Question

Two identical bodies are made of a material for which the heat capacity increases with temperature. One of these is held at a temperature of  $ 100 $  while the other one is kept at  $ 0 $ . If the two are brought in contact, then, assuming no heat loss to the environment, the final temperature that they will reach is:

A. $ 50 $ 

B. Less than  $ 50 $ 

C. More than  $ 50 $ 

D. $ 0 $ 

Answer 1

Hint: We will make use of the principle of Calorimetry which says:

When a hot body is kept in contact with a cold body, heat energy passes from the hot body to the cold body, until both the bodies attain the same temperature. If no heat energy is lost to the surroundings, then  

Heat energy lost by the hot body = Heat gained by the cold body. 

The principle of calorimetry is based on the law of conservation of energy.

Formula used:

If no heat energy is lost to the surroundings, then by the principle of calorimetry:

$ {{m}_{1}}{{c}_{1}}({{t}_{1}}-t)={{m}_{2}}{{c}_{2}}(t-{{t}_{2}}) $ 

Where  $ {{m}_{1}} $  is the mass of the hot body.

$ {{c}_{1}} $  is the specific heat capacity of the hot body.

$ {{t}_{1}} $  is the temperature of the hot body.

$ {{m}_{2}} $  is the mass of the cold body.

$ {{c}_{2}} $  is the specific heat capacity of the cold body.

$ {{t}_{2}} $  is the temperature of the cold body.

$ t $  is the final temperature of the two bodies in contact.

Complete step-by-step solution:

The two bodies are identical and made of the same material. So, their mass will be the same, that is,  $ {{m}_{1}}={{m}_{2}} $ .

As per given specific heat capacity is temperature dependent so for hotter body it will be more but specific heat for colder body will be less initially. As heat transfer takes place, the temperature of the colder body will rise and it’s specific heat capacity will also increase, on the other side specific heat capacity of the hotter body will decrease as it will lose heat and its temperature will decrease.

For initial temperature  $ c_2 > c_1 $ .

The temperature of the hot body is  $ 100 $ , so  $ {{t}_{1}}=100 $ .

The temperature of the cold body is  $ 0 $ , so  $ {{t}_{2}}=0 $ .

Now, assuming the final temperature of the two bodies is  $ t $ , when they are kept in contact there is no heat loss to the environment.

In our case we have two bodies which 

 $\Rightarrow c_2 (t_2 - t) = c_1 (t - t_1) $ 

 $\Rightarrow c_2t_2 - c_2t = c_1t-c_1t_1 $ 

 $\Rightarrow (c_1 + c_2)t = c_1t_1 + c_2t_2 $ 

 $\Rightarrow t = \dfrac{c_1t_1 + c_2t_2}{(c_1 + c_2)} $ 

Putting value of  $ t_1 $  and  $ t_2 $ -

 $\Rightarrow t = \dfrac{ 100c_2}{(c_1 + c_2)} $ 

Now as we know  $ c_2 > c_1 $ 

By Adding  $ c_2 $  both side

 $ \Rightarrow 2c_2 > c_1+c_2 $ 

 $ \Rightarrow \dfrac{c_2}{c_1+c_2} > \dfrac{1}{2} $ 

 $ \Rightarrow \dfrac{100c_2}{c_1+c_2} > \dfrac{100}{2} $ 

 $ \Rightarrow \dfrac{100c_2}{c_1+c_2} > 50 $ 

 $ \Rightarrow t > 50 $ 

Therefore, the final temperature of the two bodies is greater than  $ 50 $. Thus, option (C) is the correct answer.

Note: The unit of temperature of the two bodies should be the same. The two bodies should be in the same phase (state). If specific heat capacity is not dependent on the temperature of the body then the final temperature will be equal to  $ 50 $ .