law of Thermodynamics applied to a closed system expresses conservation of energy by accounting for changes in the system's internal energy together with changes in its kinetic and potential energies, and by equating those to heat and work crossing the system boundary.

General energy balance for a closed system

Consider a closed system which may possess internal energy, kinetic energy and potential energy. Using specific quantities (per unit mass) denote:

• u - specific internal energy
• ke - specific kinetic energy
• pe - specific potential energy

When energy crosses the system boundary it can do so in the form of heat Q or work W. The general first-law statement for the closed system is therefore:

This may be expressed on a per unit mass (or per mole) basis:

In differential form the First Law for a closed system can be written as

dU + d(KE) + d(PE) = δQ + δW

When changes in kinetic and potential energy are negligible (a common practical approximation), the relation reduces to

dU = δQ + δW

Note the notation: d denotes an exact differential of a state function, while δ denotes an inexact differential (path-dependent) for heat and work.

Sign convention for heat and work

• W > 0 if work is done on the system
• W < 0 if work is done by the system
• Q > 0 if heat is transferred to the system
• Q < 0 if heat is transferred from the system

Forms of mechanical work in closed systems

For many closed-system processes the dominant mechanical work is P dV (boundary work) where pressure acts on a deformable boundary. In these cases the work differential is

Common process constraints

• Isobaric - constant pressure
• Isochoric - constant volume
• Isothermal - constant temperature
• Adiabatic - no heat transfer

Isobaric process and enthalpy

For a process at constant pressure the First Law can be conveniently written using the property enthalpy H, defined by

H = U + PV

At constant pressure the energy balance becomes an enthalpy change equal to heat added. Integration between two equilibrium states yields:

Isochoric process

For a constant-volume process there is no boundary work. The First Law reduces to a change in internal energy equal to heat added:

Heat capacity and temperature dependence

Heat capacity quantifies the heat required to change a system's temperature. The most general differential definition is

Two important specific heat capacities for closed systems are:

• Constant-pressure heat capacityC_p defined by

• Constant-volume heat capacityC_v defined by

Using these definitions one can write the differential forms for enthalpy and internal energy

Specific heats generally depend on temperature and (for condensed phases) on pressure. For ideal gases specific heats are functions of temperature only (no intermolecular potential energy contribution). Experimentally determined values for gases are commonly expressed as polynomials in temperature:

In this expression A, B, C, D are substance-specific constants and R is the universal gas constant. Values for common gases are tabulated in standard references (see J. M. Smith, H. C. Van Ness and M. M. Abbott, Introduction to Chemical Engineering Thermodynamics, 6th ed., McGraw-Hill, 2001).

Applications to Ideal Gases

Relation between enthalpy and internal energy

For an ideal gas the equation of state is PV = RT (per mole) or pv = RT (per unit mass with appropriate gas constant). Using the definition of enthalpy

H = U + PV = U + RT

For ideal gases both U and H depend only on temperature. From the definitions of C_p and C_v it follows that

C_p - C_v = R (per mole) ... (3.11)

For liquids and solids the molar volume is small and H ≈ U, therefore C_p ≈ C_v.

Isothermal process for an ideal gas

For an isothermal process of an ideal gas the internal energy change is zero. From the First Law

dU = 0

Therefore

δQ = -δW

or

Q = -W

If only P-V work is present and the process is carried out reversibly, the work is

which integrates (for reversible isothermal change at temperature T) to

This gives the heat transfer equal in magnitude and opposite in sign to the work for isothermal, reversible processes.

Isobaric process

For isobaric process the heat transfer is related to change in enthalpy:

Adiabatic process for ideal gases

An adiabatic process has no heat transfer, so dQ = 0. For a reversible adiabatic process of an ideal gas the relation between pressure and volume is

Define the heat-capacity ratio (also called the adiabatic index)

γ = C_p / C_v

Integrating the adiabatic relation using γ yields the commonly used forms

Work for an adiabatic process can be related to temperature change. From the First Law for adiabatic reversible process

dW = dU = C_v dT

Using the integrated relations above and expressions for T in terms of P and V one can derive closed-form expressions for work in terms of initial and final states. Relevant intermediate relations are:

Polytropic processes (generalised single-form relation)

Many steady processes can be described by a polytropic relation of the form

PV^δ = constant ..... (3.22)

Other equivalent polytropic forms are

TV^δ-1 = constant ..... (3.23)

TP^(1-δ)/δ = constant ..... (3.24)

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Different values of the polytropic exponent δ reproduce special processes (for example, δ = 0 → isobaric, δ → ∞ → isochoric, δ = 1 → isothermal, δ = γ → adiabatic reversible for an ideal gas). General expressions for work and heat transfer for a polytropic process can be derived and written as:

and

Worked example - reversible isothermal compression of an ideal gas

Problem statement: A reversible, isothermal compression of an ideal gas at temperature T moves the state from volume V₁ to V₂. Calculate the work done on the gas.

Solution:

The process is isothermal so internal energy change is zero.

The reversible boundary work is the integral of P dV. Use the ideal-gas relation to substitute P = nRT/V (per mole) or P = RT/v (per unit mass).

Work done on the gas equals negative of the work done by the gas. The work done by the gas for a reversible isothermal process is

W(by) = ∫_V1^V2 P dV = ∫_V1^V2 (nRT / V) dV

Integrating the logarithmic expression gives

W(by) = nRT ln(V2 / V1)

Therefore the work done on the gas is

W(on) = -nRT ln(V2 / V1) = nRT ln(V1 / V2)

Since Q = -W(by), the heat transfer to the system equals nRT ln(V1 / V2).

Engineering applications and relevance

Understanding the First Law for closed systems and the special cases above is essential in many civil and mechanical engineering contexts. Typical applications include:

• Piston-cylinder devices such as internal combustion engines and compressors where boundary work and heat transfer determine performance and efficiency.
• Compressors and expanders (gas turbines, reciprocating compressors) modelled using adiabatic or polytropic relations to estimate work and temperature change.
• HVAC and building services calculations where isobaric and isochoric approximations are frequently used for sensible heating and cooling analyses.
• Energy balance in closed tanks and pressure vessels where internal energy and enthalpy changes determine heat addition and required isolation.
• Thermodynamic cycle analyses (Otto, Diesel, Brayton, Rankine) where component processes are approximated as isentropic, isochoric, isothermal or polytropic and linked by First-Law energy balances.

Notes on use of tabulated data and polynomial fits

When using tabulated specific heats for gases, the polynomial form

is commonly employed to integrate heat and enthalpy changes over temperature ranges. The reference by J. M. Smith, H. C. Van Ness and M. M. Abbott (2001) contains many such data and graphical representations used in engineering practice.

Summary

The First Law applied to closed systems equates changes in internal, kinetic and potential energies to heat and work interactions. By specialising the general balance to common constraints (constant pressure, constant volume, isothermal, adiabatic) and to ideal gases, closed-form relations for heat, work and state-variable changes are obtained. Heat capacities link temperature changes to energy changes and the polytropic relation provides a single mathematical framework to generate the standard process types encountered in engineering.