Question

Two bodies of specific heats ${\mathrm{c}}_1$and ${\mathrm{c}}_2$having same heat capacities are combined to form a single composite body. What is the specific heat of the composite body?

• A. $\frac{{\mathrm{c}}_1-{\mathrm{c}}_2}{2{\mathrm{c}}_1{\mathrm{c}}_2}$
• B. $\frac{{\mathrm{c}}_1+{\mathrm{c}}_2}{2{\mathrm{c}}_1{\mathrm{c}}_2}$
• C. $\frac{2{\mathrm{c}}_1{\mathrm{c}}_2}{{\mathrm{c}}_1+{\mathrm{c}}_2}$
• D. $\frac{2{\mathrm{c}}_1{\mathrm{c}}_2}{{\mathrm{c}}_1-{\mathrm{c}}_2}$

Answer 1

The correct option is C

$\frac{2{\mathrm{c}}_1{\mathrm{c}}_2}{{\mathrm{c}}_1+{\mathrm{c}}_2}$

Step 1: Given data and assumptions made

Specific heat capacity of the first body ${\mathrm{c}}_1$

Specific heat capacity of the second body ${\mathrm{c}}_2$

Let the mass of the first body be ${\mathrm{m}}_1$and the mass of the second body be ${\mathrm{m}}_2$

Let the specific heat of the composite body be c.

Step 2: Calculation of the specific heat

According to the question, two bodies of specific heats ${\mathrm{c}}_1$and ${\mathrm{c}}_2$having same heat capacities

$$\begin{gathered} {\mathrm{m}}_1{\mathrm{c}}_1={\mathrm{m}}_2{\mathrm{c}}_2 \\ {\mathrm{m}}_2=\frac{{\mathrm{m}}_1{\mathrm{c}}_1}{c_2} \end{gathered}$$

The specific heat of the composite body is:

$$\begin{gathered} \left({\mathrm{m}}_1+{\mathrm{m}}_2\right)c={\mathrm{m}}_1{\mathrm{c}}_1+{\mathrm{m}}_2{\mathrm{c}}_2 \\ \left({\mathrm{m}}_1+\frac{{\mathrm{m}}_1{\mathrm{c}}_1}{c_2}\right)c={\mathrm{m}}_1{\mathrm{c}}_1+{\mathrm{m}}_1{\mathrm{c}}_1 \\ \left({\mathrm{m}}_1+\frac{{\mathrm{m}}_1{\mathrm{c}}_1}{c_2}\right)c=2{\mathrm{m}}_1{\mathrm{c}}_1 \\ \left(\frac{{\mathrm{m}}_1{\mathrm{c}}_2+{\mathrm{m}}_1{\mathrm{c}}_1}{c_2}\right)c=2{\mathrm{m}}_1{\mathrm{c}}_1 \\ c=\frac{2{\mathrm{m}}_1{\mathrm{c}}_1}{\left(\frac{{\mathrm{m}}_1{\mathrm{c}}_2+{\mathrm{m}}_1{\mathrm{c}}_1}{c_2}\right)} \\ c=\frac{2{\mathrm{m}}_1{\mathrm{c}}_1}{\left(\frac{{\mathrm{m}}_1{\mathrm{c}}_2+{\mathrm{m}}_1{\mathrm{c}}_1}{c_2}\right)} \\ c=\frac{2c_1}{\left(\frac{c_2+c_1}{c_2}\right)} \\ c=\frac{2c_1{c}_2}{c_1+c_2} \end{gathered}$$

So, the specific heat of the composite body is $\frac{2{\mathrm{c}}_1{\mathrm{c}}_2}{{\mathrm{c}}_1+{\mathrm{c}}_2}$

Thus option C is the correct option.