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# Heat capacity, specific and molar heat capacity, latent heat, practice problems, FAQs

If you have been to the beach on a bright sunny day you have observed that land is very hot as compared to water. Have you ever thought that both receive the same amount of energy from the sun at the same time? The answer lies in heat capacity. Different substances will have different changes in temperature while providing the same amount of heat is the property we are going to discuss.

• Heat capacity
• Specific heat capacity
• Molar heat capacity
• Specific Heat for a Mixture of Non-Reacting Gases
• Latent heat
• Properties of latent heat
• Types of latent heat
• Practice problems
• FAQs

## Heat Capacity

Heat capacity is defined as the amount of heat required to raise the temperature of a mass of substance by 1 °C.

Consider a closed container filled with gas being heated by an external source as shown in the figure. In a certain time t, the amount of heat received by the gas is Q and as a result, the temperature of the gas increases by ΔT.

Then, the heat capacity C is given by,

The unit of heat capacity is JK−1.

Dimensional formula for heat capacity is [ML2T-20-1]

## Specific heat capacity

It is the amount of heat required to raise the temperature of a unit mass of substance by 1°C. It is usually assigned with solid and liquid substances and is denoted by s.

It is the heat required to raise the temperature of a unit mass of substance by ΔT. Therefore, Q = msΔT

Units of specific heat capacity is J/kg K and the dimensional formula for specific heat capacity is [M0L2T-20-1].

• Specific heat capacity at constant volume (Sv)

The amount of heat required to raise the temperature of unit mass gas through 1 °C at a constant volume is defined as the specific heat capacity at constant volume.

• Specific heat capacity at constant pressure (SP)

The amount of heat required to raise the temperature of unit mass gas through 1 °C at a constant pressure is defined as the specific heat capacity at constant pressure.

## Molar heat capacity

The amount of heat required to raise the temperature of one mole of a substance by 1 °C is known as the molar heat capacity.

Molar heat capacity,

Or

ΔQ = nCsΔT

Units of molar heat capacity is J/mol K and the dimensional formula for specific heat capacity is [M1L2T-20-1].

• Molar heat capacity at constant volume (Cv)

The amount of heat required to raise the temperature of a unit mole of gas through 1 °C at a constant volume is defined as the molar heat capacity at constant volume.

• Molar heat capacity at constant pressure (CP)

The amount of heat required to raise the temperature of a unit mole of gas through 1 °C at a constant pressure is defined as the molar heat capacity at constant pressure.

## Molar Heat for a Mixture of Non-Reacting Gases

Consider two non-reacting gases 1 and 2 of quantities n1 and n2, respectively, separated by a conducting wall. The molar heat capacities of gases 1 and 2 at constant pressure and constant volume are cp1, cp2and cV1, cV2, respectively. After some time, the wall is removed and the gases are allowed to mix. The molar heat capacity of the mixture of the gases is to be taken into consideration.

Internal energy Umix of the mixture is as follows:

Umix=U1+U2

nmix(CV)mixT=n1CV1T+n2CV2T

Similarly, the molar heat capacity of the mixture of gases at constant pressure (CP)mix is,

The gas ratio (𝛾) of the mixture is,

## Latent heat

Latent heat of any substance is defined as the heat required to change the phase of that material. The latent heat is absorbed at a constant temperature. This means that the heat absorbed is solely used for changing the phase of the substance. All the energy in the heat is absorbed by the constituent molecules or atoms to break the bond between them or to move apart.

The energy is hidden within the molecules until the phase change occurs and hence it got its name as latent heat.

If m kg of substance undergoes phase change, quantity of heat required(Q) will be,

Q=mL

Where L= Latent heat of the substance

## Properties of latent heat

• The most common units of specific latent heat are joules per gram (J/g) and kilojoules per kilogram (kJ/kg).
• Dimensional formula for latent heat is [L2T-2].
• Specific latent heat is an intensive property of matter. Its value does not depend on sample size or where within a substance the sample is taken.

## Types of latent heat

• Latent heat of fusion: When the phase change occurs from solid to liquid or vice versa, heat required is known as latent heat of fusion. Also called enthalpy of fusion, it is generally higher than specific heat which is one of the main reasons why adding ice to soft drinks keeps them cool for a longer period of time as compared to cold liquid. The specific latent heat of fusion of water is 334 kilojoules per kilogram of water. This means that to convert one kilogram of water from solid to liquid, 334 kilojoules of heat is required.

• Latent heat of vaporisation: When the phase change occurs from liquid to gas(or gas to liquid) heat required(or released) is known as latent heat of vaporisation. It is also known as enthalpy of fusion. The latent heat of vaporisation of water is approximately 2264 kilojoules per kilogram of water which is significantly greater than the latent heat of fusion of water. This is because, in the gaseous state, the particles exist with much higher energy than in the liquid state.

Heating curve

oa - Ice

a - Melting point of Ice

ab - Phase change

bc - Water

c - Boiling point of water

cd - Phase change

de - Steam

1. Initially when the ice is at -25oC, its temperature rises as heat is supplied to it. When the temperature reaches 0oC, ice starts melting and converts into water at 0oC. The heat supplied in ab region corresponds to latent heat of fusion and during this region temperature becomes constant.

2. In the region bc, the temperature of the water starts rising with heat supplied. Since the specific heat of water(1 kcal/kg oC) is more than that for ice(0.5 kcal/kg oC), therefore the slope for bc curve is less as compared to that for oa. Point c corresponds to the boiling point of water.

3. In the region cd, phase change occurs from water to steam as long as heat (latent) is supplied to the water. During this region temperature becomes constant.

4. When all the water convert into steam at 100oC, the temperature of the steam starts rising. Since the specific heat of steam(0.48 kcal/kg oC) is least of all i.e. water and ice, therefore slope will be the largest of all.

## Practice problems

Q 1. A mixture contains 1 mole of helium (cp1=2.5R, CV1=1.5R) and 1 mole of hydrogen (cp2=3.5R, CV2=2.5R). Calculate the values of CP , CV , and 𝛾 for the mixture.

Given,

Specific heat capacity of the mixture at constant volume is

Specific heat capacity of the mixture at constant pressure is

The gas ratio (𝛾) of the mixture is,

Q 2. 1 kg of ice at −20 is converted into 1 kg of water at 100 . Find the heat required (H) for the complete process.

Solution

We have,

Mass of the ice, mice= 1 kg

Specific heat of the ice, sice =0.5 cal g-1-1

Latent heat for ice, Lf = 80 cal g-1

Specific heat of the water, swater = 1 cal g-1-1

The step-by-step process from the ice at –20 to the water at 100 is as follows:

1 kg ice at −20 (Q1)1 kg ice at  0 ℃(Q2)1 kg water at  0 ℃(Q3)1 kg water at  100

Therefore, to get the water of 100 from the ice of –20 ,
Total required heat, Q=Q1+Q2+Q3

Thus, according to the given process,

Step 1: 1 kg ice at −20 1 kg ice at  0

Step 2: 1 kg ice at  0 ℃(Q2) to 1 kg water at  0

It is a fusion process and the latent heat of fusion for ice is as follows:
Lf = 80 cal g-1

Thus,

Q2 = miceLf = (1,000 ) x (80)=80 kcal

Step 3: 1 kg water at  0 1 kg water at  100

Q3 = mwaterswater(100 – (0))

Q3= (1,000 )(1)(100 )=100 kcal

Therefore, the net required heat, Q=Q1+Q2+Q3=10+80+100=190 kcal

Q 3. If the specific heat of a substance is infinite, then what does it mean?

a. Heat is given out.
b. Heat is taken in.
c. No change in temperature takes place whether heat is taken in or given out.
d. All of the above

If specific heat of a substance is infinite,

Q=mcT

Since mass of the object cannot be 0, therefore T=0

Whether heat is given to or taken from the body, the temperature of the body is not going to change.

So, the change in temperature must be 0.

Q 4. When 80 g of water at 30 oC is poured on a large block of ice at 0 oC, what is the mass of ice that melts?

a. 30 g
b. 80 g
c. 1600 g
d. 150 g

Let the mass of ice melted be mi

Heat loss =Heat gain

## FAQs

Q 1. Name of the device that can be used to measure the latent heat?\
Calorimeter.

Q 2. What is the advantage of water’s heat capacity?