Question

Very early in the study of the nature of heat it was observed that if two bodies of equal mass but different temperatures are placed in thermal contact, their specific heat capacities depend inversely on the change in temperature each undergoes on reaching its final temperature. Write a mathematical equation in modern notation to express this fact.

Short answer

As;

${\text{c}}_{\text{s2}}{\text{∆T}}_{\text{2}}{\text{= -c}}_{\text{s1}}{\text{∆T}}_{\text{1}}$

Therefore, the mathematical equation can be given as;

$\frac{{\text{c}}_{\text{s1}}}{{\text{c}}_{\text{s2}}}\text{= -}\frac{{\text{∆T}}_{\text{2}}}{{\text{∆T}}_{\text{1}}}$

Step-by-step solution

The given Information.

If two bodies of equal mass but different temperatures are placed in thermal contact, their specific heat capacities depend inversely on the change in temperature each undergoes on reaching its final temperature.

Specific heat capacity

Specific heat capacity exhibit variation with temperature and is designated by the symbol ${\mathbf{\text{c}}}_{\mathbf{\text{p}}}$. It can be acquired by dividing the heat capacity with the molar mass of a particular sample.

Mathematical equation to explain the fact

When two objects initially at different temperatures are brought into contact, energy as heat is exchanged among them until they attain a common temperature.

The amount of heat taken up by the cooler object is equal to ${\text{-q}}_{\text{1}}$, the amount of heat given up by the hotter object.

Thus,

${\text{q}}_{\text{2}}{\text{= -q}}_{\text{1}}$

And;

${\text{M}}_{\text{2}}{\text{c}}_{\text{s2}}{\text{∆T}}_{\text{2}}{\text{= -M}}_{\text{1}}{\text{c}}_{\text{s1}}{\text{∆T}}_{\text{1}}$

Here,

${\text{c}}_{\text{s1}}$is the specific heat capacity of the hot object.

${\text{c}}_{\text{s2}}$is the specific heat capacity of the cold object.

${\text{M}}_{\text{1}}$is the mass of hot object.

${\text{M}}_{\text{2}}$is the mass of cold object.

${\text{∆T}}_{\text{2}}$, ${\text{∆T}}_1$denote the changes in the temperature of cold, hot body respectively.

For two bodies of equal mass, the equation becomes;

$$\begin{gathered} {\text{c}}_{\text{s2}}{\text{∆T}}_{\text{2}}{\text{= -c}}_{\text{s1}}{\text{∆T}}_{\text{1}} \\ \frac{{\text{c}}_{\text{s1}}}{{\text{c}}_{\text{s2}}}\text{= -}\frac{{\text{∆T}}_{\text{2}}}{{\text{∆T}}_{\text{1}}} \end{gathered}$$

From the relation, it is observed that the specific heat capacities of these objects are inversely proportional to the temperature change.