Although heat is a path function but heat absorbed by the system under certain specific conditions is independ

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Q. Although heat is a path function but heat absorbed by the system under certain specific conditions is independent of path. What are those conditions? Explain.

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Concept understanding — State Function

What is a State Function? The Intuition

Imagine you're standing at the base of a hill. Your height above sea level is a number — say, 100 metres. Now, you walk to the top of the hill. Your height is now 500 metres.

Here's the key question: Does it matter how you got to the top? Did you take the steep path, the winding road, or did you get carried up by a helicopter?

The answer is no. Your height at the top is 500 metres, regardless of the path you took. Height is a state function — it depends only on where you are, not on how you got there.

Now contrast that with distance walked. If you took the winding road, you walked 2 km. If you took the steep path, you walked 500 m. The distance walked depends entirely on the path. That's not a state function — it's a path function.

Note

A state function is a property whose value depends only on the current state of the system, not on the history or path taken to reach that state.

The Precise Statement

In thermodynamics and physics, a state function (or state variable) is any property of a system that is determined entirely by the system's current equilibrium conditions — typically its temperature, pressure, volume, composition, and so on.

If you know the state of the system (say, "1 mole of ideal gas at 300 K and 1 atm"), then every state function has a fixed value. You don't need to know whether the gas was heated slowly, compressed quickly, or cooled then expanded. The value is the same.

Common State Functions in Chemistry & Physics

Property	Symbol	Why it's a state function
Pressure	$P$	Depends only on current $T$ , $V$ , $n$
Volume	$V$	Depends only on current $P$ , $T$ , $n$
Temperature	$T$	A fundamental state variable
Internal Energy	$U$	Depends only on current $P$ , $T$ , composition
Enthalpy	$H$	$H = U + PV$ — a combination of state functions
Entropy	$S$	Depends only on current state
Gibbs Free Energy	$G$	$G = H - TS$ — again, a combination

Common Path Functions (the opposite)

Property	Why it's a path function
Work ( $W$ )	Depends on how you change volume (e.g., fast vs slow)
Heat ( $Q$ )	Depends on how you transfer energy (e.g., conduction vs radiation)

The Mathematical Signature

Here's the crisp, exam-ready way to recognise a state function:

Important

For a state function $f$ , the cyclic integral is zero:

$\oint df = 0$

This means: if you go from state A to state B and back to A by any path, the net change in $f$ is zero. The value of $f$ at A is always the same when you return.

Equivalently, the change in a state function between two states is path-independent:

\Delta f = f_{\text{final}} - f_{\text{initial}}

This is a single number — no integral over a path needed.

A Concrete Example: Internal Energy

Consider a gas in a cylinder. You take it from State 1 ( $T_1 = 300$ K, $P_1 = 1$ atm) to State 2 ( $T_2 = 400$ K, $P_2 = 2$ atm).

Path A: Heat the gas at constant volume, then compress it at constant temperature.
Path B: Compress the gas at constant temperature, then heat it at constant volume.

The work done ( $W$ ) and heat transferred ( $Q$ ) will be different for Path A vs Path B. But the change in internal energy $\Delta U$ will be identical for both paths. That's because $U$ is a state function — it only cares about the starting and ending states. …

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Heat becomes a path-independent quantity when the process is carried out at constant volume (where $q_V = \Delta U$ ) or at constant pressure (where $q_P = \Delta H$ ). Under these conditions, the heat absorbed equals the change in a state function — internal energy or enthalpy — and therefore loses its path dependence.

Heat is a path function because the amount of energy transferred as heat depends on how you go from the initial state to the final state — whether you do it slowly, quickly, in one step, or in many steps. But there is a clever way out: if you constrain the process so that a particular variable (volume or pressure) stays fixed, then the heat absorbed becomes equal to the change in a state function. And a state function depends only on the initial and final states, not on the path.

Let’s see exactly how this works.

Constant volume: $q_V = \Delta U$ From the first law of thermodynamics:

\Delta U = q + W

If the volume is constant, no pressure–volume work is done: $W = -P\Delta V = 0$ . So the first law reduces to:

\Delta U = q_V

Here $q_V$ is the heat absorbed at constant volume. Since $\Delta U$ is a state function (it depends only on the initial and final states), $q_V$ must also be path-independent — it always equals the change in internal energy, no matter how the process is carried out, as long as volume stays constant.

Constant pressure: $q_P = \Delta H$ At constant pressure, the work done is $W = -P\Delta V$ . Substituting into the first law:

\Delta U = q_P - P\Delta V

Rearranging:

q_P = \Delta U + P\Delta V

The right-hand side is exactly the definition of the change in enthalpy: $\Delta H = \Delta U + \Delta(PV)$ . At constant pressure, $\Delta(PV) = P\Delta V$ , so:

q_P = \Delta H

Enthalpy $H$ is a state function, so $q_P$ is path-independent under constant pressure conditions.

Watch out

A common mistake is to think that $q = \Delta H$ always. That is only true when the pressure is constant and only $P$ – $V$ work is done. If the pressure changes during the process, $q$ is not equal to $\Delta H$ , and heat remains path-dependent. …

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Q.Although heat is a path function but heat absorbed by the system under certain specific conditions is independent of path. What are those conditions? Explain.

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