Problem 7 - Master Resource Book in JEE Main Physics

Question

If the ratio of densities of two substances is \(5: 6\) and that of the specific heats is \(3: 5\). Then, the ratio between heat capacities per unit volume is (a) \(1: 1\) (b) \(2: 1\) (c) \(1: 2\) (d) \(1: 3\)

Step-by-Step Solution

Answer

(c) 1:2

Step 1: Understanding Heat Capacity Per Unit Volume

Heat capacity per unit volume, denoted as \( C_v \), is the product of density \( \rho \) and specific heat \( c \). So for substance 1, it is \( C_{v1} = \rho_1 \cdot c_1 \), and for substance 2, it is \( C_{v2} = \rho_2 \cdot c_2 \).

Step 2: Expressing Given Ratios

We are given that the ratio of densities \( \frac{\rho_1}{\rho_2} = \frac{5}{6} \) and the ratio of specific heats \( \frac{c_1}{c_2} = \frac{3}{5} \).

Step 3: Calculating Ratio of Heat Capacities Per Unit Volume

To find the ratio of heat capacities per unit volume, we combine the ratios: \[ \frac{C_{v1}}{C_{v2}} = \frac{\rho_1 \cdot c_1}{\rho_2 \cdot c_2} = \frac{5}{6} \cdot \frac{3}{5} = \frac{3}{6} = \frac{1}{2} \].

Step 4: Conclusion on the Ratio

The ratio between the heat capacities per unit volume for the two substances is \( 1:2 \), which matches option (c).

Key Concepts

Density ratio

When comparing two substances, understanding the density ratio is key to determining how their volumes relate for the same mass. Density, denoted by \( \rho \), is mass per unit volume, indicating how tightly matter is packed within a substance.
Let's consider an example: the ratio of the densities of two substances is given as \( 5:6 \). This means that for every 5 units of something (let's say kilograms) of substance 1, you have 6 units of substance 2 when they cover the same volume.

• Higher density substance implies more mass or matter in the same amount of space.
• The given ratio \( \rho_1 : \rho_2 = \frac{5}{6} \) suggests substance 2 is less dense compared to substance 1.

Understanding density ratios helps in various calculations, including specific heat and overall heat capacity comparisons. Knowing this ratio is essential for further calculating thermal properties, like heat capacity per unit volume.

Specific heat ratio

Specific heat is all about the amount of heat energy required to raise the temperature of a unit mass of a substance by 1 degree Celsius. It's represented by \( c \). The specific heat ratio between two substances tells us how much more energy per unit mass is needed to change the temperature of each substance.
Given: The specific heat ratio is \( 3:5 \). This means that substance 1 requires 3 units of energy to accomplish what substance 2 can achieve with 5 units of energy.

• Specific heat affects how quickly a substance can change its temperature.
• A higher specific heat indicates that a substance can absorb more heat without much change in temperature.
• The ratio \( c_1 : c_2 = \frac{3}{5} \) implies substance 2 can retain heat more effectively.

These differences play out in thermal applications, where energy efficiency and temperature control are crucial. Understanding specific heat ratios becomes critical when deciding materials for building, manufacturing, and other thermal dynamics applications.

Heat capacity per unit volume

Combining density and specific heat, we can determine the heat capacity per unit volume of a substance, symbolized as \( C_v \). It represents how much heat a specific volume of the substance can retain, which is a vital part of understanding thermal energy distribution. The formula is \( C_v = \rho \cdot c \), meaning it's the product of density and specific heat. To find the ratio of heat capacities per unit volume between two substances, you multiply their respective density and specific heat ratios:

• \( C_{v1} : C_{v2} = (\rho_1 \cdot c_1) : (\rho_2 \cdot c_2) \)
• Given the ratios \( \rho_1 : \rho_2 = 5:6 \) and \( c_1 : c_2 = 3:5 \), their combined ratio becomes \( \frac{1}{2} \).

This results in heat capacities per unit volume having a ratio of \( 1:2 \), elucidating that substance 2 can store twice as much heat in the same volume as substance 1. This concept is pivotal for material science, engineering, and physics, providing insight into how materials will behave under heat in practical applications.