Calorimetry And Mixtures

Calorimetry rests on conservation of energy in an insulated system: the heat lost by the hotter body equals the heat gained by the cooler body, with no exchange to the surroundings. Let the equilibrium temperature be $T$. Heat lost by iron is $m_i{c}_i(150-T)$ and heat gained by water is $m_w{c}_w(T-20)$. Setting them equal, $200\times 0.45\times (150-T)=400\times 4.2\times (T-20)$, which gives $90(150-T)=1680(T-20)$. Expanding, $13500-90T=1680T-33600$, so $47100=1770T$ and $T\approx 26.6^{\circ }\text{C}$, closest to $26^{\circ }\text{C}$. The value $32^{\circ }\text{C}$ would follow from underestimating the water's large heat capacity. The values $45^{\circ }\text{C}$ and $58^{\circ }\text{C}$ wrongly let the small iron mass dominate the mixture. This is the standard NCERT method of mixtures using the principle of calorimetry, valid here because the container is insulated and no heat leaks to the surroundings or is absorbed by the vessel walls. The product of mass and specific heat, called the thermal capacity, decides how strongly each body resists temperature change, and the water's capacity here vastly exceeds the iron's. As a plausibility check, water has a far higher specific heat and greater mass, so the final temperature should sit very close to the water's initial $20^{\circ }\text{C}$, exactly as the modest rise to about $26^{\circ }\text{C}$ confirms.