Heat Capacity Formula: Derivation, Specific Heat Capacity, and Molar Heat Capacity

Heat capacity is a physical property of matter defined as the amount of heat required to raise the temperature of the body by one degree. It is also known as Thermal capacity.

• The SI unit of heat capacity is joule per kelvin (J/K).
• Heat capacity is an extensive property i.e. its value depends on the amount of substance that was measured.
• By dividing the heat capacity of an object by its mass, we will get its Specific heat capacity.
• Specific heat capacity is an intensive property i.e. its value does not depends on the amount of substance that was measured.

Key Terms: Heat Capacity, Specific heat capacity, Molar Specific heat capacity, Heat energy, Mayor’s Formula, SI units, Temperature.

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Heat Capacity

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Heat capacity or thermal capacity is the amount of heat needed to increase the temperature of a body by one degree. The SI unit of heat capacity is joule per kelvin (J/K).

• Heat energy is defined as the total kinetic energy of the molecules of the substance. The SI unit of heat is joule (J).
• The temperature of a substance is a physical quantity that measures the degree of hotness and coldness of the substance. The SI unit of temperature is kelvin (K).
• The specific heat capacity of a substance can be obtained by dividing the heat capacity of that substance by its mass.
• By dividing the heat capacity of a substance by its mass in moles, we will get its molar heat capacity.
• The heat capacity per unit volume is known as Volumetric heat capacity.
• The heat capacity of a building is referred to as its thermal mass in architecture and civil engineering.

Also Read: 

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Heat Capacity Formula

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The heat capacity formula gives the amount of heat energy required to raise the temperature of the body by one degree. It is given by the ratio of the amount of heat energy that is given to the object for the resulting increase in the temperature.

It can be mathematically expressed as:

\(c={\Delta Q \over \Delta T}\)

Where,

• c = heat capacity
• ΔQ = Amount of heat energy given to the object
• ΔT = change in the temperature

Also Read: Heat Formula

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Specific Heat Capacity

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If an amount of heat (ΔQ) when given to a body its temperature increases by an amount ΔT. Heat gain or loss is directly proportional to the mass (m) of the body and changes in temperature (ΔT). i.e.

ΔQ ∝ m ΔT  ⇒ ΔQ = m C ΔT

• Where C is a constant known as “specific heat capacity” or simply “specific heat” of the material of the body.
• Its value depends on the nature of the material of the body and its temperature range.

The specific heat capacity of the material of a substance is the amount of heat required to raise the temperature of the unit mass of the substance by one degree.

From the above formula, specific heat capacity can be given by

\(C={\Delta Q \over m\Delta T}\)

Where, 

• m = mass of the substance
• ΔT = increase in temperature
• ΔQ = amount of transferred heat

Multiplying the above equation by the mass (m) of the substance, we get Heat capacity (C).

\(Heat\,capacity,\,c=mC=m({\Delta Q \over m\Delta T})\)

Hence, the product of the mass of the substance to the specific heat capacity gives the Heat capacity of that substance.

Specific Heat Capacity Video Explanation

The nature of the substance and its temperature affects the specific heat capacity of the substance. 

Water is used as a coolant in automobile radiators and heaters because it has a high specific heat capacity which results in the slow heating up of the water than other substances. 

For gases, there are two specific heat capacities:

1. Specific heat capacity at constant pressure (C_P).
2. Specific heat capacity at constant volume (CV).

For an ideal gas,

C_P – C_V = R

Where

• R is the universal gas constant.
• Above relation is known as Meyer’s relation or Mayer’s Formula.

Units of Specific Heat Capacity

The units of specific heat capacity are given below:

• If m in gram (g), ΔT in ^oC, and ΔQ in calorie (cal), then the unit of specific heat capacity is (cal/g^oC).
• In SI unit of specific heat capacity is (J/kgK).

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Molar Heat Capacity

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Molar heat capacity or molar-specific heat is defined as the amount of heat required to raise the temperature of one mole of a substance by one degree. It is given by

\(Molar\,heat\,capacity\,,C_m={\Delta Q\over \mu \Delta T}\)

Where μ is the number of moles of the substance.

If m be the mass of the substance in gram, and M be the molar mass, then

\(m=\mu M \Rightarrow \mu ={m\over M}\)

Therefore, molar heat capacity can be given by

\(C_m=M{\Delta Q\over m\Delta T}\)

But, \({\Delta Q\over m\Delta T}=s,\) specific heat capacity

Hence,

Molar Heat Capacity, C_m = MC

Molar specific heat is equal to the product of the molar mass of the substance (in grams) and specific heat.

The SI unit of molar heat capacity is (J/mol K).

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Formulae Related to Heat Capacity

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The formulae related to Heat capacity are given below:

Heat capacity (c) = Δ Q/Δ T

Where

• ΔT = change in temperature 
• ΔQ = amount of transferred heat

Heat capacity (c) = mC

Where

• m is the mass of the substance 
• C is the specific heat capacity

Specific heat capacity (C) = ΔQ/mΔT

Where

• m= mass in grams
• ΔT = change in temperature
• ΔQ = amount of transferred heat

Molar specific heat capacity ( C_m ) = ΔQ/µΔT

Where

• µ = moles or the amount of substance
• ΔT = change in temperature
• ΔQ = amount of transferred heat

Relation between Celsius temperature (T_C) and Fahrenheit temperature (T_F)

T_F = (9/5)T_C + 32

The ideal gas equation

PV = µRT

Where 

• µ = number of moles 
• R = universal gas constant

Newton’s law of cooling is given by

dQ/dt = -k (T₂ – T₁)

Where

• T₁ = Temperature of the surrounding medium
• T₂ = temperature of the body

Also Read:

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Things to Remember

• Heat Capacity is the amount of heat needed to increase the temperature of a body by one degree. 
• The SI unit of heat capacity is joule per kelvin (J/K). 
• The formula for heat capacity is given by the ratio of the amount of heat energy that is given to the object for the resulting increase in the temperature.
• The transfer of heat between two different bodies due to temperature is known as conduction.
• If the dimension of a body increases with temperature, it is known as thermal expansion.
• The calorimeter is the device used to measure heat.
• If there is a change in the mass of the substance while changing from one state to another then the quantity of heat (Q) = mL or L = Q/m. where L = latent heat. 
• The electromagnetic radiation happening due to temperature is known as thermal radiation. 

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