The First Law of Thermodynamics and Ideal Processes

The first law of thermodynamics states that the change in the internal energy of a system equals the heat added to it minus the work done by it, ΔU = Q − W; it is the law of conservation of energy applied to thermal systems. In brief: for an ideal monatomic gas, U = (3/2)nRT, so ΔU = nC_vΔT with C_v = (3/2)R; the molar heat capacity at constant pressure is C_p = C_v + R. In an isochoric process, W = 0 and ΔU = Q; in an isothermal process, ΔU = 0 and Q = W; in an adiabatic process, Q = 0 and ΔU = −W; in an isobaric process, Q = nC_pΔT and W = pΔV = nRΔT.

Heating a gas can raise its temperature, make it do work, or both, and the first law keeps account. This guide states the law, its sign convention and the meaning of internal energy, applies it to the four ideal processes with heat capacities, compares the processes in a table, works through numerical examples and corrects common mistakes.

What does the first law say?

The internal energy U of a system is the total kinetic and potential energy of its molecules. It is a state function: it depends only on the current state (for an ideal gas, only on temperature), not on how it got there. Heat Q and work W are not state functions: they depend on the process. The first law:

ΔU = Q − W,

where Q is the heat added to the system (positive if it flows in) and W the work done by the system on its surroundings (positive if the system expands). (In the chemistry convention, W is the work done on the system and ΔU = Q + W.) For a complete cycle, ΔU = 0, so Q_net = W_net. The law shows that a perpetual motion machine of the first kind, one that produces work from nothing, is impossible.

What are the heat capacities of a gas?

At constant volume (no work): Q = nC_vΔT = ΔU, so for the ideal gas ΔU = nC_vΔT in every process. For a monatomic gas, C_v = (3/2)R = 12.5 J/(mol·K); for a diatomic gas at room temperature, (5/2)R = 20.8 J/(mol·K) (rotation adds two degrees of freedom).
At constant pressure: Q = nC_pΔT, with C_p = C_v + R (Mayer’s relation), since part of the heat goes into the work of expansion. Monatomic: (5/2)R; diatomic: (7/2)R.
The ratio γ = C_p ÷ C_v is 5/3 for monatomic and 7/5 for diatomic gases. It determines the adiabatic curve, pV^γ = constant, or TV^γ−1 = constant.

How does the law apply to ideal processes?

Isochoric (V constant): W = 0, so Q = ΔU. All the heat raises the temperature.
Isobaric (p constant): W = pΔV = nRΔT, Q = nC_pΔT, ΔU = nC_vΔT. For a monatomic gas, 60 percent of the heat goes into internal energy, and 40 percent into work.
Isothermal (T constant): ΔU = 0, so Q = W. The heat absorbed is converted entirely into work.
Adiabatic (no heat flow): Q = 0, so ΔU = −W. An expanding gas cools; a compressed gas heats up (the compression stroke in a diesel engine heats the air enough to ignite the fuel; a bicycle pump becomes hot).
Free expansion of a gas into a vacuum: Q = 0 and W = 0, so ΔU = 0 and (for an ideal gas) the temperature does not change.

Processes compared

First law for an ideal gas in basic processes
Process	Q	W	ΔU
Isochoric	nC_vΔT	0	= Q
Isobaric	nC_pΔT	pΔV = nRΔT	nC_vΔT
Isothermal	= W	nRT ln(V₂/V₁)	0
Adiabatic	0	−ΔU = nC_v(T₁ − T₂)	nC_vΔT

Worked example: heating a monatomic gas

1.0 mol of a monatomic ideal gas (C_v = 12.47, C_p = 20.79 J/(mol·K)) is heated from 300 K to 400 K.

At constant volume: Q = nC_vΔT = 1.0 × 12.47 × 100 = 1,247 J. W = 0. ΔU = 1,247 J.
At constant pressure: Q = nC_pΔT = 20.79 × 100 = 2,079 J. W = nRΔT = 8.314 × 100 = 831 J. ΔU = Q − W = 2,079 − 831 = 1,248 J, the same as in item 1, since ΔU depends only on ΔT.
Isothermal at 300 K with Q = 500 J absorbed: ΔU = 0, so W = 500 J. The volume increases by a factor of e^W/nRT = e^500/2494 = e^0.2005 = 1.22.
Adiabatic compression. The same gas at 300 K has its volume halved adiabatically (γ = 5/3). T₂ = T₁(V₁/V₂)^γ−1 = 300 × 2^2/3 = 300 × 1.587 = 476 K. The work done on the gas: −W = ΔU = nC_vΔT = 12.47 × 176 = 2,197 J, so W = −2,197 J (the gas gains the energy).
Check of the ratio: in the isobaric process, the fraction of heat converted to work is W ÷ Q = R ÷ C_p = 8.314 ÷ 20.79 = 0.40.

Heating 1.0 mol of a monatomic ideal gas from 300 K to 400 K, in joules. At constant volume all 1,247 J of heat raises the internal energy. At constant pressure the internal energy rises by the same 1,248 J (it depends only on the temperature change), but 831 J of extra heat leaves as work, so 2,079 J must be supplied.

How was the law formulated?

Julius Robert von Mayer proposed the conservation of energy in 1842 and calculated the mechanical equivalent of heat from the difference between the heat capacities of gases. James Joule’s paddle wheel experiments (1843 to 1850) measured it precisely: 4.18 J per calorie. Hermann von Helmholtz (1847) and Rudolf Clausius (1850) stated the first law in its general form. Joule’s free expansion experiment tested whether the internal energy of a gas depends on volume.

Which mistakes are common?

Mistake 1: Mixing up the sign convention for work

With ΔU = Q − W, W is the work done by the gas. With ΔU = Q + W, it is the work done on the gas.

Mistake 2: Using C_v for the heat in an isobaric process

Q = nC_pΔT for constant pressure. ΔU = nC_vΔT for any process.

Mistake 3: Thinking that no heat flow means no temperature change

In an adiabatic process the temperature changes because of the work.

Mistake 4: Treating heat and work as stored in the system

Heat and work are energy transfers. The system stores internal energy.