Question

Two different metal bodies A and B of equal mass are heated at a uniform rate under similar conditions. The variation of temperature of the bodies is graphically represented as shown in the figure. The ratio of specific heat capacities is : 

• A. 8 / 3
• B. 4 / 3
• C. 3 / 4
• D. 3 / 8

Hint:

In a Temperature vs Time graph with constant heating, a steeper slope indicates a lower specific heat capacity (it gets hot quickly).
Body A has a steeper slope than B, so \(c_A<c_B\). Only option D satisfies this logic.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
When a body is heated at a uniform rate, the heat supplied per unit time (\(P = dQ/dt\)) is constant.
The heat absorbed is \(dQ = mc dT\).
So, \(P = mc \frac{dT}{dt}\).
The slope of the Temperature-Time graph is \(\frac{dT}{dt}\).
Step 2: Key Formula or Approach:
\[ c = \frac{P}{m \cdot \text{Slope}} \]
Since \(P\) and \(m\) are the same for both bodies:
\[ c \propto \frac{1}{\text{Slope}} \implies \frac{c_A}{c_B} = \frac{\text{Slope}_B}{\text{Slope}_A} \]
Step 3: Detailed Explanation:
From the graph:
Slope of A = \(\frac{\Delta T_A}{\Delta t_A} = \frac{120 - 0}{3 - 0} = 40 \text{ }^\circ\text{C/s}\).
Slope of B = \(\frac{\Delta T_B}{\Delta t_B} = \frac{90 - 0}{6 - 0} = 15 \text{ }^\circ\text{C/s}\).

The ratio of specific heat capacities:
\[ \frac{c_A}{c_B} = \frac{\text{Slope}_B}{\text{Slope}_A} = \frac{15}{40} \]
Simplify the fraction by dividing by 5:
\[ \frac{c_A}{c_B} = \frac{3}{8} \]
Step 4: Final Answer:
The ratio of specific heat capacities is 3 / 8.