Heat Capacity, Specific Heat Capacity and Molar Heat Capacity: What would you observe if a swimming pool and a bathtub full of water were both exposed to the same amount of heat energy at the same temperature? The water in the bathtub would certainly rise in temperature more quickly than in the swimming pool. This is because an object’s heat capacity is determined by its mass and chemical composition. The swimming pool of water has a far higher heat capacity than the bathtub due to its much larger bulk. In this article, we will provide detailed information on Heat Capacity, Specific Heat Capacity and Molar Heat Capacity.

What is Heat Capacity?

The amount of heat required to raise the temperature of an object by \({1^{\text{o}}}{\text{C}}\) is known as the heat capacity of the object. Water is particularly resistant to temperature changes, whereas metals are not. In the case of heat absorbed by the system, this heat appears as an increase in the system’s temperature. This temperature rises in direct proportion to the amount of heat absorbed.

\({{\rm{q}}_{{\rm{coeff }}}}{\rm{ = C \times \Delta T}}\)

The magnitude of the coefficient is determined by the system’s size, composition, and nature. It can also be written as \({\rm{q = C \times \Delta T}}\) 

 Here, \({\rm{C}}\), is called the heat capacity.

What is Specific Heat Capacity?

The specific heat of a substance is defined as the amount of heat energy needed to increase the temperature of \({\rm{1g}}\) of the substance by \({{\rm{1}}^ \circ }{\rm{C}}\). It is represented as \({{\rm{C}}_{\rm{P}}}\) where \({\rm{p}}\) denotes the specific heat measured at constant pressure. It is represented as \({{\rm{C}}_{\rm{V}}}\) where \({\rm{v}}\) denotes the specific heat measured at constant volume. The units for specific heat can either be joules per gram per degree \(\left( {{\rm{J/}}{{\rm{g}}^ \circ }{\rm{C}}} \right)\) or calories per gram per degree \(\left( {{\rm{cal/}}{{\rm{g}}^ \circ }{\rm{C}}} \right)\).

In other words, the amount of heat required to raise the temperature of a unit mass of a substance by \({{\rm{1}}^ \circ }{\rm{C}}\) is known as the specific heat of a solid or liquid. In SI units, it is the amount of heat required to raise the temperature of kg of solid or liquid by \({\rm{1K}}\).

In the SI system, its unit is always given as \({\rm{JK}}{{\rm{g}}^{{\rm{ – 1}}}}{{\rm{K}}^{{\rm{ – 1}}}}\) and in the CGS system, it is \({\rm{cal}}\,{{\rm{g}}^{{\rm{ – 1}}}}{{\rm{C}}^{{\rm{ – 1}}}}\). If \(\Delta {\rm{Q}}\) is the amount of heat needed to raise the temperature of mass \({\rm{M}}\) by \({\rm{T}}\), then specific heat is given by:

\({\rm{C = }}\frac{{{\rm{\Delta Q}}}}{{{\rm{m \times \Delta T}}}}\)
or
\({\rm{\Delta Q = mC\Delta T}}\)

The specific heat capacity of water is taken to be \(1\).

What is Molar Heat Capacity?

The amount of heat needed to increase the temperature of one mole of a substance by \({\rm{1K}}\) or \({{\rm{1}}^ \circ }{\rm{C}}\) is known as the molar heat capacity of a solid or liquid. Its unit is \({\rm{J\;mo}}{{\rm{l}}^{{\rm{ – 1}}}}{{\rm{K}}^{{\rm{ – 1}}}}\). So, to raise the temperature of \({\rm{\mu }}\) moles of solid through \(\Delta {\rm{T}}\), we would need an amount of heat equal to \({\rm{\Delta Q = \mu C\Delta T}}\).

Specific Heat at Constant Pressure and Volume

When the volume of a solid remains constant on being heated through a small temperature range, then it is known as \({{\rm{C}}_{\rm{v}}}\). This is the specific heat at a constant volume.

When the pressure of solid remains constant on being heated through a small temperature range, then it is known as \({{\rm{C}}_{\rm{p}}}\). This is the specific heat at constant pressure.

Relation between \({{\rm{C}}_{\rm{p}}}\) and \({{\rm{C}}_{\rm{v}}}\)

At constant volume, we can write the equation for heat, \({\rm{q}}\) as-

\({{\text{q}}_{\text{V}}} = {{\text{C}}_{\text{V}}}\Delta {\text{T}} = \Delta {\text{U}}\)

At constant pressure, \({{\text{q}}_{\text{P}}} = {{\text{C}}_{\text{P}}}\Delta {\text{T}} = \Delta {\text{H}}\)

The difference between \({{\rm{C}}_{\rm{P}}}\) and \({{\rm{C}}_{\rm{V}}}\) can be derived for an ideal gas as-

For a mole of an ideal gas,

\({\rm{\Delta H = \Delta U + \Delta (PV)}}\)

\({\rm{ = \Delta U + \Delta (RT) = \Delta U + R\Delta T}}\)

On substituting the values of \({\rm{\Delta H}}\) and \({\rm{\Delta U}}\), we have-

\({{\rm{C}}_{\rm{P}}}{\rm{\Delta T = }}{{\rm{C}}_{\rm{V}}}{\rm{\Delta T + R\Delta T}}\)

\({{\rm{C}}_{\rm{P}}}{\rm{ = }}{{\rm{C}}_{\rm{V}}}{\rm{ + R}}\)

\({{\rm{C}}_{\rm{P}}}{\rm{ – }}{{\rm{C}}_{\rm{V}}}{\rm{ = R}}\)

where \({{\rm{C}}_{\rm{p}}}\) represents heat capacity at constant pressure, \({{\rm{C}}_{\rm{v}}}\) represents heat capacity at constant volume, \({\rm{n}}\) is the amount of substance and \({\rm{R}} = 8.314{\rm{Jmo}}{{\rm{l}}^{ – 1}}\;{\rm{K}}\) and is the molar gas constant.

Applications of Heat Capacity, Specific Heat Capacity and Molar Heat Capacity

1. The bases of cooking utensils are composed of materials with a low specific heat capacity. The handle of these utensils, on the other hand, is composed of a high specific heat material to resist the heat in and save our hands.
2. Insulators make use of materials with a high specific heat capacity. Take, for instance, wood. Houses made of wood are better suitable for areas with high or low temperatures.
3. To detect and display temperature changes quickly and accurately, sensitive thermometers must also be composed of materials with low specific heat capacity.
4. Heat storage devices are extremely useful and are often built of high specific heat capacity materials.
5. Water acts as an effective cooling agent for engines. Water is also used in houses in a cold climate because, when heated (boiled), it retains heat and warms the house due to its high specific heat capacity.

Solved Examples – Heat Capacity, Specific Heat Capacity and Molar Heat Capacity

Q.1. The molar heat capacity of water, \({{\rm{C}}_{\rm{p}}}\), is \(75.2{\rm{Jmo}}{{\rm{l}}^{ – 1}}\;{{\rm{K}}^{ – 1}}\). How much heat is needed to raise the temperature of \(36\) grammes of water from \(300\) to \(310{\rm{K}}\)?
Ans: \({\rm{Moles}}\,\,{\rm{of}}\,{{\rm{H}}_{\rm{2}}}{\rm{O = }}\frac{{{\rm{ Given \,Mass }}}}{{{\rm{ Molar\, Mass }}}}{\rm{ = }}\frac{{{\rm{36}}}}{{{\rm{18}}}}{\rm{ = 2}}{\rm{.0\;mol}}\)
Substituting the values into the formula that relates heat and heat capacity:
\({\rm{q = n}}{{\rm{C}}_{\rm{P}}}{\rm{\Delta T}}\)
\({\rm{q}} = (2.0\;{\rm{mol}})\left( {75.2\;{\rm{J}}\;{\rm{mo}}{{\rm{l}}^{ – 1}}\;{{\rm{K}}^{ – 1}}} \right)(10\;{\rm{K}})\)
\({\rm{q}} = 1504\;{\rm{J}}\)
As a result, \(1504\;{\rm{J}}\) of heat is required to raise the temperature of the water.

Summary

Heat affects different substances in different ways. On a hot day, a metal chair may become rather hot to the touch if it sits in direct sunlight. However, an equal mass of water will not get as hot when exposed to the same amount of sunlight. Why is there such a variance in temperature when the sun is the same? This is due to the fact that water can hold more heat than a metal chair.

If the same quantity of heat is applied to two different types of materials, the temperature rise in each solid may differ. The increase in temperature for different kinds of solids depends on the composition of the solid. This phenomenon is known as Specific heat capacity, making water a common coolant for machinery. Coastal climates are much more moderate than inland climates due to the specific heat capacity of water.

FAQs

Q.1. How do you convert specific heat capacity to molar heat capacity?
Ans: To calculate the molar heat capacity of a compound or element, we simply multiply the specific heat by the substance’s molar mass. For instance, the specific heat of methane \(\left( {{\rm{C}}{{\rm{H}}_4}} \right)\) is \(2.20\;{\rm{J}}/{\rm{g}} – {\rm{K}}\). To convert it to molar heat capacity, multiply the specific heat by the molar mass of methane.

Q.2. What are the differences between specific heat and molar specific heat?
Ans: The amount of heat required to increase the temperature of one gram of a substance by one degree is known as specific heat capacity. At the same time, molar heat capacity is the amount of heat required to increase the temperature of one mole of a substance by one degree.

Q.3. What is the relation between heat capacity and specific heat capacity?
Ans: \({\rm{C = mS}}\) is the relationship between Heat Capacity and Specific Heat Capacity, where \({\rm{C}}\) is the substance’s heat capacity, \({\rm{S}}\) is the substance’s specific heat capacity, and \({\rm{m}}\) is the substance’s mass.

Q.4. Is specific heat capacity always higher than molar heat capacity?
Ans: A substance’s molar heat capacity is equal to its specific heat \({\rm{c}}\) multiplied by its molar mass. As a result, its numerical value is usually lower than that of specific heat.

Q.5. What is the SI unit of molar heat capacity?
Ans: The molar heat capacity is the amount of energy required to raise the temperature of one mole of a substance by one degree; its units in the SI system are \({\rm{J}}/{\rm{mol}} \cdot {\rm{K}}\).

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