In heat transfer experiments using the method of mixtures, the relation heat lost equals heat gained is based on which fundamental principle?

Introduction / Context:
When studying heat transfer, a common laboratory technique is the method of mixtures, where a hot body and a cold body are brought into contact inside an insulated vessel. The analysis of such experiments relies on a simple but powerful idea that the heat lost by the hotter body is equal to the heat gained by the colder body, if no heat is lost to the surroundings. This question asks you to identify the named principle on which that relation is based.

Given Data / Assumptions:
- Two or more bodies with different initial temperatures are mixed or placed in contact.- The system is assumed to be thermally insulated from the surroundings.- Heat flows until all bodies reach a common final temperature.- The relation heat lost equals heat gained is valid under these conditions.

Concept / Approach:
The method of mixtures is a central topic in calorimetry, the branch of physics that deals with measurement of heat. The fundamental idea is energy conservation: in an isolated system, total energy remains constant. In thermal processes inside an insulated calorimeter, this means that energy given up by hot objects as they cool must equal the energy absorbed by colder objects as they warm. This is expressed as heat lost by hot bodies equals heat gained by cold bodies. The named principle that encapsulates this relation in the context of measurement is called the principle of calorimetry.

Step-by-Step Solution:
1. Recognise that the phrase heat lost equals heat gained is associated with mixing hot and cold substances in a calorimeter.2. Recall that calorimetry experiments aim to determine specific heat, latent heat, or final equilibrium temperature using this relation.3. Note that the underlying idea comes from conservation of energy, applied to thermal energy in an isolated system.4. Identify that this specific application and formal statement is known as the principle of calorimetry.5. Distinguish it from general thermal equilibrium, which describes equal temperatures but not explicit heat balance equations.6. Therefore, select principle of calorimetry as the correct option.

Verification / Alternative check:
In school laboratory manuals and physics textbooks, sections on the method of mixtures often begin with a statement such as according to the principle of calorimetry, in an isolated system, heat lost by hot bodies is equal to heat gained by cold bodies. This naming is consistent across many sources and is explicitly linked to the design of calorimeters and the analysis of mixing experiments. That repeated phrasing confirms the correctness of the answer.

Why Other Options Are Wrong:
- Principle of thermal equilibrium: This states that bodies in thermal contact reach the same temperature, but does not by itself give the quantitative heat balance relation.- Principle of colors: This is unrelated to heat and refers instead to optics and colour mixing.- Principle of vaporization: This concerns phase change from liquid to vapour and latent heat, not the general heat balance in mixtures.

Common Pitfalls:
Some learners may choose principle of thermal equilibrium because it sounds related to bodies reaching the same temperature. However, the question specifically refers to the equality of heat lost and heat gained, which is the defining statement of the principle of calorimetry. Remembering this exact phrase will help you quickly identify the correct principle in similar exam questions.

Final Answer:
The relation heat lost equals heat gained is based on the principle of calorimetry.