A \(10 \mathrm{~g}\) piece of iron \(\left(C=0.45 \mathrm{~J} / 8^{\circ} \mathrm{C}\right)\) at \(100^{\circ} \mathrm{C}\) is dropped into \(25 \mathrm{~g}\) of water \(\left(C=4.2 \mathrm{~J} / \mathrm{g}^{\circ} \mathrm{C}\right)\) at \(27^{\circ} \mathrm{C}\). Find temperature of the iron and water system at thermal equilibrium. (a) \(30^{\circ} \mathrm{C}\) (b) \(33^{\circ} \mathrm{C}\) (c) \(40^{\circ} \mathrm{C}\) (d) None of these

Specific heat capacity is a measure of the amount of heat energy required to raise the temperature of a unit mass of a substance by one degree Celsius (or one Kelvin). Expressed in the units of joules per gram per degree Celsius \(\frac{J}{g^\circ C}\), it represents the ability of a material to absorb heat.

Imagine you're warming up different materials, like iron and water, with a heater. You'll notice that water takes longer to heat up compared to iron, assuming you provide the same amount of heat to both. This is because water has a higher specific heat capacity than iron, which means it can absorb more heat before its temperature rises.

Let's consider an example where you have a piece of iron with a specific heat capacity of \(0.45\frac{J}{g^\circ C}\) and a mass of \(10g\). If we want to raise the temperature of this iron piece from \(20^\circ C\) to \(21^\circ C\), it would require \(10g \times 0.45\frac{J}{g^\circ C}\times (21^\circ C-20^\circ C) = 4.5J\) of heat energy.

Thermal equilibrium is a concept in thermodynamics where two objects at different temperatures reach a common temperature after being in contact for a certain period. This occurs due to the transfer of heat from the hotter object to the cooler one.

For instance, when you pour a hot beverage into a cold cup, the cup absorbs heat from the beverage. Gradually, the temperature of the beverage drops, while the temperature of the cup rises until both reach the same temperature – that's thermal equilibrium. According to the zeroth law of thermodynamics, if two systems are in thermal equilibrium with a third system, then they are in thermal equilibrium with each other.

In our given problem, we see this phenomenon taking place between iron at \(100^\circ C\) and water at \(27^\circ C\). The two will exchange heat until they stabilize at a single, intermediate temperature, at which point no more heat is transferred between them.

JEE (Joint Entrance Examination) is a prestigious entrance exam in India for students who wish to pursue engineering. The exam's chemistry section often includes problems on thermodynamics and heat transfer, which test the candidate’s understanding of concepts such as specific heat capacity and thermal equilibrium.

Problems like the one in our example are typical for JEE Physical Chemistry, where students need to apply their knowledge and problem-solving skills to determine the final equilibrium temperature of a system. The challenge lies in correctly identifying the formulas required, setting up the relevant equations, and manipulating them to solve for the desired variable.

Tip for JEE Aspirants

Paying close attention to the units and carefully managing the algebra will help to avoid simple mistakes. Practice with a variety of JEE problems can greatly enhance proficiency in solving such thermodynamics questions efficiently.