# Specific Heat Capacity and Specific Heat Capacity of Water

Let us see one important property which is required to study about the quantity of heat transferred and subsequently the change in temperature.

We can change the state of the system by transferring the energy to or from it in the form of heat, or by doing work on the body. One property of a body that may change in such a process is its temperature.

Suppose an amount of heat âˆ†Q supplied to a system changes its temperature from T to T+âˆ†T; then the change in temperature âˆ†T that corresponds to a transfer of particular amount of heat energy Q will depend on the circumstances under which heat was transferred.

For example heat can be transferred by keeping pressure as constant so that the volume can be changed or by keeping volume as constant and changing the pressure.

It is convenient to define the heat capacity of a body as the ratio of the amount of heat âˆ†Q supplied to a system in any process to its corresponding temperature change âˆ†T

Here the word "capacity" means the energy per degree of temperature change that is transferred as heat when the temperature of the body changes.

**The heat capacity per unit mass of a body is called specific heat capacity.**

where "

**c**" is known as the specific heat capacity of the body.

Both neither the heat capacity of a body nor specific heat of a material is constant and both depend on the temperature as well as on the other parameters such as pressure.

To obtain a unique value of specific heat we must indicate the specific conditions such as specific heat at constant volume or specific heat at constant pressure.

The unit of specific heat capacity is

**J kg**

^{-1}K^{-1}If the amount of substance is expressed in terms of moles Ã‚Âµ, we can define the heat capacity per mole of the substance by

Where

**is**

*C'***the molar specific heat capacity of the substance.**

The unit of

**is**

*C'***J mol**

^{-1}K^{-1}

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# Specific heat capacity of water

The ancient unit for heat was calorie. One calorie was defined to be the amount of heat required to raise the temperature of 1g of water by 1ËšC.

From the given graph it was found that the specific heat capacity of water varies slightly with temperature.

Because of this variation, we have to define the specific heat capacity of water with some particular interval.

For example, one calorie is defined as the amount of heat required to raise the temperature of 1g of water from 14.5Ëšto 15.5Ëš.

In SI units, the specific heat capacity of water is 4186 J kg

^{-1}K

^{-1}or 4.186 J g

^{-1}K

^{-1}.

Due to large specific heat capacity of water, a small change in temperature of water requires large amount of heat transfer.

We know that there are two specific heat capacities, namely Specific heat capacity at constant volume C

_{V}and Specific heat capacity at constant pressure C

_{P}.

# The difference between two specific heat capacities

Consider a gram molecule of a gas having a volume V at a pressure P and temperature T K contained in a cylinder closed by a piston.

The heat required to raise the temperature by a small amount âˆ†T at constant volume is given by

**(âˆ†Q)**...............(1)

_{V}= C_{V }âˆ†TC

_{V}is the gram molecular specific heat at constant volume.

**(âˆ†Q)**(2)

_{P}= C_{P }âˆ†T ..............C

_{P}is the gram molecular specific heat at constant pressure.

Work done

**âˆ†W = P A dx = P âˆ†V**

If volume remains constant, then

**âˆ†W = 0**

Then applying the first law of thermodynamics

**(âˆ†Q)**Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦.(3)

_{V}= (âˆ†U)_{V}From (1) and (3)

**(âˆ†U)**

_{V}= C_{V }âˆ†T(âˆ†W)

_{P}= P (âˆ†V)

_{P}

Then applying to first law of thermodynamics

**(âˆ†Q)**

_{P}= (âˆ†U)_{P}+ P (âˆ†V)_{P}**C**

_{P}âˆ†T = (âˆ†U)_{P}+ P (âˆ†V)_{P}As the internal energy of the gas depends upon only the temperature and the temperature difference in the two process is the same, therefore

(âˆ†U)

_{P}= (âˆ†U)

_{V}

C

_{P }âˆ†T - P (âˆ†V)

_{P}= C

_{V }âˆ†T

C

_{P }âˆ†T - C

_{V }âˆ†T = P (âˆ†V)

_{P}

(C

_{P }- C

_{V })âˆ†T = P (âˆ†V)

_{P}

(C

_{P }- C

_{V })âˆ†T = R âˆ†T (From ideal gas equation )

C

_{P }- C

_{V }= R

This relation is called as MayerÃ¢â‚¬â„¢s relation. According to this relation, it is clear that specific heat at constant pressure is always greater than the specific heat at constant volume.