**Specific Heat Capacity or Specific Heat [c]**

## Table of Content |

It has been observed that different bodies, of same mass required different amounts of heat to raise their temperatures to same level. It is an experimental fact that,

(a) greater the mass ‘m’ of body, greater heat is required to raise its temperature by same amount, so quantity of heat Q is, Q ∝ m

(b) greater heat is required to raise the temperature higher, Q ∝ ?T

Here ?T is the rise in temperature.

Combining the two factors together,

Q ∝ m ?T

Or, Q = cm ?T

Here c is called the “specific heat” or “specific heat capacity” of the body. It depends only on the nature of material.

c = Q / m ?T

If m = 1, ?T = 1°C, c = Q

Specific heat capacity of a material is defined as the amount of heat required to raise the temperature of a unit mass of material through 1°C.

Actually ‘c’ is the mean specific heat capacity over a temperature range ?T, since we know that the quantity of heat required to raise the temperature of material through a small interval varies with the location of the interval in the temperature scale. If ‘?Q’ is the small amount of heat required to increase the temperature by a small amount of temperature ‘?T’, the true specific heat capacity is defined as,

c = 1/m (dQ/dT)

To calculate ‘Q’ we shall have to perform integration.

Q = ∫ dQ = m ∫ c dT

‘c’ being a function of ‘T’.

**Dimension of C**

c = (energy) / (mass) (temperature)

= [M^{0}L^{2}T^{-2}K^{-1}]

**Units of specific heat:-**

kcal kg^{-1}K^{-1} or J kg^{-1}K^{-1}

**Molar Specific Heat Capacity**

Molar specific heat capacity of a substance is defined as the amount of heat required to raise the temperature of one gram molecule of the substance through one degree centigrade. It is denoted by C.

One mole of substance contains M gram of substance where M is the molecular weight of the substance.

So, C = Mc

If n is the number of moles of substance, then,

n = m/M

So, m = nM

Substituting for m in equation c = 1/m (dQ/dT), we get,

Or, Mc = 1/n (dQ/dT)

Thus, C = Mc

= 1/n (dQ/dT)

Specific heat of water is taken to be 1. This is because of the reason that we defined unit of heat (calorie) by making use of water.

**Heat Capacity or Thermal capacity**

It is defined as the amount of heat required to raise the temperature of body through 1ºC.

*Q* = *mc*Δ*T*

If Δ*T* = 1ºC, *Q* = heat capacity = *mc*

Thus, heat capacity of a body is equal to the product of mass and its specific heat capacity.

**Unit:-** kcal K^{-1} or JK^{-1}

**Water Equivalent**

Consider a body of mass ‘m’ g and water of mass ‘w’ g. Supply same quantity ‘Q’ to both of them. If both of them register same rise of temperature (θ), ‘w’ is said to be the water equivalent of the body.

Water equivalent of a body is defined as the mass of water which gets heated through certain range of temperature by the amount of heat required to raise the temperature of body through same range of temperature.

For the body, Q = mcθ

For water, Q = w×1×θ

So, wθ = mcθ

Or, w = mc

Water equivalent of a body is equal to the product of its mass and its specific heat.

**Dimension:-** [M^{1}L^{0}T^{0}]

**Units:-** kg

**Latent Heat**

When the state of matter changes, the heat absorbed or evolved is given by: *Q* = *mL*. Here *L* is called the latent heat. The magnitude of latent heat depends upon the mass of the substance.

**Dimension of latent heat:-** [M^{1}L^{2}T^{-2}]

**Units of latent heat:- **kcal or Joule

**Specific Latent Heat**

Corresponding to the two stages of conversion (fusion and vaporization), we have two categories of specific latent heat.

(a) Specific latent heat of fusion (**L_{f}**):- Specific latent heat of fusion of a substance is defined as the amount of heat required to convert 1 gram of substance from solid to liquid state, at the melting point, without any change of temperature.

(b) Specific latent heat of vaporization (**L_{v}**):- Specific latent heat of vaporization of a substance is defined as the amount of heat required to convert 1 gram of liquid into its vapors at its boiling point without any rise of temperature.

**Dimensional formula:-** M^{0}L^{2}T^{-2}

**Unit:-** *kg cal kg*^{-1 } or *J kg*^{-1}

**Problem 1:-**

In a certain solar house, energy from the Sun is stored in barrels filled with water. In a particular winter stretch of five cloud days, 5.22 GJ are needed to maintain the inside of the house at 22.0°C, Assuming that the water in the barrels is at 50.0°C, what volume of water is required?

**Concept:-**

Heat *Q* that must be given to a body of mass *m*, whose material has a specific heat *c*, to increase its temperature from initial temperature *T*_{i} to final temperature *T*_{f} is,

*Q* = *mc *(*T*_{f} - *T*_{i})

So, *m* = *Q/ c *(*T*_{f} - *T*_{i})

Density *ρ* is equal to mass *m* per unit volume *V*.

So, *ρ* = *m*/*V*

So volume *V* will be,

*V* = *m*/*ρ*

**Solution:-**

To find the volume water, first we have to find out the mass of water which is required to transfer 5.22 GJ amount of heat energy.

To find the mass *m* of water, substitute 5.22 GJ for *Q*, 4190 J/kg. K for specific heat capacity *c* of water, 50.0 ^{°} C for *T*_{f} and 22.0 ° C for *T*_{i} in the equation *m* = *Q/ c *(*T*_{f} - *T*_{i}),

*m* = *Q/ c *(*T*_{f} - *T*_{i})

= 5.22 GJ/(4190 J/kg. K) (50.0 ^{°} C-22.0 ° C)

= (5.22 GJ) (10^{9} J/1 GJ)/(4190 J/kg. K) ((50.0+273) K –(22.0+273) K)

= (5.22 ×10^{9} J)/(4190 J/kg. K) (28 K)

= 4.45×10^{4} kg

To obtain the volume *V* of water, substitute 4.45×10^{4} kg for mass *m* and 998 kg/m^{3} for density* ρ* of water in the equation *V* = *m*/*ρ*,

*V* = *m*/*ρ*

= (4.45×10^{4} kg) / (998 kg/m^{3})

= 44.5 m^{3}

From the above observation we conclude that, the volume *V* of water will be 44.5 m^{3}.

**Problem 2:-**

How much water remains unfrozen after 50.4 kJ heat have been extracted from 258 g of liquid water initially at 0°C?

**Concept:-**

The amount of heat per unit mass that must be transferred to produce a phase change is called the latent *L* for the process. The total heat transferred in a phase change is then

*Q* = *Lm*.

Here *m* is the mass of the sample that changes phase. The heat transferred during melting or freezing is called the heat of fusion.

So from the above equation* Q* = *Lm*, mass of the substance would be,

*m* = *Q* /*L*

**Solution:-**

To obtain the amount of water (*m*) which freezes, substitute 50.4 kJ for the heat *Q* and 333 ×10^{3} J/kg for latent heat of fusion of water in the equation *m* = *Q* /*L*,

*m* = *Q* /*L*

= 50.4 kJ/ (333 ×10^{3} J/kg)

= (50.4 kJ×10^{3} J/1 kJ) / (333 ×10^{3} J/kg)

= 0.151 kg

So the amount of water (*m*') which remains unfrozen will be,

*m*' =258 g – 0.151 kg

= (258 g×10^{-3} kg/1 g) –(0.151 kg)

= 0.258 kg-0.151 kg

= 0.107 kg

From the above observation we conclude that, the amount of water which remains unfrozen would be 0.107 kg.

**Problem 3:-**

An aluminum electric kettle of mass 0.560 kg contains a 2.40-kW heating element. It is filled with 0.640 L of water at 12.0°C. How long will it take (a) for boiling to begin and (b) for the kettle to boil dry? (Assume that the temperature of the kettle does not exceed 100°C at any time)

**Concept:-**

The amount of heat per unit mass that must be transferred to produce a phase change is called the latent *L* for the process. The total heat transferred in a phase change is then

*Q* = *Lm*,

Here *m* is the mass of the sample that changes phase. The heat transferred during melting or freezing is called the heat of fusion.

The heat which is given to a body of mass *m*, whose material has a specific heat *c*, to increase its temperature from initial temperature *T*_{i} to final temperature *T*_{f} will be,

*Q*= *mc* (*T*_{f }-* T*_{i})

= *mc*Δ*T*

Mass *m* is equal to density *ρ* of object times volume *V* of the object.

*m* = *ρV*

Time *t* is equal to the heat energy *Q* divided by power *P*.

*t* = *Q*/*P*

**Solution:-**

(a) The time *t* taken by the electric kettle for boiling to begin will be,

*t* = *Q*/*P*

= [*m*_{a}*c*_{a} +*ρ*_{w}*V*_{w}*c*_{w}]Δ*T*/*P *

= [*m*_{a}*c*_{a} +*ρ*_{w}*V*_{w}*c*_{w}] (*T*_{f }-* T*_{i})/*P*

Here, mass of aluminum electric kettle is *m*_{a}, specific heat of aluminum is *c*_{a}, density of water is *ρ*_{w}, volume of water is *V*_{w}, specific heat of aluminum is *c*_{w}, final temperature is *T*_{f} , initial temperature is *T*_{i} and power is *P*.

To obtain the time *t* taken by the electric kettle for boiling to begin, substitute 0.56 kg for *m*_{a}, 900 J/kg. K for *c*_{a}, 998 kg/m^{3} for *ρ*_{w}, 0.640 L for *V*_{w}, 4190 J/kg.K for *c*_{w}, 100°C for *T*_{f} and 12^{°} C for *T*_{i} and 2.40 kW for *P* in the equation *t* =[*m*_{a}*c*_{a} +*ρ*_{w}*V*_{w}*c*_{w}] (*T*_{f }-* T*_{i})/*P*,

*t* =[*m*_{a}*c*_{a} +*ρ*_{w}*V*_{w}*c*_{w}] (*T*_{f }-* T*_{i})/*P*

= [(0.56 kg) (900 J/kg. K) + (998 kg/m^{3}) (0.640 L) (4190 J/kg.K)] (100°C -12^{°} C)/2.40 kW

= [(0.56 kg) (900 J/kg. K) + (998 kg/m^{3}) (0.640 L×10^{-3} m^{3}/1 L) (4190 J/kg.K)] ((100+273) K –(12+273) K)/(2.40 kW×10^{3} W/1 kW)

= [(0.56 kg) (900 J/kg. K) + (998 kg/m^{3}) (0.640×10^{-3}m^{3}) (4190 J/kg.K)] (373 K –285 K)/(2400 W)

= 117 s

From the above observation we conclude that, the time t taken by the electric kettle for boiling to begin would be 117 s.

(b) The time *t *for the kettle to boil dry will be,

*t* = *Q*/*P*

= *ρ*_{w}*V*_{w}*L*_{w}/*P*

Here density of water is *ρ*_{w}, volume of water is *V*_{w}, latent heat of water is *L*_{w} and power is *P*.

To obtain the time *t *for the kettle to boil dry, substitute 998 kg/m^{3} for *ρ*_{w}, 0.640 L for *V*_{w}, 2256×10^{3} J/kg for *L*_{w} and 2.40 kW for *P* in the equation* t* = *ρ*_{w}*V*_{w}*L*_{w}/*P*,

*t* = *ρ*_{w}*V*_{w}*L*_{w}/*P*

= (998 kg/m^{3}) (0.640 L) (2256×10^{3} J/kg)/( 2.40 kW)

= (998 kg/m^{3}) (0.640 L×10^{-3} m^{3}/1 L) (2256×10^{3} J/kg)/( 2.40 kW×10^{3} W/1 kW)

= (998 kg/m^{3}) (0.640 ×10^{-3} m^{3}) (2256×10^{3} J/kg)/( 2400 W)

= 600 s

From the above observation we conclude that, the time t for the kettle to boil dry would be 600 s.

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