First Law of Thermodynamics Formula: ΔU = Q − W, Examples

The first law of thermodynamics formula is $\Delta U = Q - W$ : the change in a system's internal energy equals the heat Q added to it minus the work W it does on its surroundings. It is the law of conservation of energy applied to heat engines, gases and anything that can be warmed or squeezed.

Indian textbooks often write the same law as $\Delta Q = \Delta U + \Delta W$ , which is simply a rearrangement. Some courses, especially in chemistry, count W as work done on the gas and write $\Delta U = Q + W$ . The physics is identical; only the sign of W changes. Always check which convention your course uses before you substitute.

Quick facts

Physics convention
$\Delta U = Q - W$ , W done by the gas
Rearranged
$Q = \Delta U + W$
Other convention
$\Delta U = Q + W_{\text{on}}$
Work by a gas
$W = P\Delta V$ at constant pressure
Unit
joule (J) for every term
Isothermal ideal gas
ΔU = 0, so Q = W

Signs in the first law of thermodynamics formula

Sign rules with ΔU = Q − W
Quantity	Positive when	Negative when
$Q$	Heat flows into the system	Heat flows out
$W$	The gas expands and pushes on its surroundings	The gas is compressed
$\Delta U$	Internal energy rises (an ideal gas gets hotter)	Internal energy falls

For an ideal gas, internal energy depends only on temperature: $\Delta U = nC_V\Delta T$ . So any process that returns the gas to its starting temperature has ΔU = 0, whatever path it took.

Worked examples

Example 1: heating a gas that expands

A gas absorbs 500 J of heat and does 200 J of work on a piston. Find ΔU.

$\Delta U = 500 - 200 = 300\ \text{J}$

Internal energy rises by 300 J, so the gas gets warmer.

Example 2: compressing a gas

A gas is compressed by doing 150 J of work on it, while it loses 40 J of heat to the surroundings. Find ΔU.

Heat leaves, so Q = −40 J. Work is done on the gas, so W (by the gas) = −150 J.

$\Delta U = -40 - (-150) = 110\ \text{J}$

Example 3: expansion at constant pressure, volume in litres

A gas at a constant 1.0 × 10⁵ Pa expands from 2.0 L to 5.0 L while taking in 750 J of heat. Find ΔU.

ΔV = 3.0 L = 3.0 × 10⁻³ m³, so W = PΔV = (1.0 × 10⁵)(3.0 × 10⁻³) = 300 J.

$\Delta U = 750 - 300 = 450\ \text{J}$

Special cases you must know

The first law in standard processes
Process	Condition	First law becomes
Isothermal (ideal gas)	T constant, ΔU = 0	$Q = W$
Adiabatic	No heat exchange, Q = 0	$\Delta U = -W$
Isochoric	V constant, W = 0	$\Delta U = Q$
Isobaric	P constant	$Q = \Delta U + P\Delta V$
Cyclic	Returns to start, ΔU = 0	$Q_{net} = W_{net}$

In an adiabatic expansion the gas does work using its own internal energy, so it cools. That is why air rushing out of a tyre valve feels cold. In an isothermal process all the heat taken in is turned into work.

Work done by a gas and the PV diagram

When a gas expands against pressure P through a small volume change dV, it does work $dW = P\,dV$ . The total work is the area under the curve on a pressure against volume graph. For constant pressure this area is a rectangle, $W = P\Delta V$ . For an isothermal ideal gas, $W = nRT\ln\frac{V_2}{V_1}$ . The work formula page covers mechanical work in general.

Q and W depend on the path between two states. ΔU depends only on the start and end states. Exam questions test this distinction often.

How it links to other heat formulas

The heat term often comes from $Q = mc\Delta T$ or from latent heat, both on the formula of heat page. Heat can reach the system by conduction, described by the thermal conductivity relation. The first law sets the energy limit for any engine; the second law then says how much of the heat can actually become work, giving the Carnot efficiency $\eta = 1 - \frac{T_2}{T_1}$ . Units throughout are joules, the unit on the energy formula page.

Errors that cost marks

Mixing the two sign conventions in one answer.
Forgetting to make Q negative when heat is released.
Leaving volume in litres: 1 L = 10⁻³ m³.
Assuming ΔU = 0 for every process; it is zero only when the temperature of an ideal gas returns to its start.

The thermodynamics formula sheet gathers this law with the gas laws, heat transfer and engine efficiency.

Frequently asked questions

What is the first law of thermodynamics formula?

ΔU = Q − W, where Q is heat added to the system and W is work done by the system. NCERT writes it as ΔQ = ΔU + ΔW.

Why do some books write ΔU = Q + W?

They define W as work done on the system rather than by it. Switching the definition flips the sign of W, but the law is the same.

What does the first law say for an isothermal process?

For an ideal gas the temperature is fixed, so ΔU = 0 and all the heat absorbed equals the work done by the gas.

What is the first law for an adiabatic process?

No heat crosses the boundary, so Q = 0 and ΔU = −W. An expanding gas cools, a compressed gas warms.

Is the first law of thermodynamics the same as conservation of energy?

Yes. It states that energy is neither created nor destroyed, only transferred as heat or work or stored as internal energy.