Heat Capacity and Specific Heat

When we heat a substance, it starts absorbing heat up to a certain limit, after which, its temperature starts rising. This limit of a substance to tolerate or absorb heat without raising the temperature is termed as Heat Capacity. 

In this, we will see how the transferred heat of a system is measured. If heat is absorbed by the system, it is reflected as an increase in temperature. The rise in system temperature is proportional to the heat transferred to the system. If heat is represented by a symbol 'Q' and the temperature is represented by a symbol 'T' then, 

➩ { Q ∝ ∆T }

➩ { Q ＝ (coefficient)×∆T }

Heat capacity and specific heat capacity are two proportionality constants that relate to temperature change and the amount of heat and it is denoted by a symbol 'C' & 'c' respectively. Thus, the above becomes;

➩  Q ＝ C×∆T 

Or

➩  C ＝ Q / ∆T

------------------(1)

Thus, if we know the heat capacity of the system and how much the temperature of the system has risen, we can find out how much heat is given to the system. 

           According to the above equation, higher the value of heat capacity, more heat can be given to the system. And the rise in system's temperature will be less. This means that the higher the heat capacity, the less will be the rise in temperature due to heat. 

FOR EXAMPLE- 

Both sand and water are receiving the same amount of sun rays (heat energy), then 

why there is a difference in the temperature of the sand and water?

.........!!!!!!The Concept Of Heat Capacity Plays The Important Role Here!!!!!!...........

Substances with low heat capacity can be heated or cooled more rapidly than materials with high heat capacity.

Water has a high heat capacity, which means that a lot of heat will be needed to raise its temperature. While sand has a low heat capacity, which means that a small amount of heat is needed to raise its temperature. Thus, water will consume more heat energy compared to the same quantity of sand. As a result, the sand get heated more as compared to water. 

Means,

    For 1 kg of sand and 1 kg of water, the heat capacity of water will be higher than that of sand. As a result, the temperature of sand is higher than that of water. Because water absorbs more heat energy than sand.

Heat Capacity

In a simple way, "The capacity to hold heat is called heat capacity".

➩ { C ＝ Q/∆T or ∂Q/∆T }

-----------------(2)

If ∆T = 1°C then heat capacity, C = Q 

Means heat absorbed or rejected (Q) is known as Heat capacity (Hc/C)]

Thus, 

"The heat capacity of any substance (or system) refers to the amount of heat required to raise the temperature of the entire mass of that substance by one degree". 

We can also define heat capacity according to equation (2),

"The ratio of heat supplied (Q) to the body to the change in its temperature ∆T is called heat capacity (Hc/C) of that body". 

The unit of heat capacity is "Joule/K".

For Example

200g of Al requires twice the energy to change the temperature as required by 100g of Al.

From the above example of sand and water, we noticed that the heat capacity of sand is less, while the heat capacity of water is more. That means heat capacity depends on the material of the body. 

Also, one more thing I would like to add. Heat Capacity is an extensive property. Thus, the value of heat capacity is directly proportional to the amount/mass of substance. That is, the greater the amount of a substance, the greater the value of heat capacity. And the smaller the amount of a substance, the lower the value of heat capacity.

Specific Heat Capacity

The heat capacity of any substance is the amount of heat required to raise the temperature of the entire mass of a substance by 1°C. But when we have to raise the temperature of only 1 kg mass of a substance, how much heat do we need? 

This is the case where specific heat capacity comes into the picture.

Everything is same as heat capacity, we just have to take a unit mass over here.

Let's consider that;

you can see the iron block having 1 kg mass and during the initial stage its temperature is 20°C. Now, on applying some heat, its temperature rises by 1°C and it’s temperature becomes 21°C. During this phenomenon, the amount of heat supplied is known as the specific heat capacity of the body.

Thus, "The amount of heat required to raise the temperature of 1 kg of substance/system (or unit mass of substance /system) by one degree is called the specific heat of that substance". It is denoted by lower case letter 'c'. 

Specific heat capacity is an intensive property since it does not depend on the mass or size of the object i.e. when the mass of a substance increases, its specific heat capacity does not change.

For Example

The specific heat capacity of water is 4.18 J/g°C, which means that it takes 4.18 Joules of energy by 1 gram of water to increase its temperature by 1 °C.

☛ To find the amount of heat required to raise the temperature of a substance, we multiply the specific heat of that substance by the change in temperature and mass. Thus,

Heat added ＝ specific× mass×  change in

        to the                 heat                  temperature 

   system                                                                 

➩ { Q ＝ m × c × ∆T } 

Or

➩ [ c ＝ Q/ m∆T ]

-----------------(3)

In this formula, if ∆T = 1 and m = 1, then Q = C. Means heat absorbed (Q) is known as Specific heat capacity (C)

The amount of heat that a system absorbs or rejects when it is heated or cooled, can be measured by the above formula. The above formula is not applicable for situations where a phase change is encountered. The unit of specific heat is "Joule/kg-K". 

☛ There is a definite relationship between heat capacity and specific heat. The specific heat of a substance is the ratio of the heat capacity of that substance to its mass. It is given by a formula;

        ➩ { c ＝ Heat capacity/mass ＝ C/m }

                             ( ஃ Or, m×c = C )

MOLAR HEAT CAPACITY 

We can also define the heat capacity of substance on mole basis as "The amount of heat required to raise the temperature of one mol of a substance by one degree is called molar heat capacity of that substance". The molar heat capacity is the heat capacity of one mol of substance. And the unit of molar specific heat is "Joule/mol-K". The molar heat capacity is given by a formula,

                  ➩ { Q ＝ n×c×∆T }

        Here, 

              n is the number of moles

              c is the specific heat capacity

           ∆T is the change in temperature

The heat capacity related to one mole of a substance is called Molar Heat Capacity while the heat capacity related to one kilogram of substance is called Specific Heat Capacity or Specific Heat.

Types Of Specific Heat Capacity

There are two types of specific heat available at constant volume and pressure conditions i.e. specific heat at constant volume and specific heat at constant pressure. Let us now know about them in detail;

Specific Heat At Constant Volume

In this, the specific heat is measured at a constant volume. It is define as the amount of heat required to raise the temperature of one kg mass of a substance by one degrees at a constant volume. It is denoted by a symbol 'cᵥ'. It is given by a formula given below;

            ➩ { Q ＝ m × cᵥ × ∆T } 

Specific Heat At Constant Pressure

In this, the specific heat is measured at a constant pressure. It is define as the amount of heat required to raise the temperature of one kg mass of a substance by one degrees at a constant pressure. It is denoted by a symbol 'cₚ'. It is given by a formula given below;

             ➩ { Q ＝ m × cₚ × ∆T }

Relationship Between cₚ & cᵥ

According to the first law of thermodynamics;

                   ∆Q ＝ ∆U ＋ ∆W

                                ----------------(1)

And work-done in thermodynamics define as;

                      ∆W ＝ P ∆V

Thus, equation-(1) becomes;

              ∆Q ＝ ∆U ＋ P ∆V

                                  ----------------(2)

Since, the definition of enthalpy says;

               H ＝ U ＋ PV

And change in enthalpy is given as;

                      ∆H ＝ ∆U ＋ P ∆V

Thus, equation-(2) becomes;

                ∆Q ＝ ∆U ＋ P ∆V ＝ ∆H 

                                       ----------------(3) 

         Hence,   ∆Q ＝ ∆H

Now, coming back to the equation no. (2);

     Here,    ∆Q ＝ δQ ＝ cₚ ∆T

                   ∆U ＝ δU ＝ cᵥ ∆T

And, for an ideal gas;

                  PV ＝ RT   (for one mole of a gas)

And the change in ideal gas equation;

                      P∆V ＝ R∆T 

Now, equation(2) becomes;

               cₚ ∆T ＝ cᵥ ∆T ＋  R ∆T 

         Devide the above equation by ∆T;

                ➩  [ cₚ ＝ cᵥ＋ R ]

                               Or

                ➩  [ cₚ – cᵥ ＝  R ]

Here, 

     cₚ = Specific heat at constant pressure

     cᵥ = Specific heat at constant volume

☛ Like internal energy, specific heat at constant volume & at constant pressure both are independent of pressure & volume and only depends on the temperature in case of an ideal gas.

Hope you have found this article helpful!!

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