Question

Two bodies of different materials and same surface area, coated by similar material, are heated to same temperature. They are allowed to cool down in same surroundings. If \(\frac{d Q}{d t}\) represents rate of loss of heat and \(\frac{\mathrm{d} \theta}{\mathrm{dt}}\) represents rate of fall of temperature, then for both : (1) \(\frac{\mathrm{dQ}}{\mathrm{dt}}\) is same but \(\frac{\mathrm{d} \theta}{\mathrm{dt}}\) is different (2) \(\frac{\mathrm{dQ}}{\mathrm{dt}}\) is different but \(\frac{\mathrm{d} \theta}{\mathrm{dt}}\) is same (3) \(\frac{\mathrm{d} \theta}{\mathrm{dt}}\) as well as \(\frac{\mathrm{dQ}}{\mathrm{dt}}\) is same (4) \(\frac{\mathrm{d} \theta}{\mathrm{dt}}\) as well as \(\frac{\mathrm{dQ}}{\mathrm{dt}}\) is different

Short answer

(1) \(\frac{dQ}{dt}\) is the same but \(\frac{d\theta}{dt}\) is different.

Step-by-step solution

Understand the given conditions

Two bodies made of different materials have the same surface area and are heated to the same temperature. They are cooling in the same environment and are coated with the same material.

Identify the important parameters

The rate of loss of heat is given as \(\frac{d Q}{d t}\), and the rate of fall in temperature is given as \(\frac{\text{d} \theta}{\text{dt}}\).

Analyze heat loss (Newton's law of cooling)

According to Newton's law of cooling, the rate of heat loss \(\frac{d Q}{d t}\) is proportional to the temperature difference between the body's surface and its surroundings. Since both bodies have the same temperature and surface area, the rate of heat loss by radiation (which depends on surface properties) would be the same.

Analyze temperature fall

The temperature fall of an object also depends on its material, specifically its specific heat capacity. Since the materials are different, their specific heat capacities are different, leading to different rates of fall in temperature \(\frac{\text{d} \theta}{\text{dt}}\).

Conclusion

From the analysis, we conclude that \(\frac{\text{dQ}}{\text{dt}}\) is the same for both bodies, but \(\frac{\text{d} \theta}{\text{dt}}\) is different due to their different specific heat capacities.

Key concepts

rate of heat loss

Rate of heat loss refers to how quickly heat energy leaves an object. According to Newton's law of cooling, this rate is proportional to the difference in temperature between the object and its surroundings. Mathematically, it can be expressed as \(\frac{d Q}{d t} \propto (\theta - \theta_{\text{surroundings}})\). This relationship tells us that a larger temperature difference results in faster heat loss.
In the context of the exercise, both bodies are cooling in identical environments and have the same surface area. Additionally, both are coated with the same material. Given these conditions, the rate of heat loss, \(\frac{d Q}{d t}\), will be the same for both bodies.
This is because the rate of heat loss by radiation depends on surface properties, temperature difference, and surface area. Since these parameters are identical for both objects, they will lose heat at the same rate.

specific heat capacity

Specific heat capacity is a measure of how much heat energy is needed to change the temperature of a substance. It's represented by the symbol \((c)\) and is defined as the amount of heat energy required to raise the temperature of one kilogram of the material by one degree Celsius. Mathematically, it's expressed as \[ Q = mc\theta \], where:

• \(Q\) is the heat energy
• \(m\) is the mass
• \(c\) is the specific heat capacity
• \(\theta\) is the temperature change

In our exercise, the bodies are made of different materials. This means their specific heat capacities are not the same. The specific heat capacity influences how quickly each material's temperature changes as it loses heat. Materials with a higher specific heat capacity will experience a slower temperature change compared to those with a lower specific heat capacity when they lose the same amount of heat. This difference in specific heat capacity directly affects the rate of temperature fall for the two bodies, which is why \(\frac{\text{d} \theta}{\text{dt}} \) is different for them.

temperature change

Temperature change in a cooling process evaluates how the internal energy of a substance decreases over time as it loses heat. In our problem, it's denoted by the rate of fall in temperature, \(\frac{\text{d} \theta}{\text{dt}}\). This rate is determined by the material's specific heat capacity and the rate of heat loss.
Even though the bodies have the same initial temperature, surface area, and surroundings, the variation in specific heat capacities will lead to different cooling behaviors. Mathematically, the temperature change for a body over time can be described by \(\frac{d \theta}{d t} = -\frac{Q}{mc} \). This equation shows that for the same amount of lost heat (Q), the temperature change \(\frac{d \theta}{d t}\) depends inversely on the specific heat capacity \((c)\). Consequently, the body with a lower specific heat capacity will cool down faster than the one with a higher specific heat capacity, leading to different rates of fall in temperature for the two bodies.