Increase in enthalpy of the surroundings is equal to decrease in enthalpy of the system. Will the temperature

NCERT Exemplar · Q29

Q. Increase in enthalpy of the surroundings is equal to decrease in enthalpy of the system. Will the temperature of system and surroundings be the same when they are in thermal equilibrium?

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Concept understanding — First Law of Thermodynamics

First Law of Thermodynamics

The Intuition: Energy is a Bank Account

Imagine you have a bank account. You can deposit money into it, or withdraw money from it. The total amount of money in your account changes only by the net of what goes in and what comes out. You cannot create money from nothing, nor can you destroy it — it just moves.

Energy works the same way. In any physical or chemical process, energy is never created or destroyed. It is only transferred from one place to another, or converted from one form to another. This is the First Law of Thermodynamics — the law of conservation of energy, applied to systems where heat and work are the currencies.

When you heat a gas in a piston, the gas expands and pushes the piston up. The energy you put in as heat doesn't vanish — part of it stays inside the gas (raising its temperature), and part of it leaves as work done on the piston. The total energy of the universe remains constant.

The Precise Statement

\Delta U = Q - W

Where:

$\Delta U$ = change in the internal energy of the system (the energy stored inside — kinetic energy of molecules, potential energy in bonds, etc.)
$Q$ = heat added to the system (positive if heat flows into the system)
$W$ = work done by the system on the surroundings (positive if the system expands and pushes against something)

The sign convention is crucial. Many textbooks use $Q + W$ with work done on the system. The version above ( $Q - W$ ) is the most common in Indian exam syllabi (CBSE, JEE, NEET). Stick to one convention and be consistent.

Watch out

Common Mistake

Students often forget the sign of work. If a gas expands, it does positive work on the surroundings — so $W$ is positive, and $\Delta U = Q - W$ becomes smaller. If a gas is compressed, work is done on the gas — so $W$ is negative, and $\Delta U = Q - (-|W|) = Q + |W|$ , which increases internal energy.

What Each Term Means Physically

Internal energy ( $U$ ) is the total microscopic energy of the system. For an ideal gas, it depends only on temperature — higher temperature means higher $U$ . For real substances, it also depends on volume and phase.

Heat ( $Q$ ) is energy transferred due to a temperature difference. If you put a hot pan on a cold stove, heat flows from pan to stove. In thermodynamics, we always ask: who is the system? If the system is the gas, then $Q$ is positive when heat flows into the gas.

Work ( $W$ ) in thermodynamics is usually pressure-volume work: $W = \int P \, dV$ . When a gas expands against a piston, it does work on the piston. When you compress a gas, you do work on it.

A Simple Example

Take a cylinder with a movable piston, containing 1 mole of an ideal gas. You supply 500 J of heat to the gas. The gas expands and does 200 J of work on the piston.

$Q = +500$ J (heat enters the system)
$W = +200$ J (work done by the system)

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

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When system and surroundings exchange only heat at constant pressure until equilibrium, they reach the same temperature. Equal and opposite enthalpy changes guarantee energy conservation but do not by themselves define thermal equilibrium; temperature equality does.

The question probes the relationship between enthalpy transfer and thermal equilibrium. At first glance, the statement "increase in enthalpy of surroundings equals decrease in enthalpy of system" might seem to automatically imply equal temperatures, but the two ideas are distinct and worth separating.

Why enthalpy changes are equal and opposite

Enthalpy $H$ is a state function defined as $H = U + PV$ . At constant pressure, the heat exchanged is exactly the change in enthalpy:

q_p = \Delta H

When a system and its surroundings are isolated from the rest of the universe, energy conservation demands

\Delta H_{\text{system}} + \Delta H_{\text{surroundings}} = 0,

or equivalently

\Delta H_{\text{surroundings}} = -\Delta H_{\text{system}}.

This equality holds during any process in which only heat flows between them at constant pressure, regardless of whether they have reached thermal equilibrium. It is a bookkeeping statement: energy lost by one is gained by the other.

What thermal equilibrium actually means

Thermal equilibrium is the condition in which no net heat flows between system and surroundings. The zeroth law of thermodynamics tells us this happens if and only if both have the same temperature:

T_{\text{system}} = T_{\text{surroundings}}.

Temperature is the intensive property that governs the direction of heat flow. Heat spontaneously moves from higher temperature to lower until temperatures equalize.

Connecting the two ideas

During the approach to equilibrium: Suppose the system starts hotter than the surroundings. Heat flows out of the system (its enthalpy decreases) and into the surroundings (whose enthalpy increases) in equal magnitude. The system cools, the surroundings warm. …

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Q.Increase in enthalpy of the surroundings is equal to decrease in enthalpy of the system. Will the temperature of system and surroundings be the same when they are in thermal equilibrium?

Why enthalpy changes are equal and opposite

What thermal equilibrium actually means

Connecting the two ideas

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