# Dimensional Formula of Specific Heat capacity

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## Dimensions

Dimensions of the physical quantity are the powers to which the base quantities are raised to represent that quantity.

### Dimensional Formula

The dimensional formula of any physical quantity is that expression which represents how and which of the base quantities are included in that quantity.

It is written by enclosing the symbols for base quantities with appropriate power in square brackets, i.e. [ ].

E.g: Dimension formula of mass is: [M].

### Dimensional Equation

The equation obtained by equating a physical quantity with its dimensional formula is called a dimensional equation.

### Application of Dimensional Analysis

1. To convert a physical quantity from one system of the unit to the other:

It is based on a fact that the magnitude of a physical quantity remains the same whatever system is used for measurement, i.e. magnitude = numeric value(n) multiplied by unit (u) = constant,

n1u1= n1u2.

2. To check dimensional correctness of a given physical relation:

If in a given relation, the terms on both sides have the same dimensions, then the equation is dimensionally correct. This concept is best known as the principle of homogeneity of dimensions.

3. To derive the relationship between different physical quantities:

Using the principle of homogeneity of dimension the new relation among physical quantities can be derived if the dependent quantities are known.

### Limitation of This Method

1. This method can be used only if dependency is of multiplication type. The formula containing exponential, trigonometric, and logarithmic functions can not be derived using this method. A formula containing more than one term which is added or subtracted likes s = ut+ ½ at2 also cannot be derived.

2. The relation derived from this method gives no information about the dimensionless constants.

### Specific Heat Capacity

The specific heat capacity, which is denoted as Cp , of a substance is the capacity of the heat of a sample of the substance divided by the mass of the sample. Informally, it is said to be the amount of energy that must be added, in the form of heat, to a unit of mass of the substance to cause an increase of one unit in its temperature. The SI unit of specific heat capacity is joule per kelvin and kilogram, J/(K kg).

For example, at  25 °C  temperature of the specific heat capacity vary with the temperature, to raise the temperature the heat requires of 1 kg of water by 1 K (equivalent to 1 °C) is 4179.6 joules, which means that the specific heat of water is 4179.6 J·kg−1·K−1.

With temperature, the specific heat often varies and is different for each state of matter. Water has one of the highest specific heat capacities of all substances, 4182 J/(K kg) at 20 °C. However, ice just below 0 °C is only 2093 J/(K kg). The specific heat capacities of iron, hydrogen gas, and granite are about 449 J/(K kg), 790 J/(K kg), and 14300 J/(K kg), respectively. When change of state occurs, such as boiling, or melting, its specific heat is technically infinite, because the heat changes its state, instead of raising its temperature.

The specific heat capacity of a substance, especially a gas, may be significantly higher when it expands, (as it is heated at constant pressure), than when it is heated in a closed vessel that prevents expansion, at constant volume. These two values are usually denoted by the following:

The heat capacity ratio γ = Cp/Cv

The units of the specific heat capacity are J/(kg°C) or J/(kgK). The heat capacity for any substance and the specific heat are related by equation C=cm, or c=C/m. Here mass ‘m’, specific heat ‘c’, change in temperature ‘ΔT’, and heat added or released ‘Q’ are related by the equation: Q=mcΔT.

However, in some contexts, the term specific heat capacity or simply the specific heat may refer to the ratio between the specific heats of a substance at a given temperature to that of a reference substance at a temperature. Specific heat relates to other measures which are intensive of heat capacity with other denominators. If the substance amount is measured as a number of moles, one gets the molar heat capacity instead, the SI unit of which is joule per kelvin per mole, J/(K mol). If the volume is taken to be the amount of the sample as is sometimes done in engineering, one gets the volumetric heat capacity, the SI unit of which is defined as joule per kelvin per cubic meter, J/K/m3.

Scientist Joseph, doctor and professor of Medicine at Glasgow University, was one of the first scientists to use this concept in the18th-century. He measured specific heats of many substances, using the term specific heat capacity.