# Specific Heat Capacity

Have you ever noticed how the water is icy, yet the sand is heated while you’re at the beach? Why is there such a variation in temperature when the sun is the same? You must have thought about it! When we apply heat to a solid or liquid element, its temperature rises. If the same quantity of heat is applied to two distinct types of materials, the temperature rises in each solid may differ. As a result, the rise in temperature for different types of solids varies depending on the composition of the solid. Specific heat capacity is the name for these phenomena.

### What is the Heat Capacity?

Heat capacity is a measure of a system’s total internal energy. This comprises the system’s overall kinetic energy as well as the potential energy of the molecules. It has been demonstrated that a system’s internal energy may be altered by either giving heat energy to it or doing work on it. The internal energy of a system is shown to grow as the temperature rises. This rise in internal energy is affected by temperature differences, the amount of substance present, and so on.

The amount of heat energy necessary to increase the temperature of a given quantity of matter by one degree Celsius is known as

Heat Capacity. The heat capacity of a given substance varies with its size or quantity, making it an extensive property.

Mathematically, the heat capacity is given by:

**Q = C ΔT**

where Q is the amount of heat energy necessary to cause a temperature change of ΔT and C is the heat capacity of the system under consideration.

The unit of heat capacity is Joule per Kelvin (J/K) or Joule per degree Celsius (J/°C).

### Specific Heat Capacity

Specific heat capacity, in general, is a measure of how much energy it takes to change the temperature of a system. However, it is critical to understand that the energy intake must be through heating. If work is done on the system, the temperature will rise; however, attempting to compute the temperature rise using the heat capacity and the quantity of work done on it is inaccurate.

The quantity of heat energy required to increase the temperature of a substance per unit mass is referred to as its

Specific Heat Capacity. The specific heat capacity of a substance is a physical characteristic. It is also an example of an extensive property because its value is proportional to the size of the system under consideration.

Another thing to consider is the limitation that the system is held to. Because the latter does work on its surroundings as it expands, the specific heat capacity of a system kept at constant volume differs from that of a system held at constant pressure. Such discrepancies are typically overlooked when dealing with solids, but they are critical when working with gases.

A solid’s or liquid’s specific heat is an amount of heat required to increase the temperature of the unit mass of the solid by 1° C. It is represented by the symbol C.

It is an amount of heat required to increase the temperature of 1 kg of liquid or solid by 1 K in SI units. Its SI unit is always J kg^{-1} K^{-1}, and its CGS unit is always Cal g^{-1} C^{-1}.

If ΔQ is the amount of heat necessary to raise the temperature of mass m through ΔT, then formula for specific heat is:

C = ΔQ ⁄ m ΔTor

ΔQ = m c ΔT

### Molar Specific Heat

The molar specific heat of a solid or liquid is an amount of heat required to raise the temperature of one mole of solid or liquid by one degree Celsius or one degree Kelvin.

It is denoted by the letter C. Its unit is J mol^{-1} K^{-1}. To raise the temperature of μ moles of solid by ΔT, a quantity of heat equal to ΔQ = μ C T would be required.

The quantity of heat required to increase the temperature of 1 gm molecule of a substance by one degree centigrade is known as the molar specific heat capacity of the substance, abbreviated as C. Water’s specific heat is assumed to be 1. This is because we used water to define the unit of heat (calorie).

### Specific Heat of Water

The specific heat capacity of water at normal pressure and temperature is approximately 4.2 J ⁄ g °C or 1 Cal ⁄ g °C. This means that 1 gm of water requires 4.2 joules of energy to raise 1 degree Celsius. This number is actually pretty high. Even water vapour has a higher specific heat capacity than many other materials at normal temperature. Water vapour’s specific heat capacity at normal pressure and temperature is approximately 1.9 J ⁄ g °C.

Water’s temperature falls as it releases heat and rises as it absorbs heat, as it is with other liquids. But the liquid water temperature falls or rises slower than that of many other liquids. We may conclude that water absorbs heat without causing an instant temperature increase. It also keeps its temperature for a considerably longer period of time than many other substances.

We employ this property of water in the human body to keep it at a stable temperature. There would be a lot more instances of underheating and overheating if water had a lower specific heat value.

### Specific Heat at Constant Pressure or Volume

When a solid is heated over a limited temperature range, its pressure remains constant. **At constant pressure**, this is referred to as** specific heat**, abbreviated as **C _{P}**.

When a solid is heated over a short temperature range, its volume remains constant. **At constant volume**, this is referred to as **specific heat**, abbreviated as **C _{V}**.

The way gas is heated affects the behaviour of the gas, the volume and pressure change in temperature, and the amount of heat necessary to increase the temperature of 1gm of gas by 1° C. We can heat the gas with a variety of P and V values.

As a result, the specific heat value is limitless. If we don’t deliver a steady quantity of heat, the gas’s specific heat will change. As a result, we will need a constant pressure or volume of specific heat.

For an ideal gas,

C_{P}– C_{V}= n Rwhere, C

_{V}is heat capacity at constant volume, C_{P}is heat capacity at constant pressure, R is the molar gas constant, and and n is amount of substance.The value of gas constant, R = 8.3145 J mol

^{-1}K^{-1}

**C _{P} ⁄ C_{V} Ratio (Heat Capacity Ratio)**

The adiabatic index is also known as heat capacity ratio or ratio of specific heat capacities (C_{P} : C_{V}) in thermodynamics. The ratio of heat capacity at constant pressure (C_{P}) to heat capacity at constant volume (C_{V}) is defined as heat capacity ratio.

The isentropic expansion factor, commonly known as heat capacity ratio, is indicated by γ for an ideal gas (gamma). As a result, specific heat ratio, γ is equal to ratio of C

_{P}to C_{V}, i.e. γ = C_{P}⁄ C_{V}.

**Why is C _{P} Greater than C_{V}?**

The specific heats of an ideal gas are represented by C_{P} and C_{V}. This is the amount of heat required to raise the temperature of unit mass by 1° C. By the first law of thermodynamics,

**ΔQ = ΔU + ΔW**

where, ΔQ is the amount of heat that is given to the system, ΔU is the change in internal energy, and ΔW is the work done.

At constant pressure, heat is absorbed to raise internal energy and do any work on the system. On the other side, heat is absorbed just to raise internal energy at constant volume, not to do any work on the system. As a result, the specific heat under constant pressure is greater than that at constant volume, i.e. Cp > Cv.

### Applications

- Insulators make use of materials with a high specific heat capacity. Take, for instance, wood. Houses built of wood are better suitable for areas with high or low temperatures.
- Swimming pool water used to be cold in comparison to the temperature outdoors due to the high specific heat of the water.
- Cooking utensils are made of a low-specific-heat material. You can immediately heat their bottoms. This is due to their polished aluminium or copper bottoms. To maintain the heat and safeguard our hands, the handles of these utensils are made of high specific heat material.

### Sample Problems

**Problem 1: What is the advantage of water’s heat capacity?**

**Solution:**

Because water has a large heat capacity, a one-degree increase in temperature necessitates more energy. The sun emits a relatively consistent amount of energy, which causes sand to heat up faster and water to heat up slower.

**Problem 2: Calculate the heat required to raise 0.5 Kg of sand from 30°** **C to 90° C? (Specific Heat of sand = 830 J ⁄ Kg °C)**

**Solution:**

Given:

Mass of sand, m = 0.5 Kg

Temperature difference, ΔT = 90° C – 30° C = 60° C

Specific heat of sand, C = 830 J ⁄ Kg °C

The formula for specific heat capacity is given as:

C = ΔQ ⁄ m ΔT

Rearrange the formula in terms of Q.

Q = m C ΔT

= 0.5 Kg × 830 J ⁄ Kg °C× 60° C

= 24900 J.

Hence, the required heat to raise the sand temperature is

24900 J.

**Problem 3: What is the difference between heat capacity and specific heat capacity?**

**Solution:**

Specific heat capacity is the heat needed to raise a substance’s temperature by 1 degree Celsius. Similarly, heat capacity is the ratio between the energy provided to a substance and the corresponding increase in its temperature.

**Problem 4: Compute the temperature difference if 40 Kg of water absorbs 400 K J of heat?**

**Solution:**

Given:

Mass of water, m = 40 Kg

Heat transfer, Q = 400 KJ,

Specific heat of water, c = 4.2 × 10

^{3}J ⁄ Kg °CThe formula for specific heat capacity is given as:

c = ΔQ ⁄ m ΔT

Rearrange the formula in terms of ΔT.

ΔT = ΔQ ⁄ c m

= (400 × 10

^{3}) ⁄ (4.2 × 10^{3}× 40) °C= 2.38 °C

Hence, the temperature difference is

2.38 °C.

**Problem 5: What is the heat capacity ratio?**

**Solution:**

It is the ratio of two specific heat capacities, C

_{P}and C_{V}is given by: The Heat Capacity at Constant Pressure (C_{P})/ Heat capacity at Constant Volume (C_{V}).The isentropic expansion factor, commonly known as the heat capacity ratio, is indicated by γ for an ideal gas (gamma). As a result, the specific heat ratio, γ is equal to the ratio of C

_{P}to C_{V}, i.e. γ = C_{P}/ C_{V}.