# Heat Capacity

**Heat capacity or thermal capacity **is an extensive property of matter, that defines its physical property. Heat Capacity is the amount of heat that must be applied to an object in order to cause a unit change in temperature. Heat capacity is measured in Joules per Kelvin (J/K), which is its SI unit. When heat capacity is divided by the mass of the substance, gives the corresponding intensive property called Specific Heat Capacity. Moreover, heat capacity divided by the amount of substance in volumes gives Molar Heat Capacity. Hence, in this article, we’ll understand the important concepts of Heat Capacity like its definition, explanation, formula, unit, specific heat capacity, molar heat capacity

## What is Heat Capacity?

The amount of heat required to increase the temperature of a given mass of a substance by one unit without causing a phase change is known as the

heat capacity (C)orthermal capacity. It describes a substance’s ability to absorb thermal energy.

Being an extensive property, heat capacity is based on the sample’s mass and size. This suggests that, for a sample containing twice as much material as another sample, twice as much heat energy (Q) would be needed to produce the same change in temperature.

The heat capacity of various substances will determine how much their temperatures will rise. On a hot day, a metal chair left in the direct sun may get rather warm to the touch whereas equal amounts of water won’t heat up nearly as much when exposed to the same amount of sunlight. This indicates that water has a high heat capacity. Metals typically do not react well to temperature fluctuations while water does.

## Heat Capacity Formula

Mathematically, Heat Capacity for the unit mass of the substance is defined as,

C = ΔQ / ΔTwhere

ΔQis the amount of heat supplied to the substanceΔTis the change in temperature

## SI Unit of Heat Capacity

The SI unit for the heat capacity of a substance is Joule per Kelvin (J/K or J⋅K^{−1}). When the temperature is increased by 1°C is the same as an increase of 1 K, which is the same unit as J/°C.

Dimension of Heat capacity is [L^{2}M T^{−2}Θ^{−1}]

**Specific Heat Capacity**

Specific heat capacity is defined as the number of heat changes i.e. heat absorbed or rejected by a substance per unit mass in order to change its temperature by one unit.

When the same quantity of heat is applied to the same mass of different substances, the resulting temperature changes are not the same. It indicates that each substance has a distinct value for the quantity of heat absorbed or rejected to change the temperature of its unit mass by one unit. The specific heat capacity of a substance is the measurement of this quantity. It is symbolized by the letter s.

If C is the amount of heat absorbed or rejected i.e. **ΔQ** by a substance of mass **m** when it experiences a temperature change** ΔT**, then the specific heat capacity, of that substance is given by

S = C/m = (1/m) (ΔQ / ΔT)

Hence, the specific heat capacity is the property of the substance which determines the change in temperature of the given quantity of a substance which is the amount of heat absorbed or rejected. The given substance is undergoing no phase change during temperature change. It is dependent on the substance’s nature and temperature. **J kg ^{–1} K^{–1}** is the SI unit for specific heat capacity.

## Molar Heat Capacity

The amount of heat required to raise the temperature of one mole of any substance by one degree Kelvin is called the Molar Heat Capacity. Molar heat capacity and specific heat capacity are almost the same and the basic difference between them is that specific heat capacity is measured per gram whereas molar heat capacity is measured per mole.

**Unit: **Molar Heat Capacity is measured in Joule/mole.

The formula for finding molar heat capacity is,

Q = nC∆Twhere,

qis the heat suppliednis the moles of the substanceCis the molar heat capacity of the body

## Molar Heat Capacity at Constant Pressure C_{p}

The amount of heat required to raise the temperature of one mole of any substance by one degree Kelvin at constant pressure is called the Molar Heat Capacity at Constant Pressure and it is denoted by C_{p}.

The formula for calculating Molar Heat Capacity at Constant Pressure C_{p} is,

C_{p }= dH/dt (at constant pressure)where,

Crepresents the specific heat at Constant Pressure_{p}dHis the Change in Enthalpydtis the Change in Temperature

## Molar Heat Capacity at Constant Volume C_{v}

The amount of heat required to raise the temperature of one mole of any substance by one degree Kelvin at constant volume is called the Molar Heat Capacity at Constant Volume and it is denoted by C_{v}.

The formula for calculating Molar Heat Capacity at Constant Pressure C_{v} is,

C_{v }= dH/dt (at constant volume)where,

Crepresents the specific heat at Constant Pressure_{v}dHis the Change in Enthalpydtis the Change in Temperature

## Heat Capacity Ratio

The ratio between heat capacity at constant pressure (C_{p}) and heat capacity at constant volume (C_{v}) is called the Heat Capacity Ratio.

It is denoted by the Greek symbol γ.

γ = C_{p}/ C_{v}

### Relation between C_{p }and C_{v}

C_{p} and C_{v} are the specific heats of an ideal gas at constant pressure and at constant volume respectively. They indicate the amount of heat required to raise the temperature of unit mass by one degree Kelvin at constant pressure and at constant volume. The relation between C_{p }and C_{v} is given as,

C_{p }– C_{v}= Rwhere

Ris universal Gas Constant

**Also, Check**

## Solved Examples on Heat Capacity

**Example 1: A 88.3 g sample of metal at 95.24 C is added to 35.10 g of water that is initially at 17.27 °C. The final temperature of both the water and the metal is 29.20 °C. The specific heat of water is 4.184 J/(g°C). Calculate the specific heat of the metal.**

**Solution:**

Given,

Mass of metal is 88.3 g

The initial temperature of the metal is 95.24 °C.

Mass of water: 35.10 g.

The initial temperature of the water is 17.27 °C.

The final temperature of the water and the metal is 29.20 °C.

The specific heat of water is 4.184 J/(g°C).

Therefore, the expression where the energy from the hotter metal transfers to the cooler water is

−m_{o}C_{o}ΔT_{o }= m_{w}C_{w}ΔT_{w}where

m_{o }= mass of a metal object

ΔT_{o}= temperature change of metal object

C_{o }= specific heat capacity of metal object

m_{w }= mass of water

ΔT_{w }= temperature change of water

C_{w }= specific heat capacity of waterRearrange the above expression,

C

_{o }= (m_{w}C_{w}ΔT_{w})/(m_{o}ΔT_{o})Substitute the values in the above expression,

C

_{o }= [35.10 4.184(29.20−17.27)]/[88.3(29.20-95.24)]=

0.301 J/g°C

**Example 2: A 30.5 g sample of an alloy at 93.0 °C is placed into 50.0 g of water at 22.0 °C in an insulated coffee cup with a heat capacity of 9.2 J/K. If the final temperature of the system is 31.1 °C, what is the specific heat capacity of the alloy?**

**Solution:**

Heat absorbed = heat lost

then the specific heat capacity, of that substance is given by

s = (1/m)(ΔQ/ ΔT)

Rearrange the above expression,

ΔQ=smΔT

ΔQ

_{alloy}= ΔQ_{water}+ΔQ_{cup}Temperature of the water is equal to the temperature of the cup = 22.0 °C.

Temperature of the alloy is 93.0 °C.

Final Temperature is 31.1 °C.

30.5×(93.0 – 31.1)s = 9.2×(31.1-22.0) + 50.0×4.2×(31.1-22.0)

1887.95×s = 1994.72

s =

1.057 J/gK

**Example 3: The specific heat of water is 4.18 J/(g°C). Calculate the molar heat capacity of water. Express your answer to three significant figures and include the appropriate units.**

**Solution:**

Specific Heat of water is 4.18 J/(g°C).

Expression to convert gram into mole is

4.18 J/gC x (18.0 g / mole) = 75.24 J/mole C

Hence, the molar heat capacity of water is

75.24 J/mole C

## FAQs on Heat Capacity

**Question 1: What is Heat Capacity?**

**Answer:**

The change in temperature of a substance, when a given quantity of heat is absorbed or rejected by it, is characterized by a quantity called the heat capacity of that substance.

**Question 2: What is Specific Heat Capacity? **

**Answer:**

When the same quantity of heat is applied to the same mass of different substances, the resulting temperature changes are not the same. It indicates that each substance has a distinct value for the quantity of heat absorbed or rejected to change the temperature of its unit mass by one unit. This quantity is referred to as the specific heat capacity of the substance.

**Question 3: What is Molar Heat Capacity?**

**Answer:**

The amount of heat energy required to raise the temperature of 1 mole of any substance by 1-degree celsius is called Molar Heat Capacity.

**Question 4: Which substance has the highest heat capacity?**

**Answer:**

Water has the highest heat capacity among all known substances.

**Question 5: Which metal conducts heat the fastest?**

**Answer:**

Among metals, Copper and Aluminium have the highest thermal conductivity, i.e. they conduct heat at the fastest rate.

### Question 6: Why is specific heat important?

**Answer:**

The concept of specific heat is widely used to design thermal equipment. As we know that water has the highest heat capacity so it is widely used as a coolant for various activities.

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