Question

Heat capacity $ \left( {{C_P}} \right) $ is an extensive property but specific heat $ \left( c \right) $ is an intensive property. What will be the relation between $ {C_P} $ and $ c $ for 1mole of water?

Answer 1

Hint: Using the given statement that heat capacity is an extensive property but specific heat is an intensive property we can assume the relation between them with the help of thermodynamic extensive and intensive property.

Complete Step By Step Answer:
As per the problem heat capacity is an extensive property but specific heat is an intensive property and both of them is denoted as:
Heat capacity $ = {C_P} $
Specific heat $ = c $
Heat capacity is defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. Heat capacity depends upon the mass of the substance hence it is an extensive property.
Specific heat is defined as the quantity of heat required to raise the temperature of one gram of a substance by one Celsius degree. Specific heat is the heat capacity per unit mass hence it becomes independent of the amount of substance. Therefore it is an intensive property.
Now,
The heat capacity and the specific heat are related by $ C = cm $ or after rearranging we can also write $ c = \dfrac{C}{m} $ where m is the mass of the substance, c is the specific heat and C is the heat capacity.
Now the mass m, specific heat c, change in temperature $ \Delta T $ and heat added Q are heat released by the equation:
 $ Q = mc\Delta T $
 $ \Rightarrow Q = nMc\Delta T \ldots \ldots \left( 1 \right) $
Heat H is also written as,
 $ H = n{C_P}\Delta T \ldots \ldots \left( 2 \right) $
Comparing equation $ \left( 1 \right) $ and $ \left( 2 \right) $ we get,
 $ nMc = n{C_P} $
 $ \Rightarrow {C_P} = Mc $
Mass of the water is 18g.
Now for 1mole of water we can write,
Molar Heat capacity = $ 18g \times $ specific heat
Where,
Heat capacity $ = {C_P} $
Specific heat $ = c $
Mathematically we can write,
 $ {C_P} = 18g \times c \ldots \ldots \left( 1 \right) $
Specific heat of the water is given as,
 $ c = 4.18\,J\,{g^{ - 1}}\,{K^{ - 1}} $
SI unit of specific heat is joules per gram per kelvin.
Putting specific heat value in equation $ \left( 1 \right) $ we will get,
 $ {C_P} = 18g \times 4.18J{g^{ - 1}}{K^{ - 1}} $
 $ \Rightarrow {C_P} = 75.3J{K^{ - 1}} $ .

Note:
The mass of the substance is equal to the number of moles multiplied by the molar mass of the substance. Here the substance is the water molecules of one mole hence, mass is equal to the molar mass of the water which is equal to 18g.