Q.Explain why

1. Key idea: Heat transfer depends on heat capacity, not just temperature difference. When two bodies are brought in contact, the final equilibrium temperature is determined by the principle of calorimetry: heat lost = heat gained. If the bodies have different masses and specific heats, the final temperature is a weighted average, not the arithmetic mean. For example, a small hot object and a large cold object will settle much closer to the cold object’s temperature.
2. Key idea: Specific heat measures how much heat a substance can store per degree temperature rise. A coolant with high specific heat absorbs a large amount of heat without a large rise in its own temperature. This keeps the plant parts cooler and reduces the required flow rate of coolant, improving efficiency and safety.
3. Key idea: Air heats up due to friction and deformation of the tyre during driving. As the tyre rolls, work done against internal friction and road friction raises the temperature of the air inside. For a fixed volume (approximately), the ideal gas law $P \propto T$ (at constant $V$) shows that an increase in temperature directly increases pressure.
4. Key idea: Water has a much higher specific heat capacity than sand/soil. The large water body near a harbour town absorbs and releases heat slowly, moderating temperature swings. In contrast, desert sand heats and cools rapidly, leading to extreme day-night and seasonal temperature variations. Thus, harbour towns have a more temperate (milder) climate.
   ✓Final answer

   1. Final temperature depends on heat capacities, not just initial temperatures.
   2. High specific heat allows more heat absorption per unit temperature rise.
   3. Tyre air temperature rises due to friction, increasing pressure at nearly constant volume.
   4. Water’s high specific heat moderates coastal temperatures compared to desert sand.

(a) Two bodies at different temperatures $T_1$ and $T_2$ if brought in thermal contact do not necessarily settle to the mean temperature $(T_1 + T_2)/2$.

Concept & Intuition

When two bodies at different temperatures are placed in thermal contact, heat flows from the hotter to the colder until they reach a common final temperature. But the final temperature is not simply the arithmetic mean of the initial temperatures. Why? Because the amount of heat each body can store per degree temperature change — its heat capacity — determines how much its temperature shifts. A body with a large heat capacity changes temperature very little for a given heat transfer, while one with a small heat capacity changes a lot.

Step-by-step reasoning

1. Heat lost = heat gained Let the two bodies have masses $m_1$, $m_2$ and specific heats $c_1$, $c_2$. Their heat capacities are $C_1 = m_1 c_1$ and $C_2 = m_2 c_2$. If the final equilibrium temperature is $T_f$, then:

assuming $T_1 > T_2$.

2. Solve for $T_f$ Rearranging:

This is a weighted average of $T_1$ and $T_2$, with weights equal to the heat capacities.

3. When does $T_f$ equal the mean? The mean $(T_1 + T_2)/2$ occurs only when $m_1 c_1 = m_2 c_2$ — i.e., when the heat capacities are equal. In general, the body with the larger heat capacity dominates the final temperature. For example, if one body is a huge block of iron and the other is a tiny drop of water, the final temperature will be very close to the iron’s temperature, not the average.

(b) The coolant in a chemical or a nuclear plant should have high specific heat.

Concept & Intuition

A coolant’s job is to absorb heat from hot parts (like reactor cores or chemical reactors) without itself rising too much in temperature. If the coolant’s temperature rises too fast, it can’t carry away enough heat, and the plant may overheat. The property that determines how much heat a substance can absorb per unit temperature rise is its specific heat capacity.

Step-by-step reasoning

1. Heat absorbed by coolant For a mass $m$ of coolant with specific heat $c$, the heat absorbed for a temperature rise $\Delta T$ is:

2. Why high $c$ helps

   For a given $Q$ (the heat that must be removed), a higher $c$ means a smaller $\Delta T$. So the coolant stays cooler, maintaining a larger temperature difference with the hot parts, which drives more efficient heat transfer. Also, a smaller $\Delta T$ reduces thermal stress on pipes and components.

3. Practical benefit

   A high-specific-heat coolant (like water, $c \approx 4200 \, \text{J/kg·K}$) can carry away large amounts of heat without needing a huge flow rate. This makes the cooling system more compact and safer.

(c) Air pressure in a car tyre increases during driving.

Concept & Intuition

When you drive, the tyres flex repeatedly as they roll. This flexing causes internal friction in the rubber and also friction between the tyre and the road. Friction generates heat, which warms the tyre and the air inside. According to the ideal gas law, if the volume of the tyre doesn’t change much, raising the temperature increases the pressure.

Step-by-step reasoning

1. Source of heat

   The tyre deforms as it rotates — the part in contact with the road flattens, then springs back. This repeated deformation dissipates mechanical energy as heat (hysteresis in the rubber). Also, friction with the road surface adds heat.

2. Effect on the air inside

   The tyre is nearly rigid (volume change is small). The air inside behaves approximately as an ideal gas. For a fixed volume $V$ and number of moles $n$, the ideal gas law is:

So pressure $P$ is directly proportional to absolute temperature $T$.

3. Temperature rise → pressure rise As the air heats up from the tyre’s heat, $T$ increases, so $P$ increases. This is why tyre pressure is higher after a long drive than when the car was cold.

(d) The climate of a harbour town is more temperate than that of a town in a desert at the same latitude.

Concept & Intuition

“Temperate” here means moderate — not too hot in summer, not too cold in winter. The key difference is the presence of a large body of water (the sea) near the harbour town. Water has a very high specific heat compared to land (sand, rock). This means water heats up and cools down much more slowly than land.

Step-by-step reasoning

1. Specific heat comparison

   Water’s specific heat is about $4200 \, \text{J/kg·K}$, while dry sand is about $800 \, \text{J/kg·K}$. So for the same amount of solar energy, land temperature rises about 5 times faster than water.

2. Day/night and seasonal effects

   During the day, the land heats up quickly, making the desert town very hot. The sea stays relatively cool. At night, the land cools quickly, while the sea releases stored heat slowly. This moderates the harbour town’s temperature — the sea acts as a thermal buffer.

3. Sea breezes

   The temperature difference between land and sea drives local winds (sea breezes during the day, land breezes at night), which further moderate the coastal climate by bringing cooler air from the sea in the afternoon.

4. Result

   The harbour town experiences smaller daily and seasonal temperature swings (a “temperate” climate), while the desert town has extreme highs and lows.

All four parts are really one idea wearing different clothes: heat capacity — how much heat a body must exchange to change its temperature by 1° — governs everything here. Part (a) states the general rule; (b)–(d) are special cases of it, since a coolant, the tyre air, and the sea are each just a body whose heat capacity (or, for the air, whose input heat) determines how much its temperature swings. A fast way to check each part: ask 'does raising this substance's heat capacity make its temperature more or less sensitive to a given heat flow?' — more capacity always means less sensitivity, which immediately gives (b) and (d), while (c) is the one part driven by a source of heat (friction) rather than a capacity argument.